Finance/Mathematics
A First Course in Financial Mathematics
Option Valuation: A First Course in Financial Mathematics
provides a straightforward introduction to the mathematics and
models used in the valuation of financial derivatives. It examines
the principles of option pricing in detail via standard binomial and
stochastic calculus models. Developing the requisite mathematical
background as needed, the text introduces probability theory and
stochastic calculus at an undergraduate level.
Hugo D. Junghenn
Option Valuation
A First Course in
Financial Mathematics
Junghenn
Largely self-contained, this classroom-tested text offers a sound
introduction to applied probability through a mathematical finance
perspective. Numerous examples and exercises help readers
gain expertise with financial calculus methods and increase their
general mathematical sophistication. The exercises range from
routine applications to spreadsheet projects to the pricing of a
variety of complex financial instruments. Hints and solutions to
odd-numbered problems are given in an appendix.
A First Course in
Financial Mathematics
The first nine chapters of the book describe option valuation
techniques in discrete time, focusing on the binomial model. The
author shows how the binomial model offers a practical method
for pricing options using relatively elementary mathematical tools.
The binomial model also enables a clear, concrete exposition of
fundamental principles of finance, such as arbitrage and hedging,
without the distraction of complex mathematical constructs. The
remaining chapters illustrate the theory in continuous time, with
an emphasis on the more mathematically sophisticated Black–
Scholes–Merton model.
Option Valuation
Option Valuation
K14090
K14090_Cover.indd 1
10/7/11 11:23 AM
Option Valuation
A First Course in
Financial Mathematics
CHAPMAN & HALL/CRC
Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
appeal to both academics and practitioners. The inclusion of numerical code and concrete realworld examples is highly encouraged.
Series Editors
M.A.H. Dempster
Dilip B. Madan
Rama Cont
Centre for Financial Research
Department of Pure
Mathematics and Statistics
University of Cambridge
Robert H. Smith School
of Business
University of Maryland
Center for Financial
Engineering
Columbia University
New York
Published Titles
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,
Pierre Henry-Labordère
Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin Wu
Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm, Ludger Overbeck, and
Christoph Wagner
Introduction to Stochastic Calculus Applied to Finance, Second Edition,
Damien Lamberton and Bernard Lapeyre
Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,
and Gerald Kroisandt
Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller
Option Valuation: A First Course in Financial Mathematics, Hugo D. Junghenn
Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra
Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Stochastic Finance: A Numeraire Approach, Jan Vecer
Stochastic Financial Models, Douglas Kennedy
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck
Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy
Unravelling the Credit Crunch, David Murphy
Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor & Francis Group
4th, Floor, Albert House
1-4 Singer Street
London EC2A 4BQ
UK
Option Valuation
A First Course in
Financial Mathematics
Hugo D. Junghenn
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2011 by Taylor & Francis Group, LLC
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v
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Contents
xi
Preface
1 Interest and Present Value
1.1
Compound Interest
1.2
Annuities
1.3
Bonds
1.4
Rate of Return
1.5
Exercises
1
. . . . . . . . . . . . . . . . . . . . . . .
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
. . . . . . . . . . . . . . . . . . . . . . . . . .
7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2 Probability Spaces
13
2.1
Sample Spaces and Events
. . . . . . . . . . . . . . . . . . .
13
2.2
Discrete Probability Spaces
. . . . . . . . . . . . . . . . . . .
14
2.3
General Probability Spaces
. . . . . . . . . . . . . . . . . . .
16
2.4
Conditional Probability
. . . . . . . . . . . . . . . . . . . . .
20
2.5
Independence . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.6
Exercises
24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Random Variables
27
3.1
Denition and General Properties
. . . . . . . . . . . . . . .
3.2
Discrete Random Variables
3.3
Continuous Random Variables
. . . . . . . . . . . . . . . . .
32
3.4
Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . .
34
3.5
Independent Random Variables
. . . . . . . . . . . . . . . .
35
3.6
Sums of Independent Random Variables . . . . . . . . . . . .
38
3.7
Exercises
41
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Options and Arbitrage
27
29
43
4.1
Arbitrage
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Classication of Derivatives
44
4.3
Forwards
4.4
Currency Forwards
. . . . . . . . . . . . . . . . . . . . . . .
48
4.5
Futures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.6
Options
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
Properties of Options
4.8
Dividend-Paying Stocks
4.9
Exercises
. . . . . . . . . . . . . . . . . . .
46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
50
. . . . . . . . . . . . . . . . . . . . . .
53
. . . . . . . . . . . . . . . . . . . . .
55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
vii
viii
5 Discrete-Time Portfolio Processes
59
5.1
Discrete-Time Stochastic Processes.
. . . . . . . . . . . . . .
5.2
Self-Financing Portfolios
59
. . . . . . . . . . . . . . . . . . . .
5.3
Option Valuation by Portfolios
61
5.4
Exercises
. . . . . . . . . . . . . . . . .
64
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6 Expectation of a Random Variable
67
6.1
Discrete Case: Denition and Examples
. . . . . . . . . . . .
6.2
Continuous Case: Denition and Examples
6.3
Properties of Expectation
. . . . . . . . . . . . . . . . . . . .
69
6.4
Variance of a Random Variable . . . . . . . . . . . . . . . . .
71
6.5
The Central Limit Theorem
. . . . . . . . . . . . . . . . . .
73
6.6
Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
. . . . . . . . . .
7 The Binomial Model
67
68
77
7.1
Construction of the Binomial Model
. . . . . . . . . . . . . .
7.2
Pricing a Claim in the Binomial Model
7.3
The Cox-Ross-Rubinstein Formula
7.4
Exercises
77
. . . . . . . . . . . .
80
. . . . . . . . . . . . . . .
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
8 Conditional Expectation and Discrete-Time Martingales
89
8.1
Denition of Conditional Expectation
. . . . . . . . . . . . .
8.2
Examples of Conditional Expectation
. . . . . . . . . . . . .
92
8.3
Properties of Conditional Expectation
. . . . . . . . . . . . .
94
8.4
Discrete-Time Martingales
. . . . . . . . . . . . . . . . . . .
96
8.5
Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
9 The Binomial Model Revisited
89
101
9.1
Martingales in the Binomial Model
. . . . . . . . . . . . . .
9.2
Change of Probability
9.3
American Claims in the Binomial Model
9.4
Stopping Times
9.5
Optimal Exercise of an American Claim
9.6
. . . . . . . . . . . . . . . . . . . . . .
101
103
. . . . . . . . . . .
105
. . . . . . . . . . . . . . . . . . . . . . . . .
108
. . . . . . . . . . . .
111
Dividends in the Binomial Model
. . . . . . . . . . . . . . .
114
9.7
The General Finite Market Model
. . . . . . . . . . . . . . .
115
9.8
Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
10 Stochastic Calculus
119
10.1 Dierential Equations
. . . . . . . . . . . . . . . . . . . . . .
119
10.2 Continuous-Time Stochastic Processes . . . . . . . . . . . . .
120
10.3 Brownian Motion
122
. . . . . . . . . . . . . . . . . . . . . . . .
10.4 Variation of Brownian Paths
. . . . . . . . . . . . . . . . . .
123
. . . . . . . . . . . . . . . . . . .
126
. . . . . . . . . . . . . . . . . . . . . . .
126
10.5 Riemann-Stieltjes Integrals
10.6 Stochastic Integrals
10.7 The Ito-Doeblin Formula
. . . . . . . . . . . . . . . . . . . .
10.8 Stochastic Dierential Equations
. . . . . . . . . . . . . . . .
131
136
ix
10.9 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 The Black-Scholes-Merton Model
11.1 The Stock Price SDE
141
. . . . . . . . . . . . . . . . . . . . . .
11.2 Continuous-Time Portfolios
. . . . . . . . . . . . . . . . . . .
11.3 The Black-Scholes-Merton PDE
. . . . . . . . . . . . . . . .
11.4 Properties of the BSM Call Function
11.5 Exercises
139
141
142
143
. . . . . . . . . . . . .
146
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
12 Continuous-Time Martingales
12.1 Conditional Expectation
151
. . . . . . . . . . . . . . . . . . . .
12.2 Martingales: Denition and Examples
152
. . . . . . . . . . . . . .
154
. . . . . . . . . . . . . . . . .
156
12.3 Martingale Representation Theorem
12.4 Moment Generating Functions
151
. . . . . . . . . . . . .
12.5 Change of Probability and Girsanov's Theorem . . . . . . . .
158
12.6 Exercises
161
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 The BSM Model Revisited
163
13.1 Risk-Neutral Valuation of a Derivative
13.2 Proofs of the Valuation Formulas
13.3 Valuation under
P
. . . . . . . . . . . .
165
. . . . . . . . . . . . . . . . . . . . . . . .
167
13.4 The Feynman-Kac Representation Theorem
13.5 Exercises
163
. . . . . . . . . . . . . . .
. . . . . . . . .
168
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171
14 Other Options
173
14.1 Currency Options
. . . . . . . . . . . . . . . . . . . . . . . .
14.2 Forward Start Options
14.3 Chooser Options
173
. . . . . . . . . . . . . . . . . . . . .
175
. . . . . . . . . . . . . . . . . . . . . . . . .
176
14.4 Compound Options
. . . . . . . . . . . . . . . . . . . . . . .
14.5 Path-Dependent Derivatives
177
. . . . . . . . . . . . . . . . . .
178
14.5.1 Barrier Options . . . . . . . . . . . . . . . . . . . . . .
179
14.5.2 Lookback Options
. . . . . . . . . . . . . . . . . . . .
185
. . . . . . . . . . . . . . . . . . . . . .
191
14.5.3 Asian Options
14.6 Quantos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195
14.7 Options on Dividend-Paying Stocks
197
14.7.1 Continuous Dividend Stream
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
197
14.7.2 Discrete Dividend Stream . . . . . . . . . . . . . . . .
198
14.8 American Claims in the BSM Model
14.9 Exercises
. . . . . . . . . . . . . .
200
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
203
A Sets and Counting
209
B Solution of the BSM PDE
215
C Analytical Properties of the BSM Call Function
219
x
D Hints and Solutions to Odd-Numbered Problems
225
Bibliography
247
Index
249
xi
Preface
This text is intended as an introduction to the mathematics and models
used in the valuation of nancial derivatives. It is designed for an audience
with a background in standard multivariable calculus. Otherwise, the book is
essentially self-contained: The requisite probability theory is developed from
rst principles and introduced as needed, and nance theory is explained in
detail under the assumption that the reader has no background in the subject.
The book is an outgrowth of a set of notes developed for an undergraduate
course in nancial mathematics oered at The George Washington University.
The course serves mainly majors in mathematics, economics, or nance and
is intended to provide a straightforward account of the principles of option
pricing. The primary goal of the text is to examine these principles in detail via
the standard binomial and stochastic calculus models. Of course, a rigorous
exposition of such models requires a coherent development of the requisite
mathematical background, and it is an equally important goal to provide this
background in a careful manner consistent with the scope of the text. Indeed,
it is hoped that the text may serve as an introduction to applied probability
(through the lens of mathematical nance).
The book consists of fourteen chapters, the rst nine of which develop
option valuation techniques in discrete time, the last ve describing the theory in continuous time. The emphasis is on two models, the (discrete time)
binomial
model and the (continuous time)
Black-Scholes-Merton
model. The
binomial model serves two purposes: First, it provides a practical way to price
options using relatively elementary mathematical tools. Second, it allows a
straightforward and concrete exposition of fundamental principles of nance,
such as arbitrage and hedging, without the possible distraction of complex
mathematical constructs. Many of the ideas that arise in the binomial model
foreshadow notions inherent in the more mathematically sophisticated BlackScholes-Merton model.
Chapter 1 gives an elementary account of present value. Here the focus
is on risk-free investments, such money market accounts and bonds, whose
values are determined by an interest rate. Investments of this type provide a
way to measure the value of a risky asset, such as a stock or commodity, and
mathematical descriptions of such investments form an important component
of option pricing techniques.
Chapters 2, 3, and 6 form the core of the general probability portion of
the text. The exposition is self-contained and uses only basic combinatorics
and elementary calculus. Appendix A provides a brief overview of the elementary set theory and combinatorics used in these chapters. Readers with a
good background in probability may safely give this part of the text a cursory
reading. While our approach is largely standard, the more sophisticated notions of event
σ -eld
and ltration are introduced early to prepare the reader
xii
for the martingale theory developed in later chapters. We have avoided using Lebesgue integration by considering only discrete and continuous random
variables.
Chapter 4 describes the most common types of nancial derivatives and
emphasizes the role of arbitrage in nance theory. The assumption of an
arbitrage-free market, that is, one that allows no free lunch, is crucial in
developing useful pricing models. An important consequence of this assumption is the put-call parity formula, which relates the cost of a standard call
option to that of the corresponding put.
Discrete-time stochastic processes are introduced in Chapter 5 to provide
a rigorous mathematical framework for the notion of a self-nancing portfolio.
The chapter describes how such portfolios may be used to replicate options in
an arbitrage-free market.
Chapter 7 introduces the reader to the binomial model. The main result is
the construction of a replicating, self-nancing portfolio for a general European
claim. The most important consequence is the Cox-Ross-Rubinstein formula
for the price of a call option. Chapter 9 considers the binomial model from
the vantage point of discrete-time martingale theory, which is developed in
Chapter 8, and takes up the the more dicult problem of pricing and hedging
an American claim.
Chapter 10 gives an overview of Brownian motion, constructs the Ito integral for processes with continuous paths, and uses Ito's formula to solve
various stochastic dierential equations. Our approach to stochastic calculus
builds on the reader's knowledge of classical calculus and emphasizes the similarities and dierences between the two theories via the notion of variation
of a function.
Chapter 11 uses the tools developed in Chapter 10 to construct the BlackScholes-Merton PDE, the solution of which leads to the celebrated BlackScholes formula for the price of a call option. A detailed analysis of the analytical properties of the formula is given in the last section of the chapter.
The more technical proofs are relegated to appendices so as not to interrupt
the main ow of ideas.
Chapter 12 gives a brief overview of those aspects of continuous-time martingales needed for risk-neutral pricing. The primary result is Girsanov's Theorem, which guarantees the existence of risk-neutral probability measures.
Chapters 13 and 14 provide a martingale approach to option pricing, using
risk-neutral probability measures to nd the value of a variety of derivatives,
including path-dependent options. Rather than being encyclopedic, the material is intended to convey the essential ideas of derivative pricing and to
demonstrate the utility and elegance of martingale techniques in this endeavor.
The text contains numerous examples and 200 exercises designed to help
the reader gain expertise in the methods of nancial calculus and, not incidentally, to increase his or her level of general mathematical sophistication.
The exercises range from routine calculations to spreadsheet projects to the
xiii
pricing of a variety of complex nancial instruments. Hints and solutions to
the odd-numbered problems are given in Appendix D.
For greater clarity and ease of exposition (and to remain within the intended scope of the text), we have avoided stating results in their most general
form. Thus, interest rates are assumed to be constant, paths of stochastic processes are required to be continuous, and nancial markets trade in a single
risky asset. While these assumptions may be unrealistic, it is our belief that
the reader who has obtained a solid understanding of the theory in this simplied setting will have little diculty in making the transition to more general
contexts.
While the text contains numerous examples and problems involving the
use of spreadsheets, we have not included any discussion of general numerical
techniques, as there are several excellent texts devoted to this subject. Indeed,
such a text could be used to good eect in conjunction with the present one.
It is inevitable that any serious development of option pricing methods at
the intended level of this book must occasionally resort to invoking a result
that falls outside the scope of the text. For the few times that this has occurred, we have tried either to give a sketch of the proof or, failing that, to
give references, general or specic, where the reader may nd a reasonably
accessible proof.
The text is organized to allow as exible use as possible. The precursor
to the book, in the form of a set of notes, has been successfully tested in the
classroom as a single semester course in discrete-time theory only (Chapters
19) and as a one-semester course giving an overview of both discrete-time and
continuous-time models (Chapters 17, 10, and 11). It may also easily serve
as a two-semester course, with Chapters 113 forming the core and selections
from Chapter 14.
To the students whose sharp eye caught typos, inconsistencies, and downright errors in the notes leading up to the book: thank you. To the readers of
this text: the author would be grateful indeed for similar observations, should
the opportunity arise, as well as for suggestions for improvements.
Hugo D. Junghenn
Washington, D.C., USA
This page intentionally left blank
Chapter 1
Interest and Present Value
In this chapter, we consider assets whose value is determined by an interest
rate. If the asset is guaranteed, as in the case of an insured savings account or a
government bond (which, typically, has only a small likelihood of default), the
asset is said to be
risk-free . Such an asset stands in contrast to a risky asset,
for example, a stock or commodity, whose future values cannot be determined
with certainty. As we shall see in later chapters, mathematical models that
describe the value of a risky asset typically include a component involving a
risk-free asset. Therefore, our rst goal is to describe how risk-free assets are
valued.
1.1 Compound Interest
Interest
is a fee paid by one party for the use of cash assets of another.
The amount of interest is generally time dependent: the longer the outstanding
balance, the more interest is accrued. A familiar example is the interest generated by a money market account. The bank pays the depositor an amount
that is usually a fraction of the balance in the account, that fraction given in
terms of a prorated annual percentage called the
nominal rate.
n =
1, 2, . . .. Suppose the initial deposit is A0 and the interest rate per period
is i. If interest is compounded , then, after the rst period, the value of the
account is A1 = A0 + iA0 = A0 (1 + i), after the second period the value is
A2 = A1 + iA1 = A1 (1 + i) = A0 (1 + i)2 , and so on. In general, the value of
the account at time n is
Consider rst an account that pays interest at the discrete times
An = A0 (1 + i)n ,
A0
is called the
future value.
present value
or
n = 0, 1, 2, . . . .
discounted value
Now suppose that the nominal rate is
times a year. Then
i = r/m
r
(1.1)
of the account and
An
and interest is compounded
hence the value of the account after
At = A0 (1 + r/m)mt .
t
a
m
years is
(1.2)
1
Option Valuation: A First Course in Financial Mathematics
2
The distinction between the formulas (1.1) and (1.2) is that the former expresses the value of the account as a function of the number of compounding
intervals (that is, at the discrete times
a function of continuous time
t
n),
while the latter gives the value as
(in years).
In contrast to an account earning compound interest, an account drawing
simple interest
has time-t value
At = A0 (1 + tr).
In this case, interest is calculated only on the initial deposit
A0
and not on
the preceding account value.
Example 1.1.1.
Table 1.2 gives the value after two years of an account with
present value $800. The account is assumed to earn interest at an annual rate
of 12%.
Value
800(1.12)2
800(1.06)4
800(1.03)8
800(1.01)24
800(1.0003)730
Compound Method
=
$1,003.52
annually
=
$1,009.98
semiannually
=
$1,013.42
quarterly
=
$1,015.79
monthly
=
$1,016.96
daily
TABLE 1.1: Account Value in Two Years
Note that for simple interest the value of the account after two years is
800(1.24) = $992.00.
The above example suggests that compounding more frequently results in
(1+r/m)m
x = m/r in (1.2)
a greater return. This is can be seen from the fact that the sequence
is increasing in
m.
To see what happens when
so that
m → ∞,
set
rt
At = A0 [(1 + 1/x)x ] .
As
m → ∞, l'Hospital's rule shows that (1 + 1/x)x → e. In this way, we arrive
at the formula for
continuously compounded interest :
At = A0 ert .
(1.3)
Returning to Example 1.1.1 we see that, if interest is compounded continuously, then the value of the account after two years is
800e(.12)2 = $1, 016.99,
not signicantly more than for daily compounding.
The
eective interest rate re
is the simple interest rate that produces the
Interest and Present Value
3
same yield in one year as compound interest. If interest is compounded
times a year, this means that
A0 (1 + r/m)m = A0 (1 + re )
m
hence
re = (1 + r/m)m − 1.
If interest is compounded continuously, then
A0 er = A0 (1 + re )
so that
re = er − 1.
Example 1.1.2.
You just inherited $10,000, which you decide to deposit in
one of three banks, A, B, or C. Bank A pays 11% compounded semiannually, bank B pays 10.76% compounded monthly, and bank C pays 10.72 %
compounded continuously. Which bank should you choose?
Solution:
We compute the eective rate
re
for each given interest rate.
Rounding, we have
= (1 + .11/2)2 − 1
= 0.113025
= (1 + .1076/12)12 − 1 = 0.113068
= e.1072 − 1
= 0.113156
re
re
re
for Bank A,
for Bank B,
for Bank C.
Bank C has the highest eective rate and is therefore the best choice.
1.2 Annuities
An
annuity
is a sequence of periodic payments of a xed amount, say,
P.
The payments may take the form of deposits into an account, such as a pension
fund or layaway plan, or withdrawals from an account, for example, a trust
fund or retirement account.
annual rate
r
compounded
1 Suppose that the account pays interest at an
m
times per year and that a deposit (withdrawal)
is made at the end of each compounding interval. We seek the value
account at time
n,
that is, immediately after the
nth
An
of the
payment.
An is the sum of the time-n values of payments 1
j accrues interest over n − j payment periods, its
P (1 + r/m)n−j . Thus,
In the case of deposits,
through
n.
Since payment
time-n value is
An = P (1 + x + x2 + · · · + xn−1 ),
The geometric series sums to
(xn − 1)/(x − 1),
An = P
1 An
(1 + i)n − 1
,
i
x := 1 +
r
.
m
hence
i :=
r
.
m
(1.4)
account into which periodic deposits are made for the purpose of retiring a debt or
purchasing an asset is sometimes called a
sinking fund.
Option Valuation: A First Course in Financial Mathematics
4
For withdrawals we argue as follows: Let
account. The value at the end of period
nth
payment, is
An−1
plus the interest
withdrawal reduces that value by
P
n,
iAn−1
A0
be the initial value of the
just before withdrawal of the
over that period. Making the
so
An = aAn−1 − P,
a := 1 + i.
Iterating, we obtain
An = a2 An−2 − (1 + a)P = · · · = an A0 − (1 + a + a2 + · · · + an−1 )P.
Thus,
1 − (1 + i)n
i
(1 + i)n (iA0 − P ) + P
.
=
i
An = (1 + i)n A0 + P
(1.5)
Now assume that the account is drawn down to zero after
Setting
n=N
and
AN = 0
in (1.5) and solving for
A0 = P
A0
1 − (1 + i)−N
.
i
This is the initial deposit required to support exactly
P
N
withdrawals.
yields
(1.6)
N
withdrawals of amount
from, say, a retirement account or trust fund. It may be seen as the sum of
the present values of the
Solving for
P
N
withdrawals.
in (1.6) we obtain
P = A0
i
,
1 − (1 + i)−N
(1.7)
which may be used, for example, to calculate the mortgage payment for a
A0
mortgage of size
(see Example 1.2.2, below). Substituting (1.7) into (1.5)
we obtain the following formula for the time-n value of an annuity supporting
exactly
N
withdrawals:
An = A0
Example 1.2.1.
size
P
After
for
1 − (1 + i)n−N
,
1 − (1 + i)−N
n = 0, 1, . . . , N.
(1.8)
(Retirement plan). Suppose you make monthly deposits of
r, compounded monthly.
t years you wish to make monthly withdrawals of size Q from the account
into a retirement account with an annual rate
s years, drawing down the account to zero. By (1.4) and (1.6) it must then
be the case that
P
1 − (1 + i)−12s
(1 + i)12t − 1
=Q
,
i
i
i :=
r
,
12
Interest and Present Value
or
P
1 − (1 + i)−12s
=
.
Q
(1 + i)12t − 1
For a numerical example, suppose that
t = 40, s = 30,
5
(1.9)
and
r = .06.
Then
P
1 − (1.005)−360
=
≈ .084,
Q
(1.005)480 − 1
so that a withdrawal of, say,
Q = $5000
during retirement would require
monthly deposits of
P = (.084)5000 ≈ $419.
A more realistic analysis takes into account the reduction of purchasing power
due to ination. Suppose that ination is running at 3% per year or
.25%
per month. This means that goods and services that cost $1 now will cost
$(1.0025)
n
n
months from now. The present value purchasing power of the
rst withdrawal is then
5000(1.0025)−481 ≈ $1504,
while that of the last withdrawal is only
5000(1.0025)−840 ≈ $614.
For the rst withdrawal to have the current purchasing power of $5000,
Q
would have to be
5000(1.0025)481 ≈ $16, 617,
which would require monthly deposits of
P = (.084)16, 617 ≈ $1396.
For the last withdrawal to have the current purchasing power of $5000,
Q
would have to be
5000(1.0025)840 ≈ $40, 724,
requiring monthly deposits of
P = (.084)40, 724 ≈ $3421,
more than eight times the amount calculated without considering ination!
Example 1.2.2.
(Amortization). Suppose you take out a 20-year, $200,000
mortgage at an annual rate of 8% compounded monthly. Your monthly mort-
P constitute an
N = 240. Here An is
A0 = $200, 000, i = .08/12 =
n. By
gage payments
annuity with
.0067,
the amount owed at the end of month
and
(1.7), the mortgage payments are
P = 200, 000
.0067
= $1677.85.
1 − (1.0067)−240
Option Valuation: A First Course in Financial Mathematics
6
Now let
In
and
Pn
denote, respectively, the portions of the
that are interest and principle. Since
n − 1,
An−1
nth
payment
was owed at the end of month
(1.8) shows that
In = iAn−1 = iA0
1 − (1 + i)n−1−N
,
1 − (1 + i)−N
and therefore
Pn = P − In = iA0
In particular, from (1.10) we have
(1 + i)n−1−N
.
1 − (1 + i)−N
P1 = $337.86
and
(1.10)
P240 = $1666.70.
Thus
only about 20% of the rst payment goes to reducing the principle, while
almost 100% of the last payment does so.
The sequences
(Pn ), (In ),
amortization schedule
and
(An )
form the basis of what is called the
of the mortgage.
In the above annuity formulas, the compounding interval and the payment
end of the compounding
ordinary annuity. If payment is made at
interval are the same, and payment is made at the
interval, describing what is called an
the
beginning of the period, as is the case for, say, rents and
annuity due, and the formulas change accordingly.
insurance, one
obtains an
1.3 Bonds
Bonds are nancial contracts issued by governments, corporations, and
other institutions. The simplest type of bond is the
zero coupon bond.
U.S.
Treasury bills and U.S. savings bonds are common examples. The purchaser
B0 (which may be determined by bids) and receives
F , the face value of the bond, at a prescribed time T , the
maturity date. The value Bt of the bond at time t may be expressed in terms
of a continuously compounded interest rate r determined by the equation
of a bond pays an amount
a prescribed amount
B0 = F e−rT .
Bt
is then the face value of the bond discounted to time
Bt = F e−r(T −t) = B0 ert ,
Thus, during the time interval
[0, T ],
t:
0 ≤ t ≤ T.
the bond acts like a money market
account with continuously compounded interest. The time restriction may be
theoretically removed as follows: At time
the bond by buying
F/B0
T,
reinvest the proceeds
bonds, each for the amount
B0
F
from
and each with the
Interest and Present Value
7
F and maturity date 2T . At time t ∈ [T, 2T ] each
F e−r(2T −t) = B0 e−rT ert , so the bond account has value
face value
bond has value
Bt = (F/B0 )B0 e−rT ert = F e−rT ert = B0 ert , (T ≤ t ≤ 2T ).
Continuing this process we see that the formula
t≥0
over which the rate
r,
Bt = B0 ert
holds for all times
determined by the face value of the bond and the
bid, is constant.
With a
coupon bond,
one receives not only the amount
F
at time
T
but
also a sequence of payments during the life of the bond. Thus, at prescribed
t1 < t2 < · · · < tN , the bond pays an amount Cn , called a coupon, and
T one receives the face value F . The price of the bond is the total
times
at maturity
present value
B0 =
N
X
e−rtn Cn + F e−rT .
(1.11)
n=1
Note that this is the initial value of a portfolio consisting of
bonds maturing at times
t1 , t2 , . . ., tN ,
and
N +1 zero-coupon
T.
1.4 Rate of Return
Consider an investment that returns, for an initial payment of
amount
An > 0
at the end of period
n, n = 1, 2, . . . , N .
The
of the investment is dened to be that periodic interest rate
i
N
X
An (1 + i)−n .
an
for which the
present value of the sequence of returns equals the initial payment
P =
P > 0,
rate of return
P,
that is,
(1.12)
n=1
Examples of such investments are annuities and coupon bonds. For a coupon
bond that pays the amount
pays the face value
F
P =C
where
P = B0
C at each of the times n = 1, 2, . . . , N − 1
N , Equation (1.12) reduces to
1 − (1 + i)−N
+ F (1 + i)−N ,
i
is the price of the bond.
To see that Equation (1.12) has a unique solution
side by
f (i) and note that f
Since
i > −1, denote the right
(−1, ∞) and satises
is continuous on the interval
lim f (i) = 0
i→∞
P
and
at time
and
lim f (i) = ∞.
i→−1+
P > 0, the Intermediate Value Theorem implies that the equation f (i) =
i > −1. Because f is strictly decreasing, the solution is unique.
has a solution
8
Option Valuation: A First Course in Financial Mathematics
A rate of return
i
may be positive, zero, or negative. If
f (0) > P ,
that is,
the sum of the payos is greater than the initial investment, then, because
is decreasing,
i > 0.
less than the initial investment,
Example 1.4.1.
f (0) < P ,
then i < 0.
Similarly, if
f
that is, the sum of the payos is
Suppose you loan a friend $100 and he agrees to pay you
$35 at the end of the rst year, $37 at the end of the second year, and $39 at
the end of the third year, at which time the loan is considered to be paid o.
The sum of the payos is greater than 100, so the equation
39
37
35
+
= 100
+
2
(1 + i) (1 + i)
(1 + i)3
has a unique positive solution
i,
i.
One can use Newton's method to determine
or one can simply solve the equation by trial and error using a spreadsheet.
The latter approach gives
i ≈ 0.053, that is, an annual rate of about 5.3%.
Interest and Present Value
9
1.5 Exercises
1. Suppose you deposit $1500 in an account paying an annual rate of 6%.
Find the value of the account in three years if interest is compounded
(a) yearly; (b) quarterly; (c) monthly; (d) daily; (e) continuously.
2. What annual interest rate
r
would allow you to double your initial de-
posit in 6 years if interest is compounded quarterly? Continuously?
3. Find the eective interest rate if a nominal rate of 12% is compounded
(a) quarterly; (b) monthly; (c) continuously.
4. If you receive 6% interest compounded monthly, about how many years
will it take for your investment to triple?
5. If you deposit $400 at the end of each month into an account earning
8% interest compounded monthly, what is the value of the account at
the end of 5 years? 10 years?
6. You deposit $700 at the end of each month into an account earning
interest at an annual rate of
to nd the value of
r
r
compounded monthly. Use a spreadsheet
that produces an account value of $50,000 in 5
years.
7. You deposit $400 at the end of each month into an account with an
annual rate of 6% compounded monthly. Use a spreadsheet to determine
the minimum number of payments required for the account to have a
value of at least $30,000.
8. Suppose an account oers continuously compounded interest at an annual rate
r
and that a deposit of size
P
is made at the end of each
month. Show that the value of the account after
An = P
n
deposits is
ern/12 − 1
.
er/12 − 1
9. You make an initial deposit of $200,000 into an account paying 6%
compounded monthly. If you withdraw $2000 each month, how much
will be left in the account after 5 years? 10 years? When will the account
be drawn down to zero?
10. An account pays an annual rate of 8% percent compounded monthly.
What lump sum must you deposit into the account now so that in 10
years you can begin to withdraw $4000 each month for the next 20 years,
drawing down the account to zero?
10
Option Valuation: A First Course in Financial Mathematics
11. A trust fund has an initial value of $300,000 and earns interest at an
annual rate of 6%, compounded monthly. If a withdrawal of $5000 is
made at the end of each month, when will the account will fall below
$150,000? (Use a spreadsheet.)
A0 in terms of P
i that will fund a perpetual annuity, that is, an annuity for which
An > 0 for all n. What is the value of An in this case?
12. Referring to Equation (1.5), nd the smallest value of
and
13. Suppose that an account oers continuously compounded interest at an
annual rate
r
and that withdrawals of size
each month. If the initial deposit is
to zero after
N
A0
P
are made at the end of
and the account is drawn down
withdrawals, show that the value of the account after
withdrawals is
An = P
n
1 − e−r(N −n)/12
.
er/12 − 1
t = 30, s = 20, and r = .12. Find the
Q of $3000 per month. If ination
is running at 2% per year, what value of P will give the rst withdrawal
14. In Example 1.2.1, suppose that
payment amount
P
for withdrawals
the current purchasing power of $3000? The last withdrawal?
15. For a 30-year, $300,000 mortgage, determine the annual rate
r
you will
have to lock in to have payments of $1800 per month?
16. In Example 1.2.2, suppose that you must pay an inspection fee of $1000,
a loan initiation fee of $1000, and 2
points,
that is, 2% of the nominal
loan of $200,000. Eectively, then, you are receiving only $194,000 from
the lending institution. Calculate the annual interest rate
r0
you will
now be paying, given the agreed upon monthly payments of $1667.85.
17. How large a loan can you take out at an annual rate of 15% if you can
aord to pay back $1000 at the end of each month and you want to
retire the loan after 5 years?
18. Suppose you take out a 20-year, $300,000 mortgage at 7% and decide
after 15 years to pay o the mortgage. How much will you have to pay?
19. You can retire a loan either by paying o the entire amount $8000 now,
or by paying $6000 now and $6000 at the end of 10 years. Find a cuto
value
r0
such that if the nominal rate
the entire loan now, but if
r > r0 ,
r
is
< r0 ,
then you should pay o
then it is preferable to wait. Assume
that interest is compounded continuously.
20. You can retire a loan either by paying o the entire amount $8000 now,
or by paying $6000 now, $2000 at the end of 5 years, and an additional
$2000 at the end of 10 years. Find a cuto value
nominal rate
r
is
< r0 ,
r0
such that if the
then you should pay o the entire loan now,
Interest and Present Value
but if
r > r0 ,
11
then it is preferable to wait. Assume that interest is
compounded continuously.
21. Suppose you take out a 30-year, $100,000 mortgage at 6%. After 10
years, interest rates go down to 4%, so you decide to renance the remainder of the loan by taking out a new 20-year mortgage. If the cost
of renancing is 3 points (3% of the new mortgage amount), what are
the new payments? What threshold interest rate would make renancing scally unwise? (Assume that the points are rolled in with the new
mortgage.)
22. Referring to Example 1.2.2, show that
Pn = (1 + i)n−1 P1 ,
and
In =
1 − (1 + i)n−N −1
I1 .
1 − (1 + i)−N
23. Referring to Section 1.3, nd the time-t value
tm ≤ t < tm+1 , m = 0, 1, 2, . . . , N − 1,
where
Bt of a
t0 = 0.
coupon bond for
24. Find the rate of return of a 4-year investment that, for an initial investment of $1000, returns $100, $200, and $300 at the end of years 1, 2,
and 3, respectively, and, at the end of year 4, returns (a) $350; (b) $400;
(c) $550. What would the rates be if the rate of return formula is based
on continuously compounded interest?
25. Table 1.2 gives the end of year returns for two investment plans based
on an initial investment of $10,000. Determine which plan is best.
Year 1
Year 2
Year 3
Year 4
Plan A
$3000
$5000
$7000
$1000
Plan B
$3500
$4500
$6500
$1500
TABLE 1.2: End of Year Returns
26. In Exercise 25, what is the smallest return in year 1 of Plan A that would
make Plans A and B equally lucrative? Answer the same question for
year 4.
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Chapter 2
Probability Spaces
Because nancial markets are sensitive to a variety of unpredictable events,
the value of a nancial asset, such as a stock or commodity, is usually interpreted as a random quantity subject to the laws of probability. In this chapter,
we develop the probability theory needed to model the dynamic behavior of
asset prices. We assume that the reader is familiar with the notation and terminology of elementary set theory as well as basic combinatorial principles. A
review of these concepts may be found in Appendix A.
2.1 Sample Spaces and Events
A
probability is a number that expresses the likelihood of occurrence of an
experiment. The experiment can be something as simple as the
event in an
toss of a coin or as complex as the observation of stock prices over time. For
our purposes, we shall consider an experiment to be any activity that produces observable outcomes. For example, tossing a die and noting the number
of dots appearing on the top face is an experiment whose outcomes are the
integers 1 through 6. Observing the average value of a stock over the previous
week or noting the rst time the stock dips below a prescribed level are experiments whose outcomes are nonnegative real numbers. Throwing a dart is
an experiment whose outcomes may be taken as the coordinates of the dart's
landing position.
The collection of all outcomes of an experiment is called the
sample space
of the experiment. In probability theory, one starts with an assignment of
probabilities to subsets of the sample space called
events.
This assignment
must satisfy certain axioms and can be quite technical, depending on the
sample space and the nature of the events. We begin with the simplest setting,
that of a discrete probability space.
13
Option Valuation: A First Course in Financial Mathematics
14
2.2 Discrete Probability Spaces
Consider an experiment whose outcomes may be represented by a nite or
innite sequence, say,
ω1 , ω2 , . . .. Let pn denote the probability of outcome ωn .
pn may be based on relative frequency, logical
In practice, the determination of
deduction, or analytical methods and may be approximate or theoretical. For
example, suppose we take a poll of 1000 people in a certain locality and
discover that 200 prefer candidate A and 800 candidate B. If we choose a
person at random from the sample, then it is natural to assign a theoretical
probability of
.2
to the outcome that the person chosen prefers candidate
A. If, however, the person is chosen randomly from the entire locality, then
pollsters take the probability of the same outcome to be only approximately
.2. Similarly, if we ip a coin 10,000 times and nd that exactly 5143 heads
appear, we might assign the approximate probability of
.5134
to the outcome
that a single toss produces a head. On the other hand, for the idealized coin,
we would assign that same outcome a theoretical probability of .5.
However the probabilities
pn
are determined, they must satisfy the follow-
ing properties:
(a)
(b)
0 ≤ pn ≤ 1 for
P
n pn = 1.
every
n,
and
The nite or innite sequence
for the experiment. The
Ω = {ω1 , ω2 , . . .}
(p1 , p2 , . . .)
probability P(A)
is then called the
of a subset
A
probability vector
of the sample space
is dened as
P(A) =
X
pn .
(2.1)
ωn ∈A
A = ∅, then
P is called a
The sum in (2.1) is either nite or a convergent innite series. If
the sum is interpreted as having the value zero. The function
probability measure for the
discrete probability space.
experiment, and the pair
(Ω, P)
is said to be a
The following proposition summarizes the basic properties of
P.
We omit
the proof.
Proposition 2.2.1. (i) 0 ≤ P(A) ≤ 1; (ii) P(∅) = 0 and P(Ω) = 1; (iii) if
is a nite or innite sequence of pairwise disjoint subsets of Ω, then
(An )
!
P
[
n
An
=
X
P (An ) .
n
Part (iii) of the proposition is called the
additivity property
of
P.
As we
shall see, it is precisely the property needed to justify taking limits of random
quantities.
Probability Spaces
Example 2.2.2.
15
There are 10 slips of paper in a hat, two of which are labeled
with the number 1, three with the number 2, and ve with the number 3. A
slip is drawn at random from the hat, the label is noted, the slip is returned,
and the process is repeated a second time. The sample space of the experiment
may be taken to be the set of all ordered pairs
on the rst slip and
k
(j, k),
j is the number
A that the sum of
outcomes (1, 3), (3, 1), and
where
the number on the second. The event
the numbers on the slips equals 4 consists of the
(2, 2). By relative frequency arguments, the probabilities
are .1, .1, and .09, respectively, hence P(A) = .29.
Example 2.2.3.
of these outcomes
Toss a fair coin innitely often (conceptually, but not practi-
cally, possible). This produces an innite sequence of heads
experiment consists of observing the rst time an
H
H
and tails
T . Our
occurs. The sample space
Ω = {0, 1, 2, 3, . . .}, where, for example, the outcome 2 means that the
T and the second H , while outcome 0 means that H never
appears. To nd the probability vector (p0 , p1 , . . .) for the experiment, we argue as follows: Since on the rst toss the outcomes H or T are equally likely,
we should set p1 = 1/2. Similarly, the outcomes HH , HT , T H , T T of the rst
two tosses are equally likely hence p2 , the probability that T H occurs, should
−n
be 1/4. In general, we see that we should set pn = 2
By additivity,
P∞ , n ≥ 1.P
∞
−n
the probability that a head eventually comes up is
p
=
= 1,
n=1 n
n=1 2
from which it follows that p0 = 0. The probability vector for the experiment
−1 −2
is therefore 0, 2
,2 ,... .
is then
rst toss comes up
In the important special case where the sample space
outcome is equally likely,
pn = 1/|Ω|
P(A) =
where
|·|
Ω
is nite and each
and (2.1) reduces to
|A|
,
|Ω|
denotes the number of elements in a nite set. The determination
of probabilities in this case is then purely a combinatorial problem.
Example 2.2.4.
52
5 = 2, 598, 960. We
show that three of a kind (for example, three Jacks, 5, 7) beats two pair.
The total number of poker hands is
By the multiplication principle (Appendix A), the number of poker hands
with three of a kind is
13 · 4 ·
48 · 44
2
= 54, 912,
corresponding to the process of choosing a denomination for the triple, selecting three cards from that denomination, and then choosing the remaining two
cards, avoiding the selected denomination as well as pairs. The probability of
getting a hand with three of a kind is therefore
54, 912
≈ .02113.
2, 598, 960
Option Valuation: A First Course in Financial Mathematics
16
Similarly, the number of hands with two (distinct) pairs is
2
13
4
· 44 = 123, 552,
·
2
2
corresponding to the process of choosing denominations for the pairs, choosing
two cards from each of the denominations, and then choosing the remaining
card, avoiding the selected denominations. The probability of getting a hand
with two pairs is therefore
123, 552
≈ .04754,
2, 598, 960
more than twice that of getting three of a kind.
2.3 General Probability Spaces
For a discrete probability space, we were able to assign a probability to
each set of outcomes, that is, to each subset of the sample space
Ω. In general
probability spaces this may not be possible, and we must conne our assign-
Ω. To
σ -eld,
ment of probabilities to a suitably restricted collection of subsets of
have a useful and robust theory, we require that the collection form a
dened as follows.
Denition 2.3.1. A collection F of subsets of a set Ω is said to be a σ-eld
if
(a) ∅, Ω ∈ F ;
0
(b) A ∈ F ⇒ A ∈ F ; and
(c)
for any nite or innite sequence of members An of F ,
[
n
An ∈ F .
If Ω is the sample space of an experiment, then F is called an
for the experiment and members of F are called events.
event
Property (a) of Denition 2.3.1 asserts that the sure event
impossible event
∅
are always members of
F.
Ω
σ -eld
and the
Property (c) asserts that
F
is
closed under countable unions. By virtue of (b), (c), and De Morgan's law
!0
\
An =
n
F
[
A0n
,
n
is also closed under countable intersections.
The trivial collection
amples of
σ -elds.
{∅, Ω}
and the collection of all subsets of
The following examples are more interesting.
Ω
are ex-
Probability Spaces
Example 2.3.2.
P
partition
of Ω,
Ω. The
collection consisting of ∅ and all possible unions of members of P is a σ -eld.
To illustrate property (b), suppose, for example, that P = {A1 , A2 , A3 , A4 }.
The complement of A1 ∪ A3 is then A2 ∪ A4 .
Let
Ω
17
be a nite, nonempty set and
a
that is, a collection of pairwise disjoint, nonempty sets with union
Example 2.3.3.
A be any collection of subsets of Ω and let {Fλ : λ ∈ Λ}
σ -elds containing A. The intersection FA of the σ elds Fλ is again a σ -eld, called the σ -eld generated by A. It is the smallest
σ -eld containing the members of A. If Ω is nite and A is a partition of Ω,
then FA is the σ -eld of Example 2.3.2. If Ω is an interval of real numbers
and A is the collection of all subintervals of Ω, then FA is called the Borel
σ -eld of Ω and its members the Borel sets of Ω.
Let
denote the collection of all
An event
σ -eld F
may be thought of as representing the available infor-
mation in an experiment, information that is known only after an outcome
of the experiment has been observed. For example, if we are contemplating
buying a stock at time t, then the information available to us (barring insider
information) is the price history of the stock up to time t. We show later that
this information may be conveniently described by a
Once a sample space
Ω
and an event
σ -eld
σ -eld Ft .
have been specied the next
step is to assign probabilities. This is done in accordance with the following
axioms. (Compare with Proposition 2.2.1.)
Denition 2.3.4. Let Ω be a sample space and F an event σ-eld. A probability measure for (Ω, F), or a probability law for the experiment, is a function
P which assigns to each event A ∈ F a number P(A), called the probability of
A, such that the following properties hold:
(a)
0 ≤ P(A) ≤ 1;
(b)
P(Ω) = 1
(c)
and P(∅) = 0;
if A1 , A2 , . . . is a nite or innite sequence of pairwise disjoint events,
then
!
P
[
An
=
n
X
P(An ).
n
The triple (Ω, F, P) is then called a probability space.
A collection of events is said to be
each pair of distinct members
A
and
mutually exclusive
B
if
P(AB) = 0
for
in the collection. Pairwise disjoint
sets are obviously mutually exclusive, but not conversely. It is clear that the
additivity axiom (c) holds for mutually exclusive events as well.
Proposition 2.3.5. A probability measure P has the following properties:
(i)
P(A ∪ B) = P(A) + P(B) − P(AB);
Option Valuation: A First Course in Financial Mathematics
18
(ii)
(iii)
if B ⊆ A, then P(A − B) = P(A) − P(B); in particular P(B) ≤ P(A);
P(A0 ) = 1 − P(A).
Proof.
For (i), note that
AB 0 , AB ,
and
BA0 .
A∪B
is the union of the pairwise disjoint events
Therefore, by additivity,
P(A ∪ B) = P(AB 0 ) + P(AB) + P(BA0 ).
Similarly,
P(A) = P(AB 0 ) + P(AB)
P(B) = P(BA0 ) + P(AB).
and
Subtracting the last two equations from the rst gives property (i). Property
P(Ω) = 1.
(ii) follows easily from additivity, as does (iii), using
Part (i) of Proposition 2.3.5 is a special case of the inclusion-exclusion
rule. In Exercise 2, we consider versions for three and four events.
By Proposition 2.2.1, a discrete probability space is a probability space
in the general sense with event
σ -eld
consisting of all subsets of
Ω.
The
following are examples of probability spaces that are not discrete. In each
case, the underlying experiment is seen to result in a continuum of outcomes.
Accordingly, the problem of determining the appropriate
σ -eld of events and
assigning suitable probabilities is somewhat more technical.
Example 2.3.6.
Consider the experiment of randomly choosing a real num-
ber from the interval
[0, 1].
If we try to assign probabilities as in the discrete
case, we should assume that the outcomes
set
J
P(x) = p
for some
p ∈ [0, 1].
x
are equally likely and therefore
However, consider, for example, the event
that the number chosen is less than 1/2. Following the discrete case, the
probability of
J
should then be
P
x∈[0,1/2)
p, which is either 0 or +∞, if it has
meaning at all.
To avoid this problem, we take the following more natural approach: Since
we expect that half the time the number chosen will lie in the left half of
the interval
I
of
[0, 1],
we dene
P(J) = .5.
More generally, for any subinterval
the probability that the selected number
I,
x
lies in
which is the theoretical proportion of times
one may show that every Borel subset of
[0, 1]
x
I
should be the length
lands in
I.
Generalizing,
may be assigned a probability
consistent with the axioms of a probability space. Therefore, it is natural to
take the event
σ -eld
in this experiment to be the collection of all Borel sets
(see Example 2.3.3).
A that the selected number x
.d1 d2 d3 . . . with no digit dj equal to 3. Set A0 = [0, 1].
Since d1 6= 3, A must be contained in the set A1 obtained by removing from
A0 the interval [.3, .4). Similarly, since d2 6= 3, A is contained in the set
A2 obtained by removing from A1 the nine intervals of the form [.d1 3, .d1 4),
d1 6= 3. Having obtained An−1 in this way, we see that A must be contained
n−1
in the set An obtained by removing from An−1 the 9
intervals of the
As a concrete example, consider the event
has a decimal expansion
Probability Spaces
form
19
[.d1 d2 . . . dn−1 3, .d1 d2 . . . dn−1 4), dj 6= 3. Since
10−n , the additivity axiom implies that
each of these intervals
has length
P(An ) = P(An−1 ) − 9n−1 10−n = P(An−1 ) − (.1)(.9)n−1 .
Summing from
1
to
N,
we obtain
P(AN ) = 1 − (.1)
Therefore,
P(A) ≤ (.9)N
for all
N,
N
X
(.9)n−1 = (.9)N .
n=1
which implies that
P(A) = 0. Thus,
[0, 1] has a
probability one, a number chosen randomly from the interval
with
digit
3 in its decimal expansion.
Example 2.3.7.
Consider a dartboard in the shape of a square with the
origin of a coordinate system at the lower left corner and the point
(1, 1) in
(x, y)
the upper right corner. We throw a dart and observe the coordinates
of the landing spot. (If the dart lands o the board, we ignore the outcome.)
The sample space of this experiment is
Ω = [0, 1] × [0, 1].
(This is obviously
a two-dimensional version of the preceding example.) Consider the region
below the curve
y = x2 .
The area of
the time the dart will land in
A
A
is
1/3,
so we would expect that
1/3
(a fact borne out, for example, by computer
simulation.) This suggests that we dene the probability of the event
be
1/3.
A
of
A
to
More generally, the probability of any reasonable region is dened
as the area of that region. It turns out that probabilities may be assigned
to all two dimensional Borel subsets of
Ω
in a manner consistent with the
axioms.
Example 2.3.8.
In the coin tossing experiment of Example 2.2.3, we simply
noted the rst time a head appears, giving us a sample space consisting of the
nonnegative integers. If, however, we observe the entire sequence of outcomes,
then the sample space consists of all sequences of
discrete. To see why this is the case, replace
H
H 's and T 's and is no longer
and T by the digits 1 and
0, respectively, so that an outcome may be identied with the binary (base
2) expansion of a number in the interval
T HT HT HT H . . .
[0, 1].
is identied with the number
(For example, the outcome
.01010101 . . . = 1/3.)
The
sample space of the experiment may therefore be identied with the interval
[0, 1],
which is uncountable.
To assign probabilities in this experiment, we begin by giving a probability
2−n to events that prescribe exactly n outcomes. For example, the event A
that H appears on the rst and third tosses would have probability 1/4. Note
that, under the above identication, the event A corresponds to the subset of
[0, 1] consisting of all numbers with binary expansion beginning .101 or .111,
namely, the union of the intervals [5/8, 3/4) and [7/8, 1). The total length of
these intervals is 1/4, suggesting that the natural assignment of probabilities in
of
this example is precisely that of Example 2.3.6 (which is indeed the case).
Option Valuation: A First Course in Financial Mathematics
20
2.4 Conditional Probability
Suppose we assign probabilities to the events
learn that an event
B
A of an experiment and then
has occurred. One would expect that this new informa-
tion could change the original probabilities
A is called the conditional
P(A).
The altered probability of
probability of A given B and is denoted by P(A|B).
A precise mathematical denition of
P(A|B)
is suggested by the following
example.
Example 2.4.1.
Suppose that in a group of 100 people exactly 40 smoke,
and that 15 of the smokers and 5 of the nonsmokers have lung cancer. A
person is chosen at random from the group. Let
person has lung cancer and
B
A
be the event that the
the event that the person does not smoke.
Suppose we discover that the person chosen is a nonsmoker, that is, that the
A, we should
B consisting of people who don't smoke.
This gives P(A|B) = |AB|/|B| = 5/60 = .083, considerably smaller than the
original probability P(A) = |A|/100 = .2.
event
B
has occurred. Then, in computing the new probability of
restrict ourselves to the sample space
Note that in the preceding example
P(A|B) =
|AB|
|AB|/|Ω|
P(AB)
=
=
.
|B|
|B|/|Ω|
P(B)
This suggests the following general denition of conditional probability.
Denition 2.4.2. Let (Ω, F, P) be a probability space. If A and B are events
with P(B) > 0, then the conditional probability of A given B is
P(A|B) =
P(A|B)
is undened if P(B) = 0.
Example 2.4.3.
probability of
Let
B
P(AB)
.
P(B)
be the
In the dartboard experiment of Example 2.3.7, we assigned a
1/3 to the event A that the dart lands below the graph of y = x2 .
event that the dart lands in the left half of the board and C the
event that the dart lands in the bottom half. Recalling that probability in this
P(B) = P(C) = 1/2, P(AB) = 1/24,
P(BC) = 1/4. Therefore, P(A|B) = 1/12 and P(C|B) = 1/2. Knowledge
the event B changes the probability of A but not of C .
experiment is dened as area, we see that
and
of
Theorem 2.4.4 (Multiplication Rule for Conditional Probabilities). Suppose
that A1 , A2 , . . . , An are events with P(A1 A2 · · · An−1 ) > 0. Then
P(A1 A2 · · · An ) = P(A1 )P(A2 |A1 )P(A3 |A1 A2 ) · · · P(An |A1 A2 · · · An−1 ).
(2.2)
Probability Spaces
Proof.
21
n.) The condition P(A1 A2 · · · An−1 ) > 0 ensures that
n = 2, (2.2) follows from the denition of
conditional probability. Suppose (2.2) holds for n = k ≥ 2. If A = A1 A2 · · · Ak ,
then, by the case n = 2,
(By induction on
the right side of (2.2) is dened. For
P(A1 A2 · · · Ak+1 ) = P(AAk+1 ) = P(A)P(Ak+1 |A),
and, by the induction hypothesis,
P(A) = P(A1 )P(A2 |A1 )P(A3 |A1 A2 ) · · · P(Ak |A1 A2 · · · Ak−1 ).
Combining these results yields (2.2) for
Example 2.4.5.
n = k + 1.
An jar contains 5 red and 6 green marbles. We randomly
draw 3 marbles in succession without replacement. Let
that the rst marble is red,
G3
R2
R1
denote the event
the event that the second marble is red, and
the event that the third marble is green. The probability that the rst two
marbles are red and the third is green is
P(R1 R2 G3 ) = P(R1 )P(R2 |R1 )P(G3 |R1 R2 ) = (5/11)(4/10)(6/9) ≈ .12.
Theorem 2.4.6 (Total Probability Law). Let B1 , B2 , . . . be a nite or innite
sequence of mutually exclusive events whose union is Ω. If P(Bn ) > 0 for every
n, then, for any event A,
P(A) =
X
P(A|Bn )P(Bn ).
n
Proof.
AB1 , AB2 , . . . are mutually exclusive with
X
X
P(A) =
P(ABn ) =
P(A|Bn )P(Bn ).
The events
n
Example 2.4.7.
union
A,
hence
n
(Investor's Ruin) Suppose you own a stock that each day
q = 1 − p. Assume
x and that you intend to sell the stock as soon
as its value is either a or b, whichever comes rst, where 0 < a ≤ x ≤ b. What
goes up $1 with probability
p
or down $1 with probability
that the stock is initially worth
is the probability that you will sell low?
Solution: Let f (x) denote the probability of selling low, that is, of the stock
reaching
a
before
b,
given that the stock starts out at
x.
event that the stock goes up (down) the next day and
Let
A
law,
be the
P(A|S+ ) = f (x + 1), since, if the stock goes up, it's value
x + 1. Similarly, P(A|S− ) = f (x − 1). By the total probability
selling low. Then
the next day is
S+ (S− )
the event of your
P(A) = P(A|S+ )P(S+ ) + P(A|S− )P(S− ),
or
f (x) = f (x + 1)p + f (x − 1)q.
Option Valuation: A First Course in Financial Mathematics
22
p + q = 1,
Since
the last equation may be written
∆f (x) := f (x + 1) − f (x) = r∆f (x − 1),
r := q/p.
Iterating, we obtain
∆f (x + y) = r∆f (x + y − 1) = r2 ∆f (x + y − 2) = · · · = ry ∆f (x),
hence
f (x) − f (a) =
Since
f (a) = 1,
x−a−1
X
∆f (a + y) = ∆f (a)
x−a−1
X
y=0
ry .
y=0
we see that
f (x) = 1 + ∆f (a)
x−a−1
X
ry .
(2.3)
y=0
If
p = q,
noting that
f (x) = 1 + (x − a)∆f (a). Setting x = b
∆f (a) = −1/(b − a), and hence
then (2.3) reduces to
f (b) = 0,
we obtain
f (x) = 1 −
If
p 6= q ,
b−x
x−a
=
.
b−a
b−a
then (2.3) becomes
f (x) = 1 + ∆f (a)
Setting
and
x = b,
we obtain
into (2.4) gives
rx−a − 1
.
r−1
(2.4)
∆f (a) = −(r − 1)/ rb−a − 1 ,
f (x) = 1 −
and substituting this
r − 1 rx−a − 1
rb−a − rx−a
=
.
rb−a − 1 r − 1
rb−a − 1
Example 2.4.7 is a stock market version of what is usually called gambler's
ruin. The name comes from the standard formulation of the example, where
the stock's value is replaced by the winnings of a gambler. Selling low is then
interpreted as the ruin of the gambler.
The stock movement in this example is known as
random walk. We return
to this notion later.
2.5 Independence
Denition 2.5.1. Events A and B in a probability space are said to be indeif P(AB) = P(A)P(B).
pendent
Probability Spaces
23
Note that if P(B) 6= 0, then independence is equivalent to the statement
P(A|B) = P(A), which asserts that the additional information provided by B
is irrelevant to A. A similar remark holds if P(A) 6= 0.
The events B and C of Example 2.4.3 are independent, while A and B are
not. Here are some other examples.
Example 2.5.2.
Suppose in Example 2.4.5 that we draw two marbles in
succession without replacement. Then, by the total probability law, the probability of getting a red marble on the second try is
P(R2 ) = P(R1 )P(R2 |R1 ) + P(G1 )P(R2 |G1 )
= (5/11)(4/10) + (6/11)(5/10)
= 5/11
6= P(R2 |R1 ),
hence
R1
and
R2
are not independent. This agrees with our intuition, since
drawing without replacement obviously changes the conguration of marbles
in the jar. If, on the other hand, we replace the rst marble, then
P(R2 ); the events are independent. Note
type, P(R2 ) = P(R1 ), whether or not the
Example 2.5.3.
P(R2 |R1 ) =
that in general experiments of this
marbles are replaced.
Roll a fair die twice (or, equivalently, toss a pair of distin-
guishable fair dice once) and observe the number of dots on the upper face
on each roll. A typical outcome can be described by the ordered pair
where
j
and
k
(j, k),
are, respectively, the number of dots on the upper face in the
rst and second rolls. Since the die is fair, each of the 36 outcomes has the
A be the event that the sum of the dice is 7, B the event
C the event that the rst die is even. Then
P(AC) = 1/12 = P(A)P(C) but P(BC) = 1/12 6= P(B)P(C); the events A
and C are independent, but B and C are not.
same probability. Let
that the sum of the dice is 8, and
The denition of independence may be extended in a natural way to more
than two events.
Denition 2.5.4. Events in a collection A are independent if for any n and
any choice of A1 , A2 , . . ., An ∈ A,
P(A1 A2 · · · An ) = P(A1 )P(A2 ) · · · P(An ).
Example 2.5.5.
event that the
A3
j th
Aj be the
A1 , A2 , and
Toss a fair coin 3 times in succession and let
coin comes up heads,
j = 1, 2, 3.
The events
are easily seen to be independent, which explains the use of the phrase
independent trials in this and similar examples.
Option Valuation: A First Course in Financial Mathematics
24
2.6 Exercises
1. Show that
P(A) + P(B) − 1 ≤ P(AB) ≤ P(A ∪ B) ≤ P(A) + P(B).
2. Use the inclusion-exclusion rule for two events to prove the corresponding rule for three events:
P(A∪B∪C) = P(A)+P(B)+P(C)−P(AB)−P(AC)−P(BC)+P(ABC).
Formulate and prove an inclusion-exclusion rule for four events.
3. Jack and Jill run up the hill. The probability of Jack reaching the top
rst is
p,
while that of Jill is
q.
They decide to have a tournament, the
grand winner being the rst one who wins 3 races. Find the probability
that Jill wins the tournament. Assume that there are no ties (p + q
= 1)
and that the races are independent.
4. A
full house
is a poker hand with 3 cards of one denomination and 2
cards of another, for example, three kings and two jacks. Show that four
of a kind beats a full house.
5. Balls are randomly thrown one at a time at a row of 30 open-topped jars
numbered 1 to 30. Assuming that each ball lands in some jar, nd the
smallest number of throws so that there is a better than a 60% chance
that at least two balls land in the same jar.
p be the probability of
0 < p < 1. For m ≥ 2, nd the
6. Toss a coin innitely often and let
pearing on any single toss,
Pm
that
(a) a head appears on a toss that is a multiple of
(b) the
rst
m;
head appears on a toss that is a multiple of
(For example, in (a)
P2
a head approbability
m.
is the probability that a head appears on an
even toss.) Show in (b) that
lim Pm = lim− Pm = 0,
m→∞
p→1
and
lim Pm = 1/m.
p→0+
p be the probability of a head appearing on a
0 < p < 1. Find the probability that (a) both tosses come up
7. Toss a coin twice and let
single toss,
heads, given that at least one toss comes up heads; (b) both tosses come
up heads, given that the rst toss comes up heads. Can the probabilities
in (a) and (b) ever be the same?
8. A jar contains
n − 1 vanilla cookies and one chocolate cookie. You reach
into the jar and choose a cookie at random. What is the probability
that you will get the chocolate cookie on the
k th
try if you (a) eat, (b)
Probability Spaces
25
replace, each cookie you select. Show that if
n
is large compared to
k
then the ratio of the probabilities in (a) and (b) is approximately 1.
9. A hat contains six slips of paper numbered 1 through 6. A slip is drawn
at random, the number is noted, the slip is replaced in the hat, and the
procedure is repeated. What is the probability that after three draws
the slip numbered 1 was drawn exactly twice, given that the sum of the
numbers on the three draws is 8.
10. A jar contains 12 marbles: 3 reds, 4 greens, and 5 yellows. A handful
of
6
marbles is drawn at random. Let
least 3 green marbles and
P(A|B).
B
A
be the event that there are at
the event that there is exactly 1 red. Find
Are the events independent?
x is chosen randomly from the interval [0, 1]. Let A be the
x < .5 and B the event that the second and third digits of the
expansion .d1 d2 d3 . . . of x are 0. Are the events independent?
the inequality is changed to x < .49?
11. A number
event that
decimal
What if
A be the event that the rst roll comes up odd,
C the event that the sum
of the dice is odd. Show that any two of the events A, B , and C are
independent but the events A, B , and C are not independent.
12. Roll a fair die twice. Let
B
the event that the second roll is odd, and
13. Suppose that
A
and
B
are independent events. Show that in each case
the given events are independent: (a)
and
A
and
B0;
(b)
A0
B;
and
(c)
A0
B0.
14. John and Mary order pizzas. The pizza shop oers only plain, anchovy,
and sausage pizzas with no multiple toppings. The probability that John
gets a plain (resp., anchovy) is
.1
(resp.,
Mary gets a plain (resp., anchovy) is
.3
.2)
(resp.,
and the probability that
.4).
Assuming that John
and Mary order independently, use Exercise 13 to nd the probability
that neither gets a plain but at least one gets an anchovy.
odds for an event E are said to be r to 1 if E is r times as likely to
E 0 , that is, P(E) = rP(E 0 ). Odds r to s means the same thing
as odds r/s to 1, and odds r to s against means the same as odds s to
r for. A bet of one dollar on an event E with odds r to s is fair if the
15. The
occur as
bettor wins
E
s/r
dollars if
E
occurs and loses one dollar if
E0
occurs. (If
occurs, the dollar wager is returned to the bettor.) Show that, if the
odds for
dollar on
E
E
are
r
to
returns
s,
then (a)
1/P(E)
P(E) =
and (b) a fair bet of one
dollars (including the wager) if
16. Consider a race with only three horses,
the odds
r
r+s
H1 , H2 ,
and
H3 .
E
occurs.
Suppose that
against Hi winning are quoted as oi to 1. If the odds are based
solely on probabilities (determined by, say, statistics on previous races),
Option Valuation: A First Course in Financial Mathematics
26
then, by Exercise 15, the probability that horse
Hi
wins is
(1 + oi )−1 .
Assuming there are no ties,
o := (1 + o1 )−1 , (1 + o2 )−1 , (1 + o3 )−1
is a then probability vector. It is typically the case, however, that quoted
odds are based on additional factors such as the distribution of wagers
made before the race and prot margins for the bookmaker. In this
exercise, the reader is asked to use elementary linear algebra to make a
connection between quoted odds and betting strategies.
(a) Suppose that for each
i
a bet of size
bi
may be positive, negative, or 0. The vector
betting strategy.
Show that if horse
b
the betting strategy
Hi
is made on Hi . The bets
b = (b1 , b2 , b3 ) is called a
wins, then the net winnings for
may be expressed as
Wb (i) := (oi + 1)bi − (b1 + b2 + b3 ).
b is said to be a sure-win strategy, or an arbitrage,
i. Show that there is a sure-win strategy i there
(b) A betting strategy
if
Wb (i) > 0
for each
exist numbers
si < 0
such that the system
−o1 x1
x1
x1
+x2
−o2 x2
+x2
x = (x1 , x2 , x3 )
has a solution
+x3
+x3
−o3 x3
= s1
= s2
= s3
(that solution being a sure-win betting
strategy).
(c) Let
A
be the coecient matrix of the system in (b). Show that the
determinant of
(d) Suppose
A
is
D 6= 0.
A−1
D := 2 + o1 + o2 + o3 − o1 o2 o3 .
Show that

o o −1
1  2 3
1 + o3
=
D
1 + o2
1 + o3
o1 o3 − 1
1 + o1

1 + o2
1 + o1 
o1 o2 − 1
s1 , s2 , and s3 , the vector
A−1 sT is a sure-win betting strategy, where sT denotes the transpose of
s := (s1 , s2 , s3 ).
and that, for any choice of negative numbers
(e) Show that if
D 6= 0,
then a sure-win betting strategy is
b = −sgn(D)(1 + o2 + o3 + o2 o3 , 1 + o1 + o3 + o1 o3 , 1 + o1 + o2 + o1 o2 )
where sgn(D) denotes the sign of
D.
(f ) Show that there is a sure-win betting strategy i
o is not a probability
vector.
(The assertion in (f ) is a special case of the
Arbitrage Theorem, a state-
ment and proof of which may be found, for example, in [14].)
Chapter 3
Random Variables
3.1 Denition and General Properties
We saw in Chapter 2 that outcomes of some experiments may be described
by real numbers. Such outcomes are called
random variables. For a formal de-
nition, the following notation will be convenient. Given a real-valued function
X on Ω and a set A of real numbers, we shall write {X ∈ A} for the set
{ω ∈ Ω | X(ω) ∈ A}. Similarly, {X < a} denotes the set {ω ∈ Ω | X(ω) < a},
and if Y is another function on Ω, then {X ∈ A, Y ∈ B} stands for the set
{X ∈ A} ∩ {Y ∈ B}, {X ≤ Y } for the set {ω ∈ Ω | X(ω) ≤ Y (ω)}, and so
forth. For probabilities of such events, we shall write, for example, P(X ∈ A)
rather than the more cumbersome notation P({X ∈ A}). The following example should illustrate the idea.
Example 3.1.1.
The table below gives the distribution of grade-point aver-
ages for a group of 100 students, the rst row giving the number of students
having the grade-point averages listed in the second row. If
X
denotes the
no. of students
7
13
19
16
12
10
8
6
5
4
grade pt. avg.
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
grade-point average of a randomly chosen student, then, in the above notation,
P(2.5 < X ≤ 3.3) = P(X = 2.7) + P(X = 2.9) + P(X = 3.1) + P(X = 3.3)
= .16 + .12 + .10 + .08 = .46.
Denition 3.1.2. Consider an experiment with sample space Ω and event
σ -eld F . A random variable on (Ω, F) is a real-valued function X on Ω such
that for each interval J the set {X ∈ J} is a member of F . If there is a
possibility of ambiguity, we shall refer to X as an F -random variable or say
that X is F -measurable.
Remark 3.1.3.
To determine whether a function
suces to check that
{X ∈ J} ∈ F
for all intervals
X is
J of
a random variable, it
a particular type. For
27
Option Valuation: A First Course in Financial Mathematics
28
example, suppose that a function
a.
numbers
X
on
Ω
satises
Since
{X ∈ (a, b)} = {X < b} − {X ≤ a} =
our assumption implies that
{X ∈ J} ∈ F
∞
[
{X ≤ a} ∈ F
for all real
{X ≤ b − 1/n} − {X ≤ a},
n=1
{X ∈ (a, b)} ∈ F . Similar arguments show that
J . Therefore, X is a random variable.
for the remaining intervals
A simple example of a random variable is the
indicator function
of an
event, which provides a numerical way of expressing occurrence of the event.
Denition 3.1.4. The
IA on Ω dened by
indicator function
of a subset A ⊆ Ω is the function
if ω ∈ A, and
if ω ∈ A0 .
Proposition 3.1.5. IA is a random variable i A ∈ F .
Proof. If IA is a random variable, then A = {IA > 0} ∈ F .
(
IA (ω) =
A∈F
and
a ∈ R,
1
0


∅
{IA ≤ a} = A0


Ω
In each case,
Conversely, if
then
{IA ≤ a} ∈ F ,
if
if
if
a<0
0 ≤ a < 1,
a ≥ 1.
hence, by Remark 3.1.3,
and
IA
is a random variable.
The proof of the following proposition is left to the reader as an exercise.
Proposition 3.1.6. If A, B , and C are subsets of Ω, then
(i) IAB = IA IB ; (ii) IA∪B = IA + IB − IA IB ; and (iii) IA ≤ IB i A ⊆ B .
The next theorem describes a simple way of generating random variables.
Theorem 3.1.7. Let X1 , X2 ,. . ., Xn be random variables. If f (x1 , x2 , . . . , xn )
is a continuous function, then f (X1 , X2 , . . . , Xn ) is a random variable, where
f (X1 , X2 , . . . , Xn )(ω) := f (X1 (ω), X2 (ω), . . . , Xn (ω)).
Proof.
n = 1, that is, for a single random
f (x). For this, we use a standard result
that, because f is continuous, any set A of
We sketch the proof for the case
variable
X
and a continuous function
from real analysis, which asserts
the form
intervals
{x | f (x) < a} is
Jn . It follows that
a union of a sequence of pairwise disjoint open
{f (X) < a} = {X ∈ A} =
By Remark 3.1.3,
f (X)
[
{X ∈ Jn } ∈ F.
n
is a random variable.
Random Variables
29
Corollary 3.1.8. Let X and Y be random variables, α ∈ R and p > 0.
Then X + Y , αX , XY , |X|p , and X/Y (provided Y is never 0) are random
variables.
Combining Corollary 3.1.8 with Proposition 3.1.5, we obtain
Corollary 3.1.9. A linear combination
IA with Aj ∈ F is a random variable.
Pn
j=1
αj IAj
of indicator functions
j
Denition 3.1.10. The functions max(x, y) and min(x, y) denote, respectively, the larger and smaller of the real numbers x and y.
Corollary 3.1.11.
Proof.
max(X, Y )
and min(X, Y ) are random variables.
The identity
max(x, y) =
shows that
max(x, y)
|x − y| + x − y
+y
2
max(X, Y )
is continuous. Therefore, by Theorem 3.1.7,
min(X, Y )
min(x, y) = − max(−x, −y).)
is a random variable. A similar argument shows that
variable. (Or use the identity
is a random
Denition 3.1.12. The cumulative distribution function (cdf ) F of a random variable X is dened by
X
FX (x) = P(X ≤ x), x ∈ R.
Example 3.1.13.
The cdf of the number
X
of heads that come up in three
tosses of a fair coin may be described as
FX = 81 I[0,1) + 21 I[1,2) + 78 I[2,3) + I[3,∞) .
3.2 Discrete Random Variables
Denition 3.2.1. A random variable X on a probability space (Ω, F, P) is
said to be discrete if the range of X is countable. The function p dened by
X
pX (x) := P(X = x),
x ∈ R,
is called the probability mass function (pmf ) of X .
Since
pX (x) > 0
for at most countably many real numbers
we may write
P(X ∈ A) =
X
x∈A
pX (x),
x,
for
A⊆R
Option Valuation: A First Course in Financial Mathematics
30
where the sum is either nite or a convergent innite series (ignoring zero
terms). In particular,
FX (x) =
X
pX (x)
y≤x
and
X
x∈R
pX (x) = P(X ∈ R) = 1.
A linear combination of indicator functions of events is an example of a
discrete random variable. Here are some important special cases.
Example 3.2.2.
(Bernoulli Random Variable). A
Bernoulli trial
is an exper-
iment with only two outcomes, frequently called success and failure. If
the probability of a success, then a
p
is dened by setting
X =1
p is
Bernoulli random variable with parameter
if the outcome is a success and
X = 0,
if the
outcome is a failure. Thus,
pX (1) = p
and
pX (0) = 1 − p.
The number of heads that come up on a single toss of a coin is an example of
a Bernoulli random variable.
Example 3.2.3.
sisting of
N
(Binomial Random Variable). Consider an experiment con-
the number of successes in
N
n
p. Let X denote
{X = n} can occur in
pn (1 − p)N −n ,
independent Bernoulli trials, each with parameter
N
trials. Since the event
ways, and since each of these has probability
pX (n) =
N n
p (1 − p)N −n , n = 0, 1, 2, . . . , N.
n
X with this pmf is called a binomial random variable with
parameters (N, p), in symbols X ∼ B(N, p). The number of heads occurring
A random variable
in
N
consecutive coin tosses is an example of a binomial random variable.
Example 3.2.4.
(Geometric Random Variable). Suppose that an experiment
p. Let X denote
{X = k} to occur,
consists of independent Bernoulli trials with parameter
the
number of trials until the rst success. For the event
the
rst
k−1
trials must be failures and the
k th
trial a success. Thus,
pX (k) = q k−1 p, k = 1, 2, . . . ,
Note that
∞
X
k=0
so
pX
pX (k) = p
∞
X
q := 1 − p.
(3.1)
q k = 1,
k=0
is indeed a pmf. A random variable with this distribution is said to be
geometric with parameter p.
Random Variables
Example 3.2.5.
red marbles and
31
m
z ≤ N :=
(Hypergeometric Random Variable). Consider a jar with
n
white marbles. We take a random sample of size
m + n from the jar by drawing the marbles in succession without replacement.
(Equivalently, we can just draw the z marbles at once.) The sample space Ω is
N
the collection of all subsets of z marbles, hence |Ω| =
z . Let X denote the
number of red marbles in the sample. The event {X = x} can be realized by
rst choosing x red marbles from the m red marbles and then choosing z − x
white marbles from the n white marbles. Thus,
−1
m
n
N
,
P(X = x) =
x
z−x
z
where
x
must satisfy
0≤x≤m
and
(3.2)
0 ≤ z − x ≤ n. Letting p = m/N denote
q = 1 − p = n/N the fraction of
the fraction of red marbles in the jar and
white, we can write (3.2) as
pX (x) =
−1
Np
Nq
N
, max(z − n, 0) ≤ x ≤ min(z, m).
x
z−x
z
A random variable
X
with this pmf is called a
able with parameters (p, z, N ).
That
pX
(3.3)
hypergeometric random vari-
is indeed a pmf may be established
analytically or may be argued on probabilistic grounds.
The experiment described in this example is called sampling without replacement. (Sampling with replacement, that is, replacing each marble before
drawing the next, results in the binomial pmf.) For a political setting, let the
marbles represent individuals in a population of size
of size
z
N
from which a sample
is randomly selected and then polled, red marbles representing indi-
viduals in the sample who favor candidate A for political oce, white marbles
representing those who favor candidate B. Pollsters use the sample to estimate
the proportion of people in the general population who favor candidate A and
also determine the margin of error in doing so. The hypergeometric pmf may
be used for this purpose.
1 Marketing specialists apply similar techniques to
determine product preferences.
1 In practice, the more tractable normal distribution is used instead (Example 3.3.3). This
is justied by noting that for large
N
the hypergeometric distribution is nearly binomial
and, by the Central Limit Theorem, that a binomial distribution (suitably adjusted) is
approximately normal.
Option Valuation: A First Course in Financial Mathematics
32
3.3 Continuous Random Variables
Denition 3.3.1. A random variable X is said to be continuous if there exists
a nonnegative integrable function fX such that
P(X ∈ J) =
Z
fX (x) dx
J
for all intervals J . The function fX is called the probability density function
(pdf ) of X .
For example, the probability
P(a ≤ X ≤ b) =
is simply the area under the graph of
see that the probability that
X
Z
fX
b
fX (x) dx
a
between
a
and
b.
Setting
a = b,
we
takes on any particular value is zero.
X takes the form
Z x
FX (x) = P(X ≤ x) =
fX (t) dt.
The cumulative distribution function of
−∞
Dierentiating with respect to
x,
we have (at points of continuity of
fX )
FX0 (x) = fX (x).
Example 3.3.2. (Uniform Random Variable). A random variable X is said
to be uniformly distributed on the interval (α, β) if its pdf is of the form
fX (x) = (β − α)−1 I(α,β) ,
where I(α,β) is the indicator function of the interval
(a, b)
of
(α, β). For any subinterval
(α, β),
Z
P(a < X < b) =
a
b
fX (x) dx = (β − α)−1 (b − a),
which is the (theoretical) fraction of times a number chosen randomly from
the interval
(α, β)
lies in the subinterval
(a, b).
Example 3.3.3.
(Normal Random Variable). Let σ and µ be real numbers
σ > 0. A random variable X is said to have a normal distribution with
parameters µ and σ2 , in symbols X ∼ N (µ, σ2 ), if it has pdf
with
fX (x) =
2
2
1
√ e−(x−µ) /2σ .
σ 2π
(3.4)
Random Variables
If
X ∼ N (0, 1),
then
X
ϕ
for
this case, we write
is said to have the
fX
and
Φ
2
1
ϕ(x) := √ e−x /2
2π
Note that, if
X ∼ N (µ, σ 2 ),
FX .
for
and
33
standard normal distribution.
In
Thus,
1
Φ(x) := √
2π
Z
x
e−t
2
/2
dt.
−∞
then
FX (x) = Φ
x−µ
σ
,
as may be seen by making a simple substitution in the integral dening
FX .
The normal density (3.4) is the familiar bell-shaped curve, with maximum
occurring at
the value of
x = µ. The parameter σ controls the spread of the bell: the larger
σ the atter the bell. Normal random variables arise in sampling
from a large population of independent measurements such as test scores,
heights, and so forth. They will gure prominently in later chapters.
Example 3.3.4. (Exponential Random Variable). A random variable
exponential distribution with parameter λ if it has pdf
X
is
said to have an
(
λe−λx ,
fX (x) =
0,
The cdf of
X
(
x
FX (x) =
fX (y) dy =
−∞
In particular, if
s, t ≥ 0,
if
x≥0
x < 0.
(3.5)
is
Z
for
if
x ≥ 0,
then
1 − e−λx ,
0,
if
if
x≥0
x < 0.
P(X > x) = 1 − FX (x) = e−λx .
It follows that,
P(X > s + t, X > t) = P(X > s + t) = e−λ(s+t) = P(X > t)P(X > s),
which may be expressed in terms of conditional probabilities as
P(X > s + t|X > t) = P(X > s).
Equation (3.6) is the so-called
variable. If we take
X
memoryless
(3.6)
feature of an exponential random
to be the lifetime of some instrument, say, a light bulb,
then (3.6) asserts (perhaps unrealistically) that the probability of the light
bulb lasting at least
s + t hours, given that it has already lasted t hours, is the
s hours. Waiting times
same as the initial probability that it will last at least
(for example, of buses and bank clerks) are frequently assumed to be exponential random variables. It may be shown that the exponential distribution
is the only continuous distribution for which (3.6) holds.
Option Valuation: A First Course in Financial Mathematics
34
3.4 Joint Distributions
Denition 3.4.1. The joint probability
variables X and Y is dened by
mass function
of discrete random
pX,Y (x, y) = P(X = x, Y = y).
Note that the pmf
pX
may be recovered from
pX (x) = P(X = x, Y ∈ R) =
pX,Y
X
by using the identity
pX,Y (x, y),
y
where the sum is taken over all
y in the range of Y . A similar identity holds for
pY .
are called
In this context,
pX
and
pY
marginal probability mass functions.
Denition 3.4.2. The joint cumulative distribution function F
of random variables X and Y is dened by
X,Y
of a pair
FX,Y (x, y) = P(X ≤ x, Y ≤ y).
and Y are said to be jointly continuous if there exists a nonnegative integrable function fX,Y , called the joint probability density function of X and Y ,
such that
Z x Z y
X
FX,Y (x, y) =
fX,Y (s, t) dt ds.
−∞
(3.7)
−∞
It follows from (3.7) that
P(X ∈ J, Y ∈ K) =
for all intervals
J
and
K.
Z Z
ZZ
fX,Y (x, y) dx dy =
K
J
fX,Y (x, y) dx dy
J×K
More generally,
P ((X, Y ) ∈ A) =
ZZ
f (x, y) dx dy
A
for all suciently regular (e.g., Borel) sets
A ⊆ R2 .
Dierentiating (3.7) gives the following useful connection between
and
fX,Y
FX,Y :
fX,Y (x, y) =
Example 3.4.3.
Let
X
and
Y
∂2
F (x, y).
∂x∂y X,Y
be jointly continuous random variables and
Random Variables
set
Z =X +Y.
If
Az = {(x, y) | x + y ≤ z},
35
then
FZ (z) = P ((X, Y ) ∈ Az )
ZZ
=
fX,Y (x, y) dx dy
x+y≤z
Z
∞
Z
z−y
fX,Y (x, y) dx dy
=
−∞
Z ∞
−∞
Z z
−∞
−∞
=
fX,Y (x − y, y) dx dy.
Changing the order of integration in the last expression, we see that
Z
∞
fZ (x) =
−∞
fX,Y (x − y, y) dy.
The following proposition shows that, by analogy with the discrete case,
the pdfs of jointly continuous random variables may be recovered from the
marginal density functions.
Proposition 3.4.4. If X and Y are jointly continuous random variables,
then
joint pdf. In this context,
Z
and
∞
fX (x) =
fX,Y (x, y) dy
−∞
Proof.
fX
For any interval
fY
are called
and
Z
fX,Y (x, y) dx.
−∞
J,
P(X ∈ J) = P(X ∈ J, Y ∈ R) =
Z Z
∞
f (x, y) dy dx.
J
The inner integral must therefore be the density
Remark 3.4.5.
∞
fY (y) =
−∞
fX
of
X.
The above denitions and results extend to the case of nitely
many random variables
X1 , X2 , . . ., Xn .
We leave the formulations to the
reader.
3.5 Independent Random Variables
Denition 3.5.1. Random variables X1 , X2 , . . . are said to be independent if
the events {Xn ∈ Jn }, n = 1, 2, . . ., are independent for any choice of intervals
Jn .
Proposition 3.5.2. Discrete random variables X and Y are independent i
pX,Y (x, y) = pX (x)pY (y)
for all x and y.
Option Valuation: A First Course in Financial Mathematics
36
Proof.
The proof of the necessity is left to the reader. For the suciency,
J
suppose that the equation holds. Then for any intervals
P(X ∈ J, Y ∈ K) =
=
X
and
K
pX,Y (x, y)
x∈J, y∈K
X
pX (x)
x∈J
X
pY (y)
y∈K
= P(X ∈ J)P(Y ∈ K),
where the sums are taken over all
the range of
Y.
Therefore,
X
and
x ∈ J in the range
Y are independent.
of
X
and all
y∈K
in
Proposition 3.5.3. Let X and Y be random variables. Then X and Y are
independent i
for all x and y.
FX,Y (x, y) = FX (x)FY (y)
Proof.
We give a partial proof of the suciency. If the equation holds, then
P(a < X ≤ b, c < Y ≤ d) = FX,Y (b, d) − FX,Y (a, d) − FX,Y (b, c) + FX,Y (a, c)
= [FX (b) − FX (a)] [FY (d) − FY (c)]
= P(a < X ≤ b)P(c < Y ≤ d).
It follows that
P(a ≤ X ≤ b, c ≤ Y ≤ d) = lim P(a − 1/n < X ≤ b, c − 1/n < Y ≤ d)
n→∞
= lim P(a − 1/n < X ≤ b)P(c − 1/n < Y ≤ d)
n→∞
= P(a ≤ X ≤ b)P(c ≤ Y ≤ d).
Here we have used the fact that, if
A1 ⊇ A2 · · · ⊇ An · · · ,
then
P (∩∞
n=1 An ) = lim P(An ).
n→∞
Other intervals are treated in a similar fashion. Therefore,
X
and
Y
are inde-
pendent.
Corollary 3.5.4. Let X and Y be jointly continuous random variables. Then
and Y are independent i f (x, y) = f (x)f (y).
Proof. If X and Y are independent, then, by Proposition 3.5.3,
X
X,Y
Z
x
FX,Y (x, y) =
Z
fX (s) ds
−∞
which shows that
X
fX (x)fY (y)
y
−∞
Y
Z
fY (t) dt =
x
−∞
Z
y
fX (s)fY (t) dt ds,
−∞
is the joint density function. Conversely, if the
density condition holds, then reversing the argument shows that
FX (x)FY (y).
FX,Y (x, y) =
Random Variables
37
Proposition 3.5.5. If X and Y are independent random variables and g and
h are continuous functions, then g(X) and h(Y ) are independent.
Proof.
F
Let
denote the joint cdf of
orem 3.1.7, given real numbers
a
g(X) and h(Y ). As in the proof of Theb there exist sequences (Jm ) and (Kn )
and
of pairwise disjoint open intervals such that
{g(X) < a} =
[
m
{X ∈ Jm }
and
{h(Y ) < b} =
[
{Y ∈ Kn }.
n
It follows that
P(g(X) < a, h(Y ) < b) =
XX
m
=
n
XX
m
n
P(X ∈ Jm , Y ∈ Kn )
P(X ∈ Jm )P(Y ∈ Kn )
= P(g(X) < a)P(h(Y ) < b).
k = 1, 2, . . .,
In particular, for
P(g(X) < x + 1/k, h(Y ) < y + 1/k) = P(g(X) < x + 1/k)P(h(Y ) < y + 1/k),
and letting
k → ∞ shows that F (x, y) = Fg(X) (x)Fh(Y ) (y).
g(X) and h(Y ) are independent.
Therefore, by
Proposition 3.5.3,
Remark 3.5.6.
The above results have obvious extensions to the case of more
than two random variables. We leave the formulations to the reader.
Denition 3.5.7. Random variables with the same cdf are said to be identically distributed. If the random variables are also independent, then the collection is said to be iid.
Example 3.5.8.
For a sequence of independent Bernoulli trials with param-
p, let X1 be the number of trials before the rst success, and for k > 1
let Xk be the number of trials between the (k − 1)st and k th successes. The
m n
event {X1 = m, X2 = n} occurs with probability q pq p, q := 1 − p, hence
eter
∞
X
P(X2 = n) =
P(X1 = m, X2 = n) = p2 q n
m=0
∞
X
qm =
m=0
p2 q n
= pq n
1−q
= P(X1 = n).
Therefore,
X1
and
X2
are identically distributed. Since
P(X1 = m)P(X2 = n) = pq m pq n = P(X1 = m, X2 = n),
Proposition 3.5.2 shows that
X1
and
X2
are independent. An induction ar-
gument using similar calculations shows that the sequence
that
Xn + 1
is a geometric random variable with parameter
(Xn )
p.
is iid. Note
Option Valuation: A First Course in Financial Mathematics
38
3.6 Sums of Independent Random Variables
If
X
and
Y
are independent discrete random variables then
pX+Y (z) = P(X + Y = z) =
X
x
=
X
x
P(X = x, Y = z − x)
pX (x)pY (z − x).
The sum in (3.8) is called the
convolution
(3.8)
of the pmf 's
pX
and
pY .
Example 3.6.1.
with
Let X and Y be independent binomial random variables
X ∼ B(m, p) and Y ∼ B(n, p). We show that Z := X +Y ∼ B(m+n, p).
By (3.8),
pZ (z) =
X m
x
x
px q m−x
n
pz−x q n−(z−x) ,
z−x
q := 1 − p,
x satisfying the inequalities 0 ≤ x ≤ m
max(z − n, 0) ≤ x ≤ min(z, m). Simplifying, we
where the sum is taken over all integers
and
0 ≤ z − x ≤ n,
that is,
see from Example 3.2.5 that
z m+n−z
pZ (z) = p q
Therefore,
X m n m + n
=
pz q m+n−z .
x
z−x
z
x
Z ∼ B(m + n, p).
Now suppose that
X
and
Y
are jointly continuous independent random
variables. By Corollary 3.5.4, the joint density of
X
and
Y
is
fX (x)fY (y).
By
Example 3.4.3, then,
∞
Z
fX+Y (z) =
−∞
fX (x)fY (z − x) dx =
The integrals in (3.9) are called the
Example 3.6.2.
X ∼ N (µ, σ 2 )
Z
∞
−∞
convolution
fY (y)fX (z − y) dy.
of the densities
fX
(3.9)
and
fY .
X and Y be independent normal random variables with
Y ∼ N (ν, τ 2 ). We claim that
Let
and
X + Y ∼ N (µ + ν, σ 2 + τ 2 ).
To verify this set
show that
fZ = g ,
Z = X +Y
and suppose rst that
µ = ν = 0.
where
1
g(z) = √ exp −z 2 /2%2 ,
% 2π
%2 := σ 2 + τ 2 .
We need to
Random Variables
39
From (3.9),
∞
1 (z − y)2
y2
exp −
fZ (z) = a
+ 2
dy
2
σ2
τ
−∞
Z ∞
τ 2 (z − y)2 + σ 2 y 2
exp
=a
dy,
−2σ 2 τ 2
−∞
Z
where
a = (2πστ )−1 .
The expression
may be written
τ 2 (z − y)2 + σ 2 y 2
in the second integral
2τ 2 yz
τ 2 z 2 − 2τ 2 yz + %2 y 2 = %2 y 2 −
+ τ 2z2
%2
2
τ 2z
τ 4z2
2
=% y− 2
+ τ 2z2 − 2
%
%
2
τ 2z
τ 2 σ2 z2
.
= %2 y − 2
+
%
%2
Thus, for suitable positive constants
b
and
c,
τ 2 (z − y)2 + σ 2 y 2
z2
2
=
−b(y
−
cz)
−
.
−2σ 2 τ 2
2%2
It follows that
2
fZ (z) = ae
z
− 2%
2
Z
∞
2
e−b(y−cz) dy = ae
−∞
Z
∞
2
e−bu du = kg(z)
−∞
and g are both densities, k must
µ = ν = 0.
2
2
For the general case, observe that X − µ ∼ N (0, σ ) and Y − ν ∼ N (0, τ )
2
2
2
so by the rst part Z −µ−ν ∼ N (0, % ). Therefore, Z ∼ N (µ+ν, σ +τ ).
for some constant
k
2
z
− 2%
2
and for all
z.
Since
fZ
equal 1, verifying the assertion for the case
Example 3.6.3.
n. A common
Zn := Sn /Sn−1 , n ≥ 1, are iid lognormal
random variables with parameters µ and σ2 , that is, ln Zn ∼ N (µ, σ2 ). Note
that Zn − 1 is the fractional increase of the stock from day n − 1 to day n.
The probability that the price of the stock rises on each of the rst n days is
n
−µ
n
P(Z1 > 1, Z2 > 1, . . . , Zn > 1) = P (ln Z1 > 0) = 1 − Φ
σ
µ
= Φn
,
σ
Let
Sn
denote the price of a stock on day
model assumes that the ratios
1 − Φ(−x) = Φ(x) (Exercise 7).
n the price of the stock will be larger
where we have used the identity
The probability that on day
than its
40
Option Valuation: A First Course in Financial Mathematics
initial price
S0
is
P(Sn > S0 ) = P(Z1 Z2 · · · Zn > 1) = P(ln Z1 + ln Z2 + · · · + ln Zn > 0)
= 1 − P(ln Z1 + ln Z2 + · · · + ln Zn < 0)
−nµ
√
=1−Φ
σ n
√ µ n
=Φ
,
σ
the second to last equality since
(Example 3.6.2).
ln Z1 + ln Z2 + · · · + ln Zn ∼ N (nµ, nσ 2 )
Random Variables
41
3.7 Exercises
1. Determine the least number of times you need to toss a fair coin to be
99% sure that at least two heads will come up.
2. Prove Proposition 3.1.6.
3. Let
X
be a hypergeometric random variable with parameters
Show that
X1 , X2 , . . .
4. Let
(p, n, N ).
n k n−k
lim pX (k) =
p q
.
N →∞
k
be an innite sequence of random variables such that
the limit
X(ω) := lim Xn (ω)
n→∞
exists for each
ω ∈ Ω.
Verify that
{X < a} =
and hence conclude that
5. Let
Y = aX + b,
X
m=1 n=1 k≥n
X
{Xk < a − 1/m},
is a random variable.
X is a continuous random
a 6= 0. Show that
y−b
fY (y) = |a|−1 fX
.
a
where
are constants with
6. Let
∞ [
∞ \
[
In particular, nd
7. Show that
N (0, 1).
a
fX and set Y = X 2 .
√
√
f ( y) + fX (− y)
fY (y) = X
I(0,+∞) .
√
2 y
be a random variable with density
that
fY
if
X
is uniformly distributed over
1 − Φ(x) = Φ(−x).
8. Show that, if
X ∼ N (µ, σ 2 )
and
Conclude that
a 6= 0,
9. In the dartboard of Example 2.3.7, let
then
Z
to the landing position of the dart. Find
10. Let
variable and
and
Show
(−1, 1).
X ∼ N (0, 1)
i
−X ∼
aX + b ∼ N (aµ + b, a2 σ 2 ).
be the distance from the origin
FZ
and
fZ .
Z = X + Y , where X and Y are independent and uniformly
(0, 1). Show that
FZ (z) = 21 z 2 I[0,1) (z) + 1 − 21 (2 − z)2 I[1,2) (z) + I[2,∞) (z).
tributed on
b
dis-
Option Valuation: A First Course in Financial Mathematics
42
1 ≤ k ≤ n.
11. For this exercise, refer to Example 3.6.3. Let
probability that the stock
(a) increases exactly
k
times in
n
(b) increases exactly
k
consecutive
Find the
days;
times in
n
days (decreasing on the
other days); and
(c) has a run of exactly
k consecutive increases (not necessarily decreask ≥ n/2.
ing on the other days), where
12. Let
Xj , j = 1, 2, . . . , n, be independent Bernuoulli random variables
p and let 1 ≤ m < n and 1 ≤ k ≤ n. Show that
m n − m k n−k
P(Ym = j, Yn = k) =
p q
,
j
k−j
with parameter
where
max(0, k − n + m) ≤ j ≤ min(m, k).
Conclude that, for xed
k,
pX (j) := P(Ym = j|Yn = k)
is the pmf of a hypergeometric random variable
X
with parameters
(m/n, k, n).
X and Y be independent random variables.
max(X, Y ) and Fm of min(X, Y ) in terms of FX
13. Let
FM + Fm = FX + FY .
Express the cdfs
and
FY .
FM
of
Conclude that
Chapter 4
Options and Arbitrage
stocks
bonds (nancial contracts issued by governments and corporations); currencies (traded on foreign exchanges); commodities (goods, such as oil, copper, wheat, or electricity); and derivatives.
Assets traded in nancial markets fall into the following main categories:
(equity in a corporation or business);
A derivative is a nancial instrument whose value is based on that of an
underlying asset such as a stock, commodity, or currency. Derivatives provide
a way for investors to reduce the risks associated with investing. The most
common derivatives are forwards, futures, and options. In this chapter, we
show how the simple assumption of no-arbitrage may be used to derive fundamental properties regarding the value of a derivative. In later chapters, we
show how the no-arbitrage principle leads to the notion of replicating portfolio
and ultimately to the Cox-Ross-Rubinstein and Black-Scholes option pricing
formulas.
A
nancial market is a system by which are traded nitely many securities
S1 , S2 , . . ., Sd . For ease and clarity of exposition we treat only the case d = 1.
Thus, we assume that the market allows unlimited trades of a single risky
security
S
S.
With the exception of Sections 4.8, 9.6, and 14.7, we assume that
pays no dividends.
S
t will be denoted by St . The term risky
S and hence suggests a probabilistic
setting for the model. We therefore assume that St is a random variable on
1
some probability space (Ω, F, P). The set D of indices t is either a discrete
set {0, 1, . . . , N } or an interval [0, T ]. Since the initial value of the security is
known at time 0, S0 is assumed to be a constant. The collection S = (St )t∈D
is called the price process of S .
The value of a share of
at time
refers to the unpredictable nature of
For reasons that will become clear, we also assume that there is available
to investors a risk-free money market account
A
that allows unlimited trans-
actions in the form of deposits or withdrawals (including loans). As we saw in
Section 1.3, this is equivalent to the availability of a risk-free bond
B
that may
be purchased or sold (short) in unlimited quantities. We follow the convention
that borrowing an amount
−A,
where
A
A
is the same as lending (depositing) the amount
may be positive or negative.
If the price model is continuous, the risk-free asset is assumed to earn
1 In
this chapter we leave the probability space unspecied. Concrete models are devel-
oped in later chapters.
43
Option Valuation: A First Course in Financial Mathematics
44
interest compounded continuously at a constant annual rate
is discrete, we assume compounding occurs at a rate
example, if
B0
i
r.
If the model
per time interval. For
is the initial value of the bond, then the value
Bt
at time
t
is
given by
(
B0 ert
Bt =
B0 (1 + i)t
≤ t ≤ T in
= 0, 1, . . . N ).
in a continuous model (0
in a discrete model (t
years),
These conventions apply throughout the book.
4.1 Arbitrage
An
arbitrage
is an investment opportunity that arises from mismatched
nancial instruments. For example, suppose Bank A is willing to lend money
at an annual rate of 8%, compounded monthly, and Bank B is oering CDs
at an annual rate of 10%, also compounded monthly. An investor recognizes
this as an arbitrage opportunity, that is, a sure win. She simply borrows an
amount from Bank A at 8% and immediately deposits it into Bank B at 10%.
The transaction costs her nothing and results in a positive prot.
Clearly, a market cannot sustain such obvious instances of arbitrage. However, more subtle examples occur, and while their existence may be eeting
they provide employment for market analysts whose job it is to discover them.
(High-speed computers are commonly employed to ferret out and exploit arbitrage opportunities.) Lack of arbitrage in a market, while an idealized condition, is necessary for general economic stability. Moreover, as we shall see, the
assumption of no-arbitrage leads to a robust mathematical theory of option
pricing.
To construct precise nancial models, one needs a mathematical denition
of arbitrage. For now, the following will suce.
Denition 4.1.1. An arbitrage is a trading strategy resulting in a portfolio
with zero probability of loss and positive probability of gain.2
A formal denition of portfolio is given in Chapter 5. For the time being,
the reader may think of a portfolio as simply a collection of assets.
The following is the rst of several examples in the book showing how the
assumption of no-arbitrage has concrete mathematical consequences.
Example 4.1.2.
2 In
Suppose that the initial value of a single share of our security
games of chance, such as roulette, a casino has a
house advantage ).
statistical
arbitrage (the so-called
Here, in contrast to nancial arbitrage, the casino has a positive proba-
bility of loss. However, the casino has a positive expected gain, so in the long run the house
wins.
Options and Arbitrage
S
is
S0
45
and that after one time period its value goes up by a factor of
probability
p
0<d<u
and
or down by a factor of
0 < p < 1.
d
with probability
q = 1 − p,
u
with
where
In the absence of arbitrage, it must then be the
case that
d < 1 + i < u,
where
i
(4.1)
is the interest rate during the time period.
The verication of (4.1) uses the idea of selling a security
short,
that is,
borrowing and selling a security under the agreement that it will be returned
at some later date. (You are
long in the security if you actually own it.) To see
why (4.1) holds, suppose rst that
u ≤ 1 + i.
We then employ the following
trading strategy: At time 0, we sell the stock short for
S0
and invest the money
in a risk-free account paying the rate i. This results in a portfolio consisting of
−1 shares of S
and an account worth
S0 . No wealth was required to construct
S0 (1+i), which we use to buy the
the portfolio. At time 1, the account is worth
stock, returning it to the lender. This costs us nothing extra since the stock
is worth at most
uS0 ,
which, by our assumption, is covered by the account.
q that our gain is the positive
(1 + i)S0 − dS0 . The strategy therefore constitutes an arbitrage. If
1 + i ≤ d we employ the reverse strategy: At time 0 we borrow S0 dollars
and use it to buy one share of S . We now have a portfolio consisting of one
share of the stock and an account worth −S0 , and as before, no wealth was
Furthermore, there is a positive probability
amount
required to construct it. At time 1, we sell the stock and use the money to pay
back the loan. This leaves us with at least
a positive probability
p
dS0 − (1 + i)S0 ≥ 0, and there is
uS0 − (1 + i)S0 ,
that our gain is the positive amount
again implying an arbitrage. Therefore, (4.1) must hold.
The above example will play a fundamental role in the pricing of derivatives
under the binomial model (Chapter 7).
An important consequence of the assumption of no-arbitrage is the
of one price,
A and B with the same value at time
times
t < T.
law
which asserts that in an arbitrage-free market two investments
T
must have the same value at all
Indeed, if the time-t value of A were greater than that of B, one
could obtain a positive prot at time
t
by taking a short position in A and a
long position in B, and with the proceeds from selling B at time
T
cover the
obligation from shorting A.
For the remainder of the chapter, we assume that the market admits no
arbitrage opportunities. Also, for deniteness, we assume a continuous-time
price process for an asset.
Option Valuation: A First Course in Financial Mathematics
46
4.2 Classication of Derivatives
In the sections that follow, we examine various common types of derivatives. As mentioned above, a derivative is a nancial contract whose value
depends on that of another asset
time of the contract is called the
by
T.
S,
underlying. The expiration
expiration date and is denoted
called the
maturity
or
The price process of the underlying is given by
S = (St )Tt=0 .
Derivatives fall into four main categories:
A
ˆ
European,
ˆ
American,
ˆ
path independent, and
ˆ
path dependent.
European
derivative is a contract that the holder may exercise only at
T . By contrast, an American derivative may be exercised at any
time τ ≤ T . A path-independent derivative has payo (that is, value) at
exercise time τ which depends only on Sτ , while a path-dependent derivative
has payo at τ which depends on the entire history S[0,τ ] of the price process.
maturity
We begin our analysis with the simplest types of European path independent derivatives: forwards and futures.
4.3 Forwards
A
forward contract
between two parties places one party under the obliga-
tion to buy an asset at a future time
price,
T
for a prescribed price
K,
the
forward
and requires the other party to sell the asset under those conditions.
long position,
short position. The payo for
The party that agrees to buy the asset is said to assume the
while the party that will sell the asset has the
the party in the long position is
position is
K − ST .
ST − K ,
The forward price
while that for the party in the short
K
is set so that there is no cost to
either party to enter the contract. (This is in contrast to options, as we shall
see below.)
The following examples illustrate how forwards may be used to hedge
against unfavorable changes in commodity prices.
Example 4.3.1.
A farmer expects to harvest and sell his wheat crop six
months from now. The price of wheat is currently $8.70 per bushel and
he predicts a crop of 10,000 bushels. He calculates that he would make
Options and Arbitrage
47
an adequate prot even if the price dropped to $8.50. To hedge against a
larger drop in price, he takes the short position in a forward contract with
K = $8.50 × 10, 000 = $85, 000.
At time
T
(six months from now) he is under
the obligation to sell his wheat to the party in the long position for $8.50 per
bushel. His payo is the dierence between the forward price
K
and the price
of wheat at maturity. Thus, if wheat drops to $8.25, his payo is $.25 per
bushel; if it rises to $8.75, his payo is
Example 4.3.2.
−$.25
per bushel.
An airline company needs to buy 100,000 gallons of jet
fuel in three months. It can buy the fuel now at $4.80 per gallon and pay
storage costs, or wait and buy the fuel three months from now. The company
decides on the latter strategy, and to hedge against a large increase in the
cost of jet fuel, it takes the long position in a forward contract with
$4.90 × 100, 000 = $490, 000.
K =
In three months, the airline is obligated to buy,
and the fuel company to sell, 100,000 gallons of jet fuel for $4.90 a gallon.
The company's payo is the dierence between the price of fuel then and the
strike price. If in three months the price rises to, say, $4.96 per gallon, then
the company's payo is $.06 per gallon. If it falls to $4.83, its payo is
per gallon.
Since there is no cost to enter a forward contract, the initial value
F0
−$.07
of the
forward is zero. However, as time passes the forward may acquire a non-zero
value. Indeed, the no-arbitrage assumption implies that the (long position)
value
Ft
of a forward at time
t
is given by
Ft = St − e−r(T −t) K,
t = T . Suppose, however,
Ft < St − e−r(T −t) K holds. We
0 ≤ t ≤ T.
(4.2)
t < T
This is clear for
that at some time
inequality
then take a short position on
the
the security and a long position on the forward. This provides us with cash
St − Ft > 0,
which we deposit in a risk-free account at rate
r.
we discharge our obligation to buy the security for the amount
it to the lender. Since our account has grown to
returning
(St − Ft )er(T −t) ,
(St − Ft )er(T −t) − K . As this amount
−r(T −t)
an arbitrage. If Ft > St − e
K , we
have cash in the amount
our strategy constitutes
At maturity,
K,
we now
is positive,
employ the
reverse strategy, taking a short position on the forward and a long position
on the security. This requires cash in the amount
the rate
r.
At time
T,
St − Ft , which we borrow at
K
we discharge our obligation to sell the security for
and use this to settle our debt. This leaves us with the positive cash amount
K − (St − Ft )er(T −t) ,
again implying an arbitrage and hence verifying (4.2).
One can also obtain (4.2) using the the law of one price, one investment
consisting of a long position in the forward, the other consisting of a long
position in the stock and a short position in a bond with face value
investments have the same value at maturity ((4.2) with
must have the same value for all
Setting
t=0
t ≤ T.
t = T)
in (4.2) and solving the resulting equation
K.
The
and therefore
F0 = 0
for
K,
we
Option Valuation: A First Course in Financial Mathematics
48
see that
K = S0 erT .
(4.3)
Substituting (4.3) into (4.2) yields the alternate formula
Ft = St − ert S0 ,
0 ≤ t ≤ T.
(4.4)
4.4 Currency Forwards
foreign exchange market
Currencies are traded over the counter on the
(FX), which determines the relative values of the currencies. The main purpose of the FX is to facilitate trade and investment so that business among
international institutions may be conducted eciently with minimal regard to
currency issues. The FX also supports currency speculation, typically through
hedge funds. In this case, currencies are bought and sold without the traders
actually taking possession of the currency.
An
exchange rate
species the value of one currency relative to another,
as expressed, for example, in the equation 1 euro = 1.44226 US dollars. Like
stock prices, FX rates can be volatile. Rates may be inuenced by various
economic factors, including government budget decits or surpluses, balance
of trade levels, ination levels, political conditions, and market perceptions.
Because of this volatility there is a risk associated with currency trading and
therefore a need for currency derivatives.
A forward contract whose underlying is a foreign currency is called a
rency forward.
T.
dollars of one euro at time
tablished at time
= Qt
cur-
Consider a currency forward that allows the purchase in US
t
dollars. We let
Let
Kt
denote forward price of the euro es-
and suppose that the time-t rate of exchange is 1 euro
rd
and
re
denote, respectively, the dollar and euro inter-
est rates. To establish a formula relating
Kt and Qt ,
t:
consider the following
possible investment strategies made at time
ˆ
Kt e−rd (T −t) dollars in a
Kt , which is used to
Enter into the forward contract and deposit
US bank. At time
T,
the value of the account is
purchase the euro.
ˆ
Buy
in a
e−re (T −t) euros for e−re (T −t) Qt dollars and deposit the euro amount
European bank. At time T , the account has exactly one euro.
As both strategies yield the same amount at maturity, the law of one price
ensures that they must have the same value at time
Kt e
Solving for
Kt ,
−rd (T −t)
= Qt e
−re (T −t)
t,
that is,
.
we see that the proper time-t forward price of a euro is
Kt = Qt e(rd −re )(T −t) .
Options and Arbitrage
49
In particular, the forward price at time zero is
K = K0 = Q0 e(rd −re )T .
This expression should be contrasted with the forward price
S0 erd t
of a stock.
In the latter case, only one instrument, the dollar account, makes payments.
In the case of a currency, both accounts make payments.
4.5 Futures
A
futures contract,
like a forward, is an agreement between two parties
to buy or sell a specied asset for a certain price at a certain time in the
future. There are important dierences, however. Unlike forward contracts,
future contracts are usually traded on exchanges rather than negotiated by
the parties. Also, a futures price, unlike a forward price, is negotiated daily,
and the daily dierences are received by the long holder of the contract. The
process is implemented by brokers via
margin accounts, which have the eect
of protecting the parties against default.
Example 4.5.1.
Suppose the farmer in Example 4.3.1 takes the short position
on a futures contract on day 0. On each day
with delivery date
T = 180
j,
a futures price
Fj
for wheat
days is quoted. The price depends on the current
prospects for a good crop, the expected demand for wheat, and so forth. The
F1 − F0 on day one,
F2 −F1 on day two, and so on until delivery day T , when he receives FT −FT −1 ,
3
where FT is the spot price of wheat that day. Some of these amounts may
long holder of the contract (the wheat buyer) receives
be negative, in which case a payout is required. The total amount received by
the buyer is
T
X
j=1
(Fj − Fj−1 ) = FT − F0 .
T , the buyer has cash in the amount of FT − F0 , pays FT , and receives
F0 . Since this amount is known on day 0,
F0 acts like a forward price. The dierence is that the payo FT − F0 is paid
On day
his wheat. The net cost to him is
gradually rather than at delivery.
It may be shown that under the assumption of constant interest rates, a
futures price and a forward price are the same. (See, for example, [7].)
3 The
spot price
of a commodity is its price ready for delivery.
Option Valuation: A First Course in Financial Mathematics
50
4.6 Options
Options are derivatives similar to forwards and futures but have the additional feature of limiting an investor's loss to the cost of the option. Speci-
option
cally, an
holder and the writer,
but not the obligation, to buy or sell a partic-
is a contract between two parties, the
which gives the former the right,
ular security under terms specied in the contract. An option has value, since
the holder is under no obligation to exercise the contract and could gain from
long
position, while the writer of the option has the short position. Each of the two
the transaction if she does so. The holder of the option is said have the
basic types of options, the call option and the put option, comes in two styles,
American and European. We begin with the denition of the European call
option.
A
European call option
prescribed time
T,
prescribed security
is a contract with the following conditions: At a
the holder (that is, buyer) of the option may purchase a
S
for a prescribed amount
K,
the
strike price.
For the
holder, the contract is a right, not an obligation. On the other hand, the
writer (seller) of the option
does
have a potential obligation: he
must
sell the
asset if the holder chooses to exercise the option. Since the holder has a right
with no obligation, the option has a value and therefore a price, called the
premium.
The premium must be paid by the holder at the time of opening
of the contract. Conversely, the writer, having assumed a nancial obligation,
must be compensated.
To nd the payo of the option at maturity
ST > K ,
T,
we argue as follows: If
the holder will exercise the option and receive the payo
On the other hand, if
ST ≤ K ,
ST − K .
the holder will decline to exercise the option,
since otherwise he would be paying
K − ST
more than the security is worth.
The payo for the holder of the call option may therefore be expressed as
(ST − K)+ ,
where, for a real number
x,
x+ := max(x, 0).
in the money at time t if St > K , at the money if
out of the money if St < K .
A European put option has a denition analogous to that of a call option:
The option is said to be
St = K ,
and
it is a contract that allows the holder, at a prescribed time
sell
an asset for a prescribed amount
K.
T
in the future, to
The holder is under no obligation to
exercise the option, but if she does so the writer must buy the asset. Whereas
the holder of a call option is betting that the asset price will rise (the wager
being the premium), the holder of a put option is counting on the asset price
falling in the hopes of buying low and selling high.
An argument similar to the call option case shows that the payo of a
European put option at maturity is
(K − ST )+ .
Here the option is in the
Options and Arbitrage
money at time
t
if
St < K ,
at the money if
51
St = K ,
and out of the money if
St > K .
Figure 4.1 shows the payos, graphed against
ST ,
for the holder of a call
and the holder of a put. The writer's payo is the negative of the holder's
payo: the transaction is a zero-sum game.
Call Payoff
Put Payoff
K
ST
K
K
ST
FIGURE 4.1: Call and Put Option Payos
Options, like forwards and futures, may be used to hedge against price
uctuations. For instance, the farmer in Example 4.5.1 could buy a put option
that guarantees a price
K,
K
for his harvest in six months. If the price drops below
he would exercise the option. Similarly, the airline company in Example
4.3.2 could take the long position in a call option that gives the company the
right to buy fuel at a pre-established price
K.
Options may also be used for speculation. A third party in the airline
example might take the long position in a call option, hoping that the price
of fuel will go up. Of course, in contrast to forwards and futures, option-based
hedging and speculation strategies have a cost, namely, the price of the option.
The determination of that price is a primary goal of this book.
Note that, while the holder of the option has only the price of the option to
lose, the writer stands to take a signicantly greater loss. To oset this loss, the
writer could take the money received for the option to start a portfolio with
maturity value sucient to settle the claim of the holder. Such a portfolio is
called a
hedging strategy. For example, the writer of a call option could take the
long position in one share of the security. This requires borrowing an amount
c,
the price of the security minus the income received from selling the call. At
time
T,
the writer's net prot is
ST − cerT ,
the value of the security less the
loan repayment. If the option is exercised, the writer can use the portfolio to
settle his obligation of
ST − K .4
The writer has successfully hedged the short
position of the call. In the next chapter, we consider dynamic hedges with
time-dependent units of the security and a risk-free bond.
Put and call options may be used in combinations to achieve a variety of
payos and hedging eects. For example, consider a portfolio that consists of
a long position in a call option with strike price
4 That S − cerT ≥ S − K
T
T
below.
K1
and a short position in a
is a consequence of the put-call parity formula, discussed
52
Option Valuation: A First Course in Financial Mathematics
Payoff
K2 − K1
K1
ST
K2
FIGURE 4.2: Bull Spread Payo
K2 > K1 , each with underlying S and maturity
T . Such a portfolio is called a bull spread. Its payo (ST − K1 )+ − (ST − K2 )+
call option with strike price
gains in value as the stock goes up. Reversing positions in the calls produces
a
bear spread, which benets from a decrease in stock value. Figure 4.2 shows
the payo of bull spread graphed against
ST .
Of more relevance to the investor than the payo is the
prot
of a portfo-
lio, that is, the payo minus the cost of starting the portfolio. Consider, for
example, the prot from a
straddle,
which is a portfolio with long positions
in a call and a put with the same strike price, maturity, and underlying. The
c of the call and the put,
(ST − K)+ + (K − ST )+ − c = |ST − K| − c, graphed in
start-up cost of the portfolio is the combined cost
hence the prot is
Figure 4.3. The straddle is seen to benet from a movement in either direction
away from the strike price.
Profit
K −c
K
ST
−c
FIGURE 4.3: Straddle Prot
So far we have considered only European options, characterized by the
fact that the contracts may be exercised only at maturity. By contrast, American options may be exercised at any time up to and including the expiration
date. American options are more dicult to analyze as there is the additional
complexity introduced by the holder's desire to nd the optimal time to exercise the option and the writer's need to construct a hedging portfolio that
will cover the claim at any time
t ≤ T.
Nevertheless, as we shall see, some
Options and Arbitrage
53
properties of American options may be readily deduced from those of their
European counterparts. We consider American options in Chapters 9 and 14.
In recent years, an assortment of more complex derivatives has appeared.
These are frequently called
exotic options
and are distinguished by the fact
that their payos are no longer simple functions of the value of the underlying
at maturity. Prominent among these are the path-dependent options, whose
payos depend not only on the value of the underlying at maturity but also
Asian
options, lookback options, and barrier options. The payo of an Asian option
on earlier values as well. The main types of path-dependent options are
depends on an average of the values of
S,
while that of a lookback option is
a function of the maximum or minimum values of
payo that depends on whether
St
S.
A barrier option has
crosses a prescribed level. These options
are examined in detail in Chapter 14.
We end this section with a brief discussion of swaps and swaptions. A
swap
is a contract between two parties to exchange nancial assets or cash streams
(payment sequences) at some specied time in the future. The most common of
currency swaps, exchanges of one currency for another, and interestrate swaps, where a cash stream with a xed interest rate is exchanged for one
with a oating rate. A swaption is an option on a swap. It gives the holder
these are
the right to enter into a swap at some future time. A detailed analysis of
swaps and swaptions may be found in [7]. A
credit default swap
(CDS) is a
contract that allows one party to make a series of payments to another party
in exchange for a future payo if a specied loan or bond defaults. A CDS is,
in eect, insurance against default but is also used for hedging purposes and
by speculators, as was famously the case in the subprime mortgage crisis of
2007 (see [11]).
4.7 Properties of Options
T,
(St )Tt=0 . We dee
note the cost of a European (resp., American) call option by C0 (respectively,
a
C0 ) and that of a European (resp., American) put option by P0e (respectively,
P0a )
The options considered in this section are assumed to have maturity
strike price
K
and underlying one share of
S
with price process
The following proposition asserts that an American call option, despite its
greater exibility, has the same value as that of a comparable European call
option.
Proposition 4.7.1. It is never advantageous to exercise an American call
option early. In particular, C0e = C0a .
Proof.
Clearly
C0a ≥ C0e ,
and the only way to have
C0a > C0e
is for there to be
Option Valuation: A First Course in Financial Mathematics
54
an advantage to exercise the American option early. We will show that this is
not the case.
S at
t < T,
Suppose an investor holds an American option to buy one share of
time
T
for the price
K.
If she exercises the option at some time
St − K , which, if invested in a risker(T −t) (St − K) at maturity. But suppose
instead that she sells the stock short for the amount St , invests the proceeds,
and then purchases the stock at time T (returning it to the lender). If ST ≤ K ,
she will pay the market price ST ; if ST > K , she will exercise the option
and pay the amount K . Under this strategy, she will therefore have cash
er(T −t) St − min(ST , K) at time T . Since this amount is at least as large as
er(T −t) (St − K), the second strategy is generally superior to the rst. Since
she will immediately realize the payo
free account, will yield the amount
the second strategy did not require that the option be exercised early, there
is no advantage in doing so.
Hereafter, we denote the cost of either a European or American call option
by
C0 .
The next result uses an arbitrage argument to obtain an important
connection between
P0e
and
C0 .
Proposition 4.7.2 (Put-Call Parity Formula). Consider a call option (either
European or American) and a European put option, each with strike price K ,
maturity T , and underlying one share of S . Then
Proof.
S0 + P0e − C0 = Ke−rT .
(4.5)
S0 + P0e − C0 6= Ke−rT leads to an
e
−rT
arbitrage. Suppose rst that S0 + P0 − C0 < Ke
. We then buy one share
e
of S , buy one put option, and sell one call option. This costs us S0 + P0 − C0 ,
e
rT
which we borrow and repay in the amount (S0 + P0 − C0 )e
at maturity.
If
We show that the assumption
ST ≤ K ,
then the call option we sold will not be exercised, but we can
exercise our put option and sell the security for
K.
If
ST > K ,
the put option
is worthless but the call option we sold will be exercised, requiring us to sell
the security for
K . In either case, we will realize the cash amount K . Since our
K > (S0 + P e − C)erT , the amount received exceeds
assumption implies that
the amount owed and our strategy constitutes an arbitrage.
If
S0 +P0e −C0 > Ke−rT , we use the reverse strategy: sell short one share of
the security, sell one put option, and buy one call option. An argument similar
to that in the rst paragraph shows that this strategy is also an arbitrage.
One can also establish (4.5) by using the law of one price. Consider two
portfolios, one consisting of a long holding of the put, the other consisting of
a long holding of the call, a bond with face value
K
and maturity
T,
and a
short position on one share of the security. The portfolios have initial values
P0e and C0 + e−rT K − S0 , respectively. Since the nal values (K − ST )+ and
(ST − K)+ + K − ST are equal, the initial values must also be equal, giving
P0e = C0 + B0 − S0 , which is (4.5).
Proposition 4.7.2 shows that the price of a European put option may be
Options and Arbitrage
expressed in terms of the price
C0
55
of a call. To nd
C0 ,
one uses the notion of
self-nancing portfolio, described in the next chapter. We shall see later that
C0
depends on a number of factors, including
ˆ
the initial price of
ˆ
the strike price (the smaller the value of
S
(a large
S0
suggests that a large
K
ST
is likely);
the greater prot for the
holder);
ˆ
the volatility of
ˆ
the expiration date;
ˆ
the interest rate (which aects the discounted value of the strike price).
S;
The exact quantitative dependence of
C0
on these factors will be examined in
detail in the context of the Black-Scholes-Merton model in Chapter 11.
4.8 Dividend-Paying Stocks
In the foregoing, we have assumed that the underlying asset of an option
pays no dividends. In this section, we illustrate how dividends can aect option price properties by proving a version of the put-call parity formula for
dividend-paying stocks.
We begin by observing that when a stock pays a dividend the stock's value
is immediately reduced by the amount of the dividend. Indeed, suppose the
t1 > 0. If the stock's value x immediately
St1 − D1 , an arbitrageur could buy the stock for
St1 , get the dividend, and then sell the stock for x, realizing a prot of D1 +
x − St1 > 0. On the other hand, if x were less than St1 − D1 she could sell
the stock short for St1 , buy the stock immediately after t1 for x, and return
it along with the dividend (as is required) for a prot of St1 − x − D1 > 0.
stock pays a dividend
after
t1
D1
at time
were greater than
To derive a put-call parity formula for a dividend-paying stock, we assume
that a dividend
D
Dj
is paid at time
tj ,
where
0 < t 1 < t2 < · · · < t n ≤ T .
Let
denote the present value of the dividend stream:
D := e−rt1 D1 + e−rt2 D2 + · · · + e−rtn Dn .
Consider two portfolios, one consisting of a long holding of a put with maturity
T , strike price K , and underlying one share of the stock; the other consisting of
a long holding of the corresponding call, a bond with face value K and maturity T , bonds with face values Dn and maturity times tn , n = 1, 2, . . . , N , and
a short position on one share of the stock. The initial values of the portfolios
are
P0e
and
C0 + Ke−rT + D − S0 ,
respectively. At maturity, the value of the
56
Option Valuation: A First Course in Financial Mathematics
rst portfolio is
(K − ST )+ .
Assuming that the dividends are deposited into
an account and recalling that in the short position the stock as well as the
dividends with interest must be returned, we see that the value of the second
portfolio at maturity is
+
(ST − K) + K +
N
X
n=1
Dn e
r(T −tn )
−
ST +
N
X
!
Dn e
r(T −tn )
n=1
= (K − ST )+ .
Since the portfolios have the same value at maturity, the law of one price
ensures that the portfolios have the same initial value. It follows that
S0 + P0e − C0 = Ke−rT + D,
which is the
put-call parity formula for dividend-paying stocks. Note that the
formula reduces to the non-dividend-paying version if each
Dn = 0.
Options and Arbitrage
57
4.9 Exercises
1. Show that
S0 − Ke−rT ≤ C0 ≤ S0
P0e ≤ Ke−rT .
and
2. (Generalization of Exercise 1.) Show that
and
e
rT
P0e
+ min(ST , K) ≤ K .
C0 + e−rT min(ST , K) ≤ S0
3. Complete the proof of Proposition 4.7.2 by showing that the assumption
S0 + P0e − C0 > Ke−rT
leads to an arbitrage.
4. Consider two call options
C
strike price, and with prices
maturity
T
and
C
0
and
C0
C0
with the same underlying, the same
C00 , respectively. Suppose that C
T < T . Explain why C00 ≤ C0 .
and
has maturity
P
5. Consider two American put options
P has
P00 ≤ P0 .
and
P0
with the same underlying,
K , and with prices P0 and P00 , respectively. Suppose
0
0
maturity T and that P has maturity T < T . Explain why
the same strike price
that
has
0
ST of a portfolio that is (a) long in a stock
S ; (b) short in S and long in a call on S ; (c)
long in S and short in a put on S ; (d) short in S and long in a put
on S . Assume the calls and puts are European with strike price K and
maturity T .
6. Graph the payos against
S
and short in a call on
7. Graph the payos against
ST
for an investor who (a) buys one call and
two puts; (b) buys two calls and one put. Assume the calls and puts
are European with strike price
K,
maturity
T
and underlying
portfolios in (a) and (b) are called, respectively, a
strip
S.
and a
(The
strap.)
What can you infer about the strategy underlying each portfolio?
8. Let
0 < K1 < K2 .
Graph the payo against
(a) holds one call with strike price
K1
ST
for an investor who
and writes one put with strike
K2 ; (b) holds one put with strike price K1 and writes one call with
K2 . Assume that the options are European and have the
same underlying and maturity T .
price
strike price
9. Graph the payo against
strike price
K1
ST
for an investor who holds one put with
and one call with strike price
K2 > K1 . Assume that the
options are European and have the same underlying and same expiration
date
T.
(The portfolio in this exercise is known as a
strangle.)
10. Consider two call options with the same underlying and same maturity
T,
C0
K > K.
one with price
strike price
C0 ≥ C00 .
0
and strike price
K,
the other with price
C00
and
Give a careful arbitrage argument to show that
Option Valuation: A First Course in Financial Mathematics
58
11. Consider two European put options with the same underlying and same
maturity
P00
T,
that
P0 and strike price K , the other with
K 0 > K . Give a careful arbitrage argument to
one with price
and strike price
price
show
P0 ≤ P00 .
C00 and C000 denote the costs of call options with strike prices K 0
00
0
00
and K , respectively, where 0 < K < K , and let C0 be the cost of a
0
00
call option with price K := (K + K )/2. If all options have the same
0
00
maturity and underlying, show that C0 ≤ (C0 + C0 )/2.
12. Let
Hint:
C0 > (C00 + C000 )/2. Write two options with strike
0
00
price K , buy one with strike price K and another with strike price K ,
0
00
giving you cash in the amount 2C0 − C0 − C0 > 0. Consider the cases
0
00
obtained by comparing ST with K , K , and K .
Assume that
The portfolio described in the hint is called a
the portfolio has a positive cost
function of the portfolio.
buttery spread. Assuming
c := C00 + C000 − 2C0 ,
graph the prot
13. Referring to Exercises 10 and 11, show that
max(P00 − P0 , C0 − C00 ) ≤ (K 0 − K)e−rT .
14. A
capped option
is
a
pre-established amount
min ((ST − K)+ , A).
standard
A.
option
with
prot
capped
at
a
The payo of a capped call option is
Which of the following portfolios has time-t value
the same as that of a capped call option: strip, strap, straddle, strangle,
bull spread, or bear spread?
15. Show that a bull spread can be created from a combination of positions
in put options and a bond.
Chapter 5
Discrete-Time Portfolio Processes
The notion of self-nancing, replicating portfolio is a key component of option
valuation models. A portfolio is an example of a stochastic process, that is, a
random variable changing with time. We begin with a brief discussion of this
important notion.
5.1 Discrete-Time Stochastic Processes.
Experiments consisting of a sequence of trials may be viewed dynamically,
that is, changing in time. If the outcome of the
n)
nth trial (the outcome at time
is described by a random variable, then the resulting sequence of random
variables provides a mathematical model of the experiment. This idea leads
to the formal notion of
stochastic process.
Denition 5.1.1. A (discrete-time) stochastic (or random) process on a probability space is a nite or innite sequence X = (Xn ) = (Xn )n≥0 of random
variables Xn . 1 If each Xn is a constant, then the process X is said to be de1
2
d
terministic. If X , X , . . . , X are stochastic processes and Xn is the random
1
2
d
vector (Xn , Xn , . . . , Xn ), then (Xn )n≥0 is called a d-dimensional stochastic
process.
Example 5.1.2.
Let
Sn
denote the price of a stock on day
n.
On day 0,
the price of the stock is known but future prices are not and therefore are
usually taken to be random variables. The sequence
S = (Sn )
is a stochastic
process that models the price movement of the stock. A portfolio consisting
d-dimensional stochastic process
j
where Sn denotes the price of stock j on day n.
of
d
stocks gives rise to a
(S 1 , S 2 , . . . , S d ),
As noted above, a stochastic process may be thought of as a mathematical
description of an experiment evolving in time. A related concept is the evolution or ow of information revealed during the experiment. This is described
mathematically by a
1 The
ltration.
sequence may begin at indices other than
0.
59
Option Valuation: A First Course in Financial Mathematics
60
Denition 5.1.3. A (discrete-time) ltration on a probability space (Ω, F, P)
is a nite or innite sequence (Fn ) = (Fn )n≥0 of σ-elds with
F0 ⊆ F1 ⊆ · · · ⊆ Fn ⊆ · · · ⊆ F.
A stochastic process (Xn ) is said to be adapted to a ltration (Fn ) if for each n
the random variable Xn is Fn -measurable. A d-dimensional stochastic process
is adapted to a ltration if each component process is adapted.
A ltration
(Fn )
may be thought of as representing the accumulation of
information over time. If
(Xn )
all relevant information about
is adapted to the ltration, then
Xn .
Fn
includes
The following example should illustrate
this idea.
Example 5.1.4.
tails
T
where
T 's
A coin is tossed
N
H or
ω1 ω2 . . . ωN ,
ω = ω1 ω2 . . . ωn of H 's and
times and the outcomes heads
are observed. The sample space consists of all sequences
ωn = H
or
T.
For each xed sequence
n tosses, let Aω denote the set of all sequences
ν = ωn+1 ωn+2 . . . ωN is a sequence of H 's and T 's
representing the outcomes of the last N − n tosses. For example, if N =
4, then AT H = {T HHH, T HHT, T HT H, T HT T }. We show that the sets
Aω generate a ltration that describes the ow of information during the
appearing in the rst
of the form
ων ,
where
experiment.
Before the rst toss, we have no knowledge of the outcomes of the experiment. The
σ -eld F0
corresponding to this void of information is
After the rst toss, we know whether the outcome was
H
or
T
F0 = {∅, Ω}.
but have no
σ -eld F1 describing the information gained on the rst toss is {∅, Ω, AH , AT }. After the second toss, we
know which of the outcomes ω1 ω2 = HH , HT , T H , T T has occurred, but we
have no knowledge about the impending third toss. The σ -eld F2 describing
the information gained on the rst two tosses consists of ∅, Ω, and all unions of
N
the sets Aω1 ω2 . Continuing in this way, we obtain a ltration (Fn )n=0 , where
Fn (n ≥ 1) is the σ -eld consisting of all unions of (that is, generated by)
the pairwise disjoint sets Aω (see Example 2.3.2). Note that after N tosses
we have complete information, demonstrated by the fact that FN contains all
subsets of Ω.
Now let Xn denote the number of heads appearing in the rst n tosses. The
stochastic process (Xn ) is easily seen to be adapted to (Fn ). For example, the
event {X4 = 2} is the union of AHHT T , AHT HT , AHT T H , AT T HH , AT HT H ,
and AT HHT and hence is a member of F4 . Conversely, knowing that the event
AHT HT has occurred tells us that X4 = 2 (and that X1 = X2 = 1 and
X3 = 2).
information regarding subsequent tosses. The
Filtrations, such as that of Example 5.1.4, which contain only the information generated by a random process are of sucient importance to warrant
special terminology and notation.
Discrete-Time Portfolio Processes
61
Denition 5.1.5. Let (Xn )n≥0 be a stochastic process. For each n, denote by
FnX = σ(Xj : j ≤ n)
the σ-eld generated by all events of the form {Xj ∈ J}, where j ≤ n and J
is an arbitrary interval of real numbers. (FnX ) is called the natural ltration
for (Xn ) or the ltration generated by the process (Xn ). It is the smallest
ltration to which (Xn ) is adapted.
A concept related to adaptability is predictability, dened as follows.
Denition 5.1.6. A stochastic process X = (Xn ) is said to be predictable
(Fn ) if Xn is Fn−1 measurable for each n ≥ 1. A
d-dimensional stochastic process is predictable if each component process is
predictable.
with respect to a ltration
Every predictable process is adapted, but not conversely. The dierence may
be understood as follows: For an adapted process,
in
Fn .
Xn
is determined by events
For a predictable process, it is possible to determine
Xn
by events in
Fn−1 .
Example 5.1.7.
Marbles are randomly drawn one at a time without replace-
ment from a jar initially containing red and white marbles of equal number.
Before each draw a player places a bet on the outcome red or white. By keeping track of the number of marbles of each color in the jar after each draw,
the player may make an informed decision as to the size of the wager. For
example, if the rst
n−1
draws resulted in more red marbles than white, the
gambler should bet on white for the
nth
draw, the size of the bet determined
by the ratio of white to red marbles left in the jar. As in the coin toss example,
(Fn ).
Xn must be determined
the wager process (Xn ) is predictable. On the
number of red balls in the rst n draws, then
the ow of information in this example may be modeled by a ltration
Because the gambler is not prescient, the
by events in
Fn−1 . Therefore,
Yn denotes the
other hand, if
(Yn )
nth
wager
is adapted to the ltration but is not predictable.
5.2 Self-Financing Portfolios
Suppose that the price of a security S is given by a stochastic process S =
(Sn )N
n=0 on a probability space (Ω, F, P), where S0 is constant. We may assume
S
S
that F = FN , where (Fn ) is the natural ltration for S . Note that because
S0 is constant F0S is the trivial σ -eld {∅, Ω}. The ltration (FnS ) models the
accumulation of stock price information in the discrete time interval [0, N ].
Let B denote a risk-free bond earning compound interest at the rate i per
period. We assume that the market allows unlimited transactions in S and B .
Option Valuation: A First Course in Financial Mathematics
62
The value of the bond at time 0 is taken to be 1 unit, so that the value of
the security is measured in terms of the initial value of the bond. (Later we
introduce the notion of
the security against the
process
B = (Bn )N
n=0
discounted price process, which measures the value of
current value of the bond.) In contrast to S , the price
of
B
is deterministic; indeed,
Bn = (1 + i)n .
Denition 5.2.1. A portfolio or trading strategy for (B, S) is a twodimensional predictable stochastic process (φ, θ) = ((φn , θn ))N
n=1 on (Ω, F, P),
where φn and θn denote, respectively, the number of units of B and shares of
S held at time n. The value Vn of the portfolio at time n is dened as
V0 = φ1 + θ1 S0 ,
Vn = φn Bn + θn Sn , n = 1, 2, . . . , N.
The stochastic process V = (Vn )N
n=0 is called the value or wealth process of
the portfolio, and V0 is the initial investment or initial wealth.
The idea behind a trading strategy is this: At time 0, the number of units
φ1
of
B
and shares
θ1
of
S
are chosen to satisfy the initial wealth equation of
Denition 5.2.1. These are constants, as the value
the value of the portfolio before the price
Sn
S0
is known. At time
n ≥ 1,
is known (and before the bond's
new value is noted) is
φn Bn−1 + θn Sn−1 .
When
Sn
becomes known the portfolio has value
φn Bn + θn Sn .
(5.1)
S and units of B may be adjusted, the
S0 , S1 , . . ., Sn of the stock. Predictability of
At this time, the number of shares of
strategy based on the price history
the portfolio process is the mathematical property underlying this procedure;
the new values
FnS . After
(n, n + 1) is
by
φn+1
and
θn+1
are determined using only information provided
readjustment, the value of the portfolio during the time interval
φn+1 Bn + θn+1 Sn .
At time
n + 1,
(5.2)
the process is repeated.
Now suppose that each readjustment of the portfolio is accomplished without changing its current value, that is, without the removal or infusion of
wealth. Shares of
S
and units of
B
may be bought or sold, but the net value
of the transactions is zero. Mathematically, this simply means that the quantities in (5.1) and (5.2) are equal. This is the notion of
self-nancing portfolio.
Using the notation
∆xn := xn+1 − xn ,
we may express this idea formally as follows.
Denition 5.2.2. A portfolio (φ, θ) is said to be self-nancing if
Bn ∆φn + Sn ∆θn = 0, n = 1, 2, . . . , N − 1.
(5.3)
Discrete-Time Portfolio Processes
63
In Theorem 5.2.5 below, we give several alternate ways of characterizing a
self-nancing portfolio. One of these uses the idea of a
discounted process.
Denition 5.2.3. Let X = (Xn )Nn=0 be a stochastic process. The discounted
process X̃ is dened by
X̃n = (1 + i)−n Xn ,
Remark 5.2.4.
The process
the current value of the bond
be a
n = 0, 1, . . . , N.
X̃ measures the current value of X in terms of
B . In this context, the bond process is said to
numeraire. Converting from one measure of value to another is referred
change of numeraire.
to as a
Theorem 5.2.5. For a trading strategy (φ, θ) with value process V , the following statements are equivalent:
(i)
(ii)
(iii)
(φ, θ)
is self-nancing;
∆Vn = φn+1 ∆Bn + θn+1 ∆Sn ,
n = 0, 1, . . . N − 1;
satises the recursion equations
V
Vn+1 = θn+1 [Sn+1 − (1 + i)Sn ] + (1 + i)Vn
= θn+1 Sn+1 + (1 + i)[Vn − θn+1 Sn ],
(iv)
(v)
∆Ṽn = θn+1 ∆S̃n , n = 0, 1, . . . , N − 1;
Pn−1
φn = V0 − j=0 S̃j ∆θj , n = 1, 2, . . . , N ,
Proof.
For
n = 0, 1, . . . , N − 1,
n = 0, 1, . . . , N − 1;
where θ0 := 0.
let
Yn = Bn ∆φn + Sn ∆θn = φn+1 Bn + θn+1 Sn − Vn .
Then
Vn = φn+1 Bn + θn+1 Sn − Yn ,
hence
∆Vn = φn+1 Bn+1 + θn+1 Sn+1 − (φn+1 Bn + θn+1 Sn − Yn )
= φn+1 ∆Bn + θn+1 ∆Sn + Yn .
Since the portfolio is self-nancing i
Yn = 0
for all
n,
(i) and (ii) are seen to
be equivalent.
From the equations
we have
φn+1 Bn = Yn + Vn − θn+1 Sn
and
Bn+1 = (1 + i)Bn ,
Vn+1 = φn+1 Bn+1 + θn+1 Sn+1
= (1 + i) [Yn + Vn − θn+1 Sn ] + θn+1 Sn+1
= θn+1 [Sn+1 − (1 + i)Sn ] + (1 + i)(Vn + Yn ).
The last equation shows that (i) and (iii) are equivalent. That (iii) and (iv)
are equivalent follows immediately from the denition of discounted process.
Option Valuation: A First Course in Financial Mathematics
64
To show that (i) and (v) are equivalent, assume rst that
(φ, θ)
is self-
nancing. By (5.3),
Bj ∆φj = −Sj ∆θj ,
−1
Multiplying by Bj
n
X
j=1
−j
= (1 + i)
∆φj = −
n
X
j = 1, 2, . . . , N − 1.
and summing, we have
S̃j ∆θj ,
j=1
The left side of this equation collapses to
φn+1 = φ1 −
Finally, noting that
have
n
X
S̃j ∆θj ,
j=1
n = 1, 2, . . . , N − 1.
φn+1 − φ1
n = 1, 2, . . . , N − 1.
φ1 = V0 − θ1 S0 = V0 − S0 ∆θ0
φn+1 = V0 −
n
X
S̃j ∆θj ,
j=0
hence
(recalling that
θ0 = 0),
we
n = 0, 1, . . . , N − 1,
which is (v). Since the steps in this argument may be reversed, (i) and (v) are
equivalent.
Corollary 5.2.6. Given a predictable process θ and initial wealth V0 , there
exists a unique predictable process φ such that the trading strategy (φ, θ) is
self-nancing.
Proof. The process φ given in part (v) of the theorem is clearly predictable.
Remarks 5.2.7.
The quantity
Vn − θn+1 Sn
in part (iii) of Theorem 5.2.5 is
the cash left over from the transaction of buying
n and
value Vn+1
time
θn+1
shares of the stock at
therefore represents the value of the bond account. Thus, the new
of the portfolio results precisely from the change in the value of
(n, n + 1).
V and S ,
by θ and the
the stock and the growth of the bond account over the time interval
Part (iv) implies that
θ
may be expressed uniquely in terms of
φ is completely determined
V0 . It follows that, in the self-nancing case, the trading strategy
and part (v) asserts that the process
initial wealth
(φ, θ)
may be determined uniquely from the value process. (See Exercise 4 in
this regard.)
5.3 Option Valuation by Portfolios
Self-nancing portfolios may be used to establish the fair value of a derivative. To describe the method with sucient generality, we make the following
denition.
Discrete-Time Portfolio Processes
65
Denition 5.3.1. A contingent claim is an FNS -random variable H . A hedging strategy or hedge for H is a self-nancing trading strategy with value
process V satisfying VN = H . If a hedge for H exists, then H is said to be
attainable and the hedge is called a replicating portfolio. A market is complete
if every contingent claim is attainable.
European options are the most common examples of contingent claims.
The holder of the option has a claim against the writer, namely, the value
(payo ) of the option at maturity. In the case of a call option, that claim
(SN − K)+ ; for a put option the claim is (K − SN )+ . Both are obviously
S
FN -random variables.
Recall that an arbitrage is a trading strategy with zero probability of loss
is
and positive probability of net gain. We may now give a precise denition in
terms of the value process of a portfolio.
Denition 5.3.2. An arbitrage is a trading strategy (φ, θ) whose value process
V satises P(VN ≥ V0 ) = 1 and P(VN > V0 ) > 0.
To see the implications of completeness, suppose that we write a European
contract with payo
H
in a complete and arbitrage-free market. At maturity
we are obligated to cover the claim, which, by assumption, is attainable by a
V . Our strategy is to sell the conV0 and use this amount to start the portfolio. At time N , our portfolio
value VN , which we use to cover our obligation. The entire transaction
self-nancing portfolio with value process
tract for
has
costs us nothing since the portfolio is self-nancing; we have hedged the short
position of the contract. It is natural then to dene the time-n value of the
contract to be
Vn ;
any other value would result in an arbitrage. (This is an-
other instance of the law of one price.) We summarize this discussion in the
following theorem.
Theorem 5.3.3. In a complete and arbitrage-free market, the time-n value of
a European contingent claim H with maturity N is Vn , where V is the value
process of a self-nancing portfolio with nal value VN = H . In particular,
the fair price of the claim is V0 .
In Chapter 7, we illustrate Theorem 5.3.3 for the special case of a security
that follows the binomial model.
S
66
Option Valuation: A First Course in Financial Mathematics
5.4 Exercises
1. A hat contains three slips of paper numbered 1, 2, and 3. The slips are
randomly drawn from the hat one at a time without replacement. Let
Xn denote the number on the nth slip drawn. Describe the sample space
Ω of the experiment and the natural ltration (Fn )3n=1 associated with
(Xn )3n=1 .
2. Rework Exercise 1 if the second slip is replaced.
3. Complete the proof of Theorem 5.2.5 by showing that (v) implies (i).
4. Show that for
n = 0, 1, . . . , N − 1,
θn+1 =
∆Ṽn
,
∆S̃n
φn+1 =
and
∆Vn ∆S̃n − ∆Ṽn ∆Sn
.
i(1 + i)−1 Sn+1 − iSn
These equations explicitly express the trading strategy in terms of the
stock price and value processes.
5. The
gain
of a portfolio
(φ, θ)
in the time interval
(φn Bn + θn Sn ) − (φn Bn−1 + θn Sn−1 )
= φn ∆Bn−1 + θn ∆Sn−1 ,
The gain up to time
n is the sum Gn
(n − 1, n]
is dened as
n = 1, 2, . . . , N.
of the gains over the time intervals
(j − 1, j], 1 ≤ j ≤ n:
G0 = 0
and
Gn =
n
X
(φj ∆Bj−1 + θj ∆Sj−1 ) , n = 1, 2, . . . N.
j=1
(Gn )N
n=0
is called the
gains process
folio is self-nancing i for each
initial value plus the gain
Gn ,
of the portfolio. Show that the port-
n the time-n value of the portfolio is its
that is,
Vn = V0 + Gn ,
n = 1, 2, . . . , N.
Chapter 6
Expectation of a Random Variable
Expectation is a probabilistic interpretation and generalization of the notion
of weighted average. For example, suppose that we repeatedly toss a pair of
distinguishable fair coins. If
X
denotes the total number of heads that come
up on each toss, then, in the long run,
X
takes on the values 0, 1, and 2 with
X , that
(.25)0 + (.5)1 +
random variable X .
relative frequencies .25, .5, and .25, respectively. The average value of
is, the average number of heads in the long run, is therefore
(.25)2 = 1.
This idea may be made precise for a general
For our purposes, however, it is sucient to consider two special cases: discrete
and continuous random variables.
6.1 Discrete Case: Denition and Examples
Denition 6.1.1. The expectation (expected value, mean) of a discrete random variable X on a probability space (Ω, F, P) is dened as
EX =
X
xpX (x),
x∈R
provided the sum on the right exists.
Since
X
is discrete, the expression on the right (ignoring zero terms) is either
a nite sum or an innite series. If the series diverges, then
EX
is undened.
For the remainder of the chapter, we shall tacitly assume that all expectations
in any given discussion exist.
Example 6.1.2.
The table below gives the course grade distribution of a
class of 100 students. Let
where
X=4
X
be the grade of a student chosen at random,
no. of students
15
25
40
12
8
grade
A
B
C
D
F
if the student received an A,
X=3
for a B, and so forth. From
67
Option Valuation: A First Course in Financial Mathematics
68
Denition 6.1.1, the class average is seen to be
E X = 4(.15) + 3(.25) + 2(.4) + 1(.12) + 0(.8) = 2.27.
Example 6.1.3.
and
pX (x) = 0
A∈F
Let
for all other
X = IA . Since pX (1) = P(A), pX (0) = P(A0 ),
values of x, we see that
and
E IA = P(A).
Example 6.1.4.
(q
:= 1 − p),
EX =
If
we have
X ∼ B(n, p),
pX (k) =
n
k
pk q n−k
n
n−1
X
X n − 1
n k n−k
k
p q
= np
pk q n−1−k = np(p+q)n−1 = np.
k
k
k=1
k=0
Example 6.1.5.
Since
then, recalling that
Let
X
be a geometric random variable with parameter
p.
pX (n) = pq n−1 ,
EX = p
∞
X
nq
n−1
n=1
Remark 6.1.6.
∞
∞
X
d n
d X n
d
q
1
=p
q =p
q =p
= .
dq
dq
dq
1
−
q
p
n=1
n=1
In a discrete probability space, the term
xpX (x)
may be
expanded as
x
X
ω:X(ω)=x
Summing over
tion
Ω,
x
X
P(ω) =
X(ω)P(ω).
ω:X(ω)=x
and noting that the pairwise disjoint sets
{X = x}
parti-
we obtain the following useful characterization of expected value in a
discrete space:
EX =
X
X(ω)P(ω).
ω∈Ω
6.2 Continuous Case: Denition and Examples
Denition 6.2.1. The expectation (expected value, mean) of a continuous
random variable X on a probability space is dened as
Z
∞
EX =
xfX (x) dx.
−∞
If the integral diverges, then
EX
is undened. As in the discrete case, we shall
tacitly assume in what follows that all stated expectations exist.
Expectation of a Random Variable
Example 6.2.2.
Let
X
69
be uniformly distributed on the interval
E X = (β − α)−1
Z
β
x dx =
α
(α, β). Then
α+β
.
2
In particular, the average value of a number selected randomly from the interval
(0, 1)
is
1/2.
Example 6.2.3.
Let
X ∼ N (0, 1). Then
Z ∞
1 2
1
EX = √
xe− 2 x dx = 0,
2π −∞
since the integrand is an odd function. More generally, if
X ∼ N (µ, σ 2 ),
then
∞
1 x−µ 2
1
√
xe− 2 ( σ ) dx
σ 2π −∞
Z ∞
1 2
1
√
=
(σy + µ)e− 2 y dy
2π −∞
Z ∞
Z ∞
1 2
σ
µ
− 21 y 2
√
√
=
ye
dy +
e− 2 y dy
2π −∞
2π −∞
= µ.
Z
EX =
6.3 Properties of Expectation
The following theorem is useful for computing the expectation of more
complex random variables.
Theorem 6.3.1 (Law of the Unconscious Statistician I).
random variable and h(x) is any function, then
E h(X) =
X
(i)
If X is a discrete
h(x)pX (x),
x
where the sum is taken over all x in the range of X .
(ii) If X is a continuous random variable and h(x) is a continuous function,
then
Z ∞
E h(X) =
h(x)fX (x) dx.
−∞
Proof.
We prove only (i); for a proof for (ii), see, for example, [5]. For
range of
h(X),
we have
P(h(X) = y) =
X
x
P(X = x, h(x) = y) =
X
x:h(x)=y
pX (x)
y
in the
Option Valuation: A First Course in Financial Mathematics
70
hence
E h(X) =
X
y
y
X
pX (x) =
X
Example 6.3.2.
Let
X ∼ N (0, 1)
and let
1
EX = √
2π
Z
n
If
n
h(x)pX (x) =
y x:h(x)=y
x:h(x)=y
Theorem 6.3.1,
X
X
h(x)pX (x).
x
n
be a nonnegative integer. By
∞
xn e−x
2
/2
dx.
−∞
is odd, then the integrand is an odd function hence
even,
2
E Xn = √
2π
∞
Z
xn e−x
2
/2
E X n = 0.
If
n
is
dx,
0
and an integration by parts yields
2
E X n = (n − 1) √
2π
Z
∞
xn−2 e−x
2
/2
0
dx = (n − 1)E X n−2 .
Iterating we see that for any even positive integer
n,
E X n = (n − 1)(n − 3) · · · 3 · 1.
In particular,
E X 2 = 1. E X n
is called the
nth
moment
of
X.
Theorem 6.3.1 extends to the case of functions of more than one variable.
We state a version for two variables.
Theorem 6.3.3 (Law of the Unconscious Statistician II). (i) If X and Y are
discrete random variables and h(x, y) is any function, then
E h(X, Y ) =
X
h(x, y)pX,Y (x, y),
x,y
where the sum is taken over all x in the range of X and y in the range of Y .
(ii) If X and Y are jointly continuous random variables and h(x, y) is a
continuous function, then
Z
∞
Z
∞
E h(X, Y ) =
h(x, y)fX,Y (x, y) dx dy.
−∞
−∞
Theorem 6.3.4. If X and Y are discrete or jointly continuous random variables and α, β ∈ R, then
(i) (unit property)
(ii) (linearity)
E 1 = 1;
E(αX + βY ) = αE X + βE Y ;
Expectation of a Random Variable
(iii) (order property)
X ≤ Y ⇒ EX ≤ EY ;
(iv) (absolute value property)
Proof.
71
and
|E X| ≤ E |X|.
Part (i) is clear. We prove (ii) only for the discrete case. By Theo-
rem 6.3.3,
E(αX + βY ) =
X
(αx + βy)pX,Y (x, y)
x,y
=α
X X
X X
x
pX,Y (x, y) + β
y
pX,Y (x, y)
=α
X
x
y
y
xpX (x) + β
x
X
x
ypY (y)
y
= αE X + βE Y.
Z = Y −X . Then, by Example 3.4.3,
z < 0,
For the continuous version of (iii), set
Z
is a continuous, nonnegative random variable hence, for
Z
z
−∞
fZ (t) dt = P(Z ≤ z) = 0.
Dierentiating with respect to
z,
we see that
EY − EX = EZ =
Z
0
fZ (z) = 0
for
z < 0.
Therefore,
∞
zfZ (z) dz ≥ 0.
Part (iv) follows from (iii) and the inequality
±E X = E(±X) ≤ E |X|.
Theorem 6.3.5. Let X and Y be discrete or jointly continuous independent
random variables. Then E(XY ) = (E X)(E Y ).
Proof. We prove only the jointly continuous case. By Corollary 3.5.4,
fX,Y (x, y) = fX (x)fY (y)
hence, by Theorem 6.3.3,
Z
∞
E(XY ) =
−∞
Z ∞
=
Z
∞
xyfX (x)fY (y) dx dy
Z ∞
xfX (x) dx
yfY (y) dy
−∞
−∞
−∞
= (E X)(E Y ).
6.4 Variance of a Random Variable
Denition 6.4.1. Let X be a discrete or continuous random variable with
mean µ := E X . The variance and standard deviation of X are dened, re-
Option Valuation: A First Course in Financial Mathematics
72
spectively, as
√
and σ(X) = VX.
V X = E(X − µ)2
Variance is a convenient measure of how much on average a random variable deviates from its mean. By linearity of expectation, we have the alternate
characterization
V X = E X 2 − 2µE X + µ2 = E X 2 − µ2 = E X 2 − E2 X,
where we have used the shorthand notation
E2 X
for
(E X)2 .
Theorem 6.4.2. (i) For real numbers α and β , V(αX + β) = α2 V X .
(ii) If X and Y are independent, then V(X + Y ) = V X + V Y .
Proof. By linearity,
E(αX + β)2 = α2 E X 2 + 2αβµ + β 2
and
2
E2 (αX + β) = (αµ + β) = α2 µ2 + 2αβµ + β 2 ,
µ = E X . Subtracting these equations proves part (i).
X and Y are independent, then, by Theorem 6.3.5,
where
If
E(X + Y )2 = E(X 2 + 2XY + Y 2 ) = E X 2 + 2(E X)(E Y ) + E Y 2 .
Also,
2
E2 (X + Y ) = (E X + E Y ) = E2 X + 2(E X)(E Y ) + E2 Y.
Subtracting these equations yields (ii).
Example 6.4.3.
E X = p = E X2
If
X
hence
Example 6.4.4.
is Bernoulli random variable with parameter
p,
then
V X = p(1 − p).
The variance of a binomial random variable
Y ∼ B(n, p)
may be calculated directly from the denition but it is easier to use the fact
that
Y
has the same distribution as a sum
Bernoulli random variables
Xj
X1 + X2 + · · · + Xn of independent
p. Then, by Theorem 6.4.2 and
with parameter
Example 6.4.3,
V Y = V X1 + V X2 + · · · + V Xn = np(1 − p).
Example 6.4.5.
Let
X1 , X2 , . . .
be a sequence of iid random variables such
−1 with probabilities p and 1 − p, respecYn = X1 + X2 + · · · + Xn . P
Then E Yn = n(2p − 1). Moreover,
n
Zj := (Xj + 1)/2 is Bernoulli and Yn = 2 j=1 Zj − n, so by Example 6.4.4
that
Xj
takes on the values 1 and
tively, and set
and Theorem 6.4.2,
The random variable
Yn
V Yn = 4np(1 − p).
may be interpreted as the position of a particle
p and one step to the left with
1 − p. The stochastic process (Yn )n≥1 is called a random walk.
moving one step to the right with probability
probability
Expectation of a Random Variable
Example 6.4.6.
Example 6.3.2,
X ∼ N (µ, σ 2 ), then Y = (X − µ)/σ ∼ N (0, 1)
E Y = 0 and E Y 2 = 1. Therefore,
If
73
hence, by
V X = V(σY + µ) = σ 2 V Y = σ 2 .
6.5 The Central Limit Theorem
The Central Limit Theorem (CLT) is one of the most important results
in probability theory. It conveys the remarkable fact that the distribution
of a (suitably normalized) sum of a large number of iid random variables is
approximately that of a standard normal random variable. It explains why
the data from so many populations exhibits a bell-shaped curve. Proofs of the
CLT may be found in standard texts on advanced probability.
Theorem 6.5.1 (Central Limit Theorem). Let X1 , X2 , . . . be a sequence of
iid
Pn random variables with mean µ and standard deviation σ , and let Yn =
j=1 Xj . Then
lim P
n→∞
Remarks 6.5.2.
Yn − nµ
√
≤x
σ n
= Φ(x).
It follows from the CLT that
Yn − nµ
√
lim P a ≤
≤ b = Φ(b) − Φ(a).
n→∞
σ n
In the special case that the random variables
p ∈ (0, 1), Yn ∼ B(n, p)
Xj
(6.1)
are Bernoulli with parameter
and (6.1) becomes
!
Yn − np
lim P a ≤ p
≤ b = Φ(b) − Φ(a).
n→∞
np(1 − p)
(see Example 6.4.3). This result is known as the
(6.2)
DeMoivre-Laplace Theorem.
One can use (6.2) to obtain the following approximation for the pmf of the
binomial random variable
Yn :
For
k = 0, 1, . . . , n,
P(Yn = k) = P(k − .5 < Yn < k + .5)
Yn − np
k + .5 − np
k − .5 − np
< √
<
=P
√
√
npq
npq
npq
k + .5 − np
k − .5 − np
≈Φ
−
Φ
.
√
√
npq
npq
(In the rst equality, we made a correction to compensate for using a continuous distribution to approximate a discrete one.) In particular, for
p = .5
we
74
Option Valuation: A First Course in Financial Mathematics
have the approximation
P(Yn = k) ≈ Φ
Example 6.5.3.
2k + 1 − n
√
n
−Φ
2k − 1 − n
√
n
.
(6.3)
Suppose we ip a fair coin 50 times. By (6.3), the probability
that the coin comes up heads 25 times (rounded to four decimal places) is
Φ
1
√
50
−Φ
−1
√
50
= .1125.
The actual probability (rounded to four decimal places) is
50
50
1
= .1123
P(X = 25) =
2
25
Expectation of a Random Variable
75
6.6 Exercises
r
1. A jar contains
w
red and
white marbles. The marbles are drawn one
Y
at a time and replaced. Let
denote the number of red marbles drawn
before the second white one. Find
EY
in terms of
r
and
w.
(Use Exam-
ples 3.5.8 and 6.1.5.)
2. Pockets of a roulette wheel are numbered 1 to 36, of which
and
18
18
are red
black. There are also green pockets numbered 0 and 00. If a $1
bet is placed on black, the gambler wins $2 (hence has a prot of $1)
if the ball lands on black, and loses $1 otherwise (similarly for red).
Suppose a gambler employs the following betting strategy: She initially
bets $1 on black. If black appears, she takes her prot and quits. If she
loses, she bets $2 on black, quitting and taking her prot of $1 if she
wins, otherwise betting $4 on the next spin. She continues in this way,
quitting if she wins a game, doubling the bet on black otherwise. She
decides to quit after the
prot is
1 − (2p)N ,
N th game, win or lose. Show that her expected
p = 20/38. What is the expected prot for a
where
roulette wheel with no green pockets? (The general betting strategy of
doubling a wager after a loss is called a
martingale.)
3. Show that the variance of a geometric random variable
p
4. A
is
q/p2 ,
X
with parameter
q = 1 − p.
where
Poisson random variable X with parameter λ > 0
has distribution
n
λ −λ
e , n = 0, 1, 2, . . . .
n!
variance of X . (Poisson random
pX (n) =
Find the expectation and
variables are
used for modeling the random ow of events such as motorists arriving
at toll booths or calls arriving at service centers.)
5. A jar contains
r
red and
w
white marbles. The marbles are drawn ran-
domly one at a time until the drawing has produced two marbles of the
same color. Find the expected number of marbles drawn if, after each
draw, the marble is (a) replaced; (b) discarded.
6. A hat contains
labeled
X
2.
a
slips of paper labeled with the number
be the number on the rst slip drawn and
second. Show that (a)
X
and
Y
A1 , A2 , . . . , An
aj ∈ R.
and
b
slips
Y
the number on the
are identically distributed, and (b)
E(XY ) 6= (E X)(E Y ).
7. Let
1
Two slips are drawn at random without replacement. Let
Show that
be independent events and set
VX =
n
X
j=1
a2j P (Aj )P (A0j ).
X =
Pn
j=1
aj IAj ,
Option Valuation: A First Course in Financial Mathematics
76
8. Let
Y
9. Let
X
be a binomial random variable with parameters
culate
10. Find
11. Let
Y
and
E|X|
X
be independent and uniformly distributed on
4XY
2
X +Y2+1
E
and
Show that
if
(n, p). Find E 2Y .
[0, 1].
Cal-
.
X ∼ N (0, 1).
Y be independent random variables with E X = E Y = 0.
E (X + Y )2 = E X 2 + E Y 2 and E (X + Y )3 = E X 3 + E Y 3 .
What can you say about higher powers?
12. Let
X
Y
and
be independent jointly continuous random variables, each
with an even density function. Show that if
13. Express the integrals (a)
of
Φ.
Rb
a
14. A positive random variable
lognormal with2 parameters
E X 2 = e2(µ+σ
15. Find
16. Let
VX
X
if
and
tributed on
17. Show that
18. Find the
X
)
eαx ϕ(x) dx
n
is odd,
Rb
and (b)
a
E (X + Y )n = 0.
eαx Φ(x) dx
in terms
X such that ln X ∼ N (µ, σ 2 ) is said to be
2
µ and σ 2 . Show that E X = eµ+σ /2 and
.
is uniformly distributed on the interval
(α, β).
Y be independent random variables with X uniformly dis[0, 1] and Y ∼ N (0, 1). Find the mean and variance of Y eXY .
E2 X ≤ E X 2 .
nth
moment of a random variable
X
with density
1 −|x|
.
2e
fX (x) =
19. Show that, in the notation of Example 3.2.5, the expectation of a hypergeometric random variable with parameters
20. Find the expected value of the random variable
(p, z, N )
Z
is
pz .
in Exercise 3.9.
21. Use the Central Limit Theorem to estimate the probability that the
number
Y
of heads appearing in 100 tosses of a fair coin is (a) exactly
50; (b) lies between 40 and 60. (A spreadsheet with a built in normal
cdf is useful here.) Find the exact probability for part (a).
22. A true-false exam has 54 questions. Use the CLT to approximate the
probability of getting a passing score of 35 or more correct answers
simply by guessing on each question.
Chapter 7
The Binomial Model
To nd an explicit expression for the value of an option one needs a concrete
mathematical model for the value of the underlying asset. In this chapter,
we construct the geometric binomial model for stock price movement and
use it to determine the value of a general European claim. An important
consequence is the Cox-Ross-Rubinstein formula for the price of a call option.
The valuation analysis in this chapter is based on the notion of self-nancing
portfolio described in Chapter 5.
7.1 Construction of the Binomial Model
Consider a (non-dividend-paying) stock
S
with current price
u with
0 < d < u.
the price changes each time period by a factor
factor
d
with probability
q := 1 − p,
where
S0
probability
such that
p
The symbols
or by a
u
and
d
are meant to suggest the words up and down, and we shall use these terms
to describe the price movement from one period to the next, even though
may be less than
1
(the prices drift downward) or
d
greater than
1
u
(the prices
drift upward).
Ω be the set of
ω = (ω1 , ω2 , . . . , ωN ), where ωn = u if the stock moves up
during the nth time period and ωn = d if the stock moves down. Thus, Ω =
Ω1 × Ω2 × · · · × ΩN , where Ωn = {u, d} represents the possible movements at
time n. Dene a probability measure Pn on Ωn by
(
p if ωn = u, and
Pn (ωn ) =
q if ωn = d.
We model the stock's random behavior as follows: Let
all sequences
Using the measures
Pn
we dene a probability measure
P
on subsets
A
of
Ω
by
X
P(A) =
ω∈A
where
ω = (ω1 , ω2 , . . . , ωN ).
P1 (ω1 )P2 (ω2 ) · · · PN (ωN ),
For example, the probability that the stock rises
the rst period and falls the next is
P1 (u)P2 (d)P3 (Ω3 ) · · · PN (ΩN ) = pq.
77
Option Valuation: A First Course in Financial Mathematics
78
An ⊆ Ωn , n = 1, 2, . . . , N ,
More generally, if
then
P(A1 × A2 × · · · × AN ) = P1 (A1 )P2 (A2 ) · · · PN (AN ).
The probability measure
movements.
P
P
is called the
(7.1)
therefore models the independence of the stock
product
of the measures
Pn .
As usual,
E
denotes
the corresponding expectation operator.
The price of the stock at time
(
Sn (ω) =
n
is a random variable
uSn−1 (ω)
dSn−1 (ω)
if
if
Sn
on
Ω
such that
ωn = u,
ωn = d.
Iterating, we see that
Sn (ω) = ωn Sn−1 (ω) = ωn ωn−1 Sn−2 (ω) = · · · = ωn ωn−1 · · · ω1 S0 .
Now let
Xj = 1
(7.2)
j th time period and Xj = 0 if
Yn := X1 + X2 + · · · + Xn counts
time period from 0 to n, hence from
if the stock goes up in the
the stock goes down. The random variable
the number of upticks of the stock in the
(7.2)
Sn = uYn dn−Yn S0 =
u Yn
d
dn S0 .
(7.3)
P, the Xj 's are independent Bernoulli random varip, hence Yn is a binomial random variable with
N
stochastic process S := (Sn )n=0 is called a geometric
Under the probability law
ables on
Ω
parameters
with parameter
(n, p).
The
binomial price process.
p
p
u3 S0
q
u2 dS0
p
u2 dS0
q
ud2 S0
p
u2 dS0
q
ud2 S0
p
ud2 S0
q
d3 S0
u2 S0
uS0
p
q
udS0
S0
q
p
udS0
dS0
q
2
d S0
FIGURE 7.1: 3-Step Binomial Tree
The Binomial Model
79
Figure 7.1 shows the possible stock movements for three time periods. The
probabilities along the edges are conditional probabilities; specically,
P(Sn = ux|Sn−1 = x) = p,
and
P(Sn = dx|Sn−1 = x) = q.
Other conditional probabilities may be found by multiplying probabilities
along edges and adding. For example,
P(Sn = udx|Sn−2 = x) = 2pq,
since there are two paths leading from vertex
Example 7.1.1.
x
to vertex
udx.
The probability that the price of the stock at time
n is larger
than its initial price is found from (7.3) by observing that
Sn > S0 ⇐⇒
If
d > 1, a
u Yn
is negative and
dn > 1 ⇐⇒ Yn > a :=
d
P(Sn > S0 ) = 1.
If
d < 1,
n ln d
.
ln d − ln u
n
X
n
X
n j n−j
P(Sn > S0 ) =
P(Yn = j) =
p q
,
j
j=m+1
j=m+1
m := bac is the greatest integer in a. For example, if n = 100, p = .5,
u = 1.2, and d = .8, then P(Sn > S0 ) ≈ .14. If u is increased to 1.25 or if p is
increased to .55, the probability goes up to .46. Similarly, if d is decreased to
.75 or p is decreased to .44, the probability goes down to .01.
where
We model the ow of stock price information by the natural ltration
S
(FnS )N
n=0 . Note that because S0 is constant, F0 = {∅, Ω}. It is easy to see that
S
for n ≥ 1 the σ -eld Fn consists of Ω, ∅, and all unions of sets of the form
Aη := {η} × Ωn+1 × · · · ΩN ,
η = (η1 , η2 , . . . , ηn ) represents a particular market scenario up through
n. For example, if N = 4, F2S is generated by the sets {η} × Ω3 × Ω4 ,
where η = (u, u), (u, d), (d, u), or (d, d).
S
The following notational convention will be convenient: If Z is a FN random variable on Ω that depends only on the rst n coordinates of ω , we will
suppress the last N −n coordinates in the notation Z(ω1 , ω2 , . . . , ωN ) and write
S
instead Z(ω1 , ω2 , . . . , ωn ). Such a random variable is Fn -measurable since the
event {Z = z} is of the form A × Ωn+1 × · · · × ΩN and hence is a union of
the sets Aη , where η ∈ A. Moreover, since Pn+1 (Ωn+1 ) = · · · = PN (ΩN ) = 1,
Remark 6.1.6 implies the following truncated form of the expectation of Z :
X
EZ =
Z(ω)P(ω)
where
time
ω∈Ω
=
X
(ω1 ,...,ωn )
=
X
(ω1 ,...,ωn )
Z(ω1 , . . . , ωn )P1 (ω1 ) · · · Pn (ωn )Pn+1 (Ωn+1 ) · · · PN (ΩN )
Z(ω1 , . . . , ωn )P1 (ω1 ) · · · Pn (ωn ).
(7.4)
Option Valuation: A First Course in Financial Mathematics
80
7.2 Pricing a Claim in the Binomial Model
H when
(Sn )N
n=0 , as
In this section, we determine the fair price of a European claim
the underlying stock
S
has a geometric binomial price process
described in the preceding section. Let B be a risk-free bond with price process
Bn = (1 + i)n . According to Theorem 5.3.3, if the binomial model is arbitragefree, then the proper value of the claim at time n is that of a self-nancing,
replicating portfolio (φ, θ) based on (B, S) with nal value H . For the time
being, however, we do not make the assumption that the model is arbitragefree.
To construct a self-nancing, replicating portfolio, we start by dening
VN = H
and then work backward, using the characterization of self-nancing
portfolio given in part (iii) of Theorem 5.2.5, namely,
Vn+1 = θn+1 Sn+1 + (1 + i)[Vn − θn+1 Sn ],
For a given
n = 0, 1, . . . , N − 1.
(7.5)
ω = (ω1 , ω2 , . . . , ωn ), Equation (7.5) evaluated at ω may be written
as a system
θn+1 (ω)Sn+1 (ω, u) + (1 + i)[Vn (ω) − θn+1 (ω)Sn (ω)] = Vn+1 (ω, u)
θn+1 (ω)Sn+1 (ω, d) + (1 + i)[Vn (ω) − θn+1 (ω)Sn (ω)] = Vn+1 (ω, d).
(Recall our convention of displaying only the time-relevant coordinates of
Ω.) The idea is to solve the system for Vn (ω) in terms of Vn+1 (ω, u)
Vn+1 (ω, d). With VN already dened as H , backward induction may be
N
used to construct a process V = (Vn ) and from it a process (θn )n=1 which
satises (7.5) and hence generates a self-nancing, replicating portfolio for H .
We begin by solving the above system for θn+1 (ω). Subtracting the equations, using Sn+1 (ω, ωn+1 ) = ωn+1 Sn (ω), we have
members of
and
θn+1 (ω) =
Vn+1 (ω, u) − Vn+1 (ω, d)
,
(u − d)Sn (ω)
Equation (7.6) is referred to as the
in the above system for
n = 0, 1, . . . , N − 1.
(7.6)
delta hedging rule. Solving the rst equation
(1+i)Vn (ω) and using the delta hedging rule, we obtain
(1 + i)Vn (ω) = Vn+1 (ω, u) + θn+1 (ω)Sn (ω)(1 + i − u)
1+i−u
u−d
1+i−d
u−1−i
= Vn+1 (ω, u)
+ Vn+1 (ω, d)
.
u−d
u−d
= Vn+1 (ω, u) + [Vn+1 (ω, u) − Vn+1 (ω, d)]
Equation (7.7) expresses
ward induction process.
Vn
uniquely in terms of
Vn+1 ,
(7.7)
completing the back-
The Binomial Model
81
V = (Vn )N
n=0 satises (7.5), where the process
N
(θn )n=1 is dened by (7.6). Now dene a process φ = (φn )N
n=1 by
By construction,
φn+1 (ω) = (1 + i)−n [Vn (ω) − θn (ω)Sn (ω)],
Since
φn
and
θn
depend only on the rst
n−1
n = 0, 1, . . . , N − 1.
time steps, the process
θ =
(7.8)
(φ, θ)
is predictable and hence is a trading strategy. Note that Equations (7.6) and
(7.8) dene
θ1
φ1
and
as constants, in accordance with the portfolio theory of
Chapter 5. From (7.5) with
n
replaced by
n − 1,
we have
Vn = θn Sn + (1 + i)[Vn−1 − θn Sn−1 ] = θn Sn + φn Bn ,
which shows that
V
n = 1, 2 . . . , N,
is the value process for the portfolio. We have constructed
a unique self-nancing, replicating strategy for the claim
H.
The above results are summarized in the following theorem:
Theorem 7.2.1. Given a European claim H in the binomial model there
exists a unique self-nancing trading strategy (φ, θ) with value process V satisfying VN = H . Furthermore, V is given by the backward recursion scheme
Vn (ω) = (1 + i)−1 [Vn+1 (ω, u)p∗ + Vn+1 (ω, d)q ∗ ] ,
ω = (ω1 , ω2 , . . . , ωn ),
where
p∗ :=
1+i−d
u−d
(7.9)
n = N − 1, N − 2, . . . , 0,
and q∗ :=
The strategy (φ, θ) is expressed in terms of V by
u−1−i
.
u−d
(7.6)
and
(7.10)
.
(7.8)
(p∗ , q ∗ ) is a probability vector i u and d satisfy
the inequalities 0 < d < 1 + i < u. In this case, we may construct a prob∗
ability measure P on Ω in exactly the same manner as P was constructed,
∗
∗
but with p and q replaced, respectively, by p and q . Denoting the corre∗
sponding expectation operator by E , we have the following consequence of
It is easy to verify that
Theorem 7.2.1:
Corollary 7.2.2. If 0 < d < 1 + i < u, then (p∗ , q∗ ) is a probability vector
and the discounted value process Ṽ = (Ṽn )N
n=0 satises
E∗ Ṽn = E∗ Ṽn+1 ,
In particular, V0 = (1 + i)−N E∗ H .
n = 0, 1, . . . , N − 1.
Option Valuation: A First Course in Financial Mathematics
82
Proof.
Using (7.4) with
P
replaced by
X
(1 + i)E∗ Vn = (1 + i)
ω=(ω1 ,...,ωn )
X
=
ω=(ω1 ,...,ωn )
and (7.9) we have
Vn (ω)P∗1 (ω1 ) · · · P∗n (ωn )
[Vn+1 (ω, u)p∗ + Vn+1 (ω, d)q ∗ ] P∗1 (ω1 ) · · · P∗n (ωn )
X
=
P∗
ω=(ω1 ,...,ωn+1 )
Vn+1 (ω)P∗1 (ω1 ) · · · P∗n+1 (ωn+1 )
= E∗ Vn+1 .
Dividing by
(1 + i)n+1
Remarks 7.2.3.
completes the proof.
(a) For
d < 1 + i < u,
the random variables
Xj
dened
in Section 7.1 are still independent Bernoulli random variables with respect
P∗ ,
Yn = X1 + X2 + · · · + Xn is a
(n, p∗ ). The probabilities p∗ and
q ∗ are called the risk-neutral probabilities and P∗ is the risk-neutral probability
measure. Note that, in contrast to the law P, which reects the perception of
∗
the market, P is a purely mathematical construct.
∗
∗
(b) Using the identity up +dq = 1+i, we see from (7.3) and independence
to the new probability measure
hence
binomial random variable with parameters
that
E∗ Sn = S0 dn E∗
Thus,
E∗ Sn
u Y n
d
= S0 dn E∗ n
u X1
d
= S0 dn
u
d
p∗ + q ∗
n
= (1 + i)n S0 .
is the time-n value of a risk-free account earning compound in-
terest at the rate
i.
A similar calculation shows that
n
E Sn = (up + dq) S0 = (1 + j)n S0 ,
where
j := up + dq − 1. Since
j > i, which
would expect that
there is risk involved in buying a stock, one
is equivalent to
p > p∗ .
Corollary 7.2.4. The binomial model is complete. Moreover, it is arbitragefree i the inequality d < 1 + i < u holds. In this case, the proper price of a
claim H is (1 + i)−N E∗ H .
Proof. Theorem 7.2.1 shows that the model is complete, and we have already
seen that the inequalities
d < 1+i < u
follow from the assumption that the
market is arbitrage-free (Example 4.1.2).
Assume that the inequalities hold. To show that the binomial model is
arbitrage-free, suppose there exists a self-nancing trading strategy with value
U = (Un ) such that U0 = 0, Un ≥ 0 for all n, and P(UN > 0) > 0. Then
{UN > 0} 6= ∅, and since P∗ (ω) > 0 for all ω it follows that E∗ UN > 0. On
the other hand, if we take H = UN in Theorem 7.2.1 then, by uniqueness, U
must be the process V constructed in that theorem hence, by Corollary 7.2.2,
E∗ UN = (1 + i)N U0 = 0. This contradiction shows that the binomial model
process
must be arbitrage-free. The last assertion of the corollary follows from Theorem 5.3.3.
The Binomial Model
SN = uYN dN −YN S0
Since
and
83
YN ∼ B(N, p∗ ),
Corollary 7.2.4 and the
law of the unconscious statistician imply the following general formula for the
price of a claim.
Corollary 7.2.5. If d < 1 + i < u and the claim H is of the form f (SN ) for
some function f (x), then the proper price of the claim is
V0 = (1 + i)
−N
∗
−N
E f (SN ) = (1 + i)
N X
N
j=0
Example 7.2.6.
For a forward we take
j
f S0 uj dN −j p∗ j q ∗ N −j .
f (x) = x−K
in Corollary 7.2.5. Since
N X
N
E f (SN ) = S0
(uj dN −j S0 − K)p∗ j q ∗ N −j
j
j=0
∗
N N X
X
N
N ∗ j ∗ N −j
∗ j
∗ N −j
= S0
(up ) (dq )
−K
p q
j
j
j=0
j=0
= S0 (up∗ + dq ∗ )N − K(p∗ + q ∗ )N
= S0 (1 + i)N − K,
V0 = S0 − K(1 + i)−N . Recalling that there is no cost to enter
N
contract we have K = S0 (1 + i) . This is the discrete-time analog of
we see that
the
Equation (4.3), which was obtained by a general arbitrage argument.
7.3 The Cox-Ross-Rubinstein Formula
To apply the results of Section 7.2 to call options, dene a function
Ψ(m, N, p̃) =
N X
N
j=m
j
p̃j (1 − p̃)N −j ,
0 < p̃ < 1,
Ψ
by
m = 0, 1, 2, . . . N.
Ψ(m, N, p̃) = 1 − Θ(m − 1, N, p̃), where Θ( · , N, p̃) is the cdf of a bi(N, p̃). The following result expresses
of a call option in terms of Ψ.
Note that
nomial random variable with parameters
the cost
Theorem 7.3.1 (Cox-Ross-Rubinstein (CRR) Formula). If d < 1 + i < u,
then the cost C0 of a call option with strike price K to be exercised after N
time steps is
C0 = S0 Ψ(m, N, p̂) − (1 + i)−N KΨ(m, N, p∗ ),
(7.11)
Option Valuation: A First Course in Financial Mathematics
84
where
p̂ :=
p∗ u
1+i
q̂ := 1 − p̂ =
,
q∗ d
1+i
,
(7.12)
and m is the smallest integer ≥ 0 for which S0 um dN −m > K . Specically
m := (bac + 1)+ ,
a :=
ln (K) − ln (dN S0 )
.
ln u − ln d
If m > N (which occurs i K ≥ uN S0 ), the right side of
as zero.
Proof.
By Corollary 7.2.5 applied to the function
C0 = (1 + i)−N
N X
N
j=0
If
j
(7.11)
is interpreted
f (x) = (x − K)+ ,
(S0 uj dN −j − K)+ p∗ j q ∗ N −j .
S0 uN ≤ K , then S0 uj dN −j −K ≤ S0 uj uN −j −K ≤ 0 for all j
hence
(7.13)
C0 = 0.
Accordingly, the right side of (7.11) is interpreted as zero.
S0 uN > K . Then there must be a smallest integer m
m N −m
in the set {0, 1, . . . , N } for which S0 u d
> K . Moreover, since u/d > 1,
j N −j
j N −j
S0 u d
is increasing in j hence S0 u d
> K for j ≥ m. It follows that
Now assume that
N X
N
C0 = (1 + i)−N
j=m
S0 uj dN −j − K p∗ j q ∗ N −j
j
N N ∗ j ∗ N −j
X
X
q d
N ∗ j ∗ N −j
N
p u
−N
− (1 + i) K
p q
= S0
j
1
+
i
1
+
i
j
j=m
j=m
= S0
N X
N
j=m
The inequality
j
j N −j
p̂ q̂
−N
− (1 + i)
N X
N ∗ j ∗ N −j
K
p q
.
j
j=m
u > 1 + i implies that p̂ < 1. Since p̂ + q̂ = 1, (7.11) follows.
Example 7.3.2.
C0 of a call, as calculated by the
P0 = C0 −S0 +(1+i)−N K of the corresponding put
(put-call parity formula) for various values of K , u, and d, where S0 = $20.00
and the nominal rate is taken to be r = .10. We consider daily uctuations
of the stock so i = .10/365 ≈ .00027. The options are assumed to expire
in 90 days. The table shows that C0 typically increases as the spread u − d
(volatility of the stock price) increases. The table also shows that C0 decreases
as K increases. This is clear from (7.13) and is to be expected, as a larger K
Table 7.1 gives the price
CRR formula, and the price
results in a smaller payo for the holder, making the option less attractive.
The Binomial Model
K
u
d
$18.00
1.1
.9
$18.00
1.5
.9
$18.00
1.1
.4
$18.00
1.5
.4
C0
85
P0
K
u
d
$8.15
$5.71
$22.00
1.1
.9
$6.87
$8.34
$13.89
$11.45
$22.00
1.5
.9
$13.25
$14.71
$16.99
$14.55
$22.00
1.1
.4
$16.63
$18.10
$19.92
$17.49
$22.00
1.5
.4
$19.92
$21.38
TABLE 7.1: Variation of
C0
and
P0
with
C0
K , u,
and
P0
d.
Option Valuation: A First Course in Financial Mathematics
86
7.4 Exercises
1. Prove Equation 7.1.
2. Let
η ∈ Ω1 × Ω2 × . . . × Ω n .
Show that
P({η} × Ωn+1 × Ωn+2 × . . . × ΩN ) = pYn (η) q n−Yn (η) .
n = 1, 2, . . . , N , dene a
Zn (ω) = ωn . Show that
3. For
by
random variable
Zn
in the binomial model
Zn = (u − d)Xn + d;
(a)
Sn = Zn Zn−1 · · · Z1 S0 ;
(b)
Xn = (Sn − dSn−1 )/(u − d)Sn−1 .
(c)
Conclude from (c) that
FnS = FnX .
4. Find the probability (in terms of
n
and
p)
that the price of the stock
in the binomial model goes down at least twice during the rst
n
time
periods.
For the remaining exercises, assume that 0 < d < 1 + i < u.
5. In the one-step binomial model, the Cox-Ross-Rubinstein formula reduces to
C0 = (1 + i)−1 (S0 u − K)+ p∗ + (S0 d − K)+ q ∗ .
(a) Show that if
S0 d < K < S0 u
then
∂C0
>0
∂u
Conclude that for
spread
u−d
u > K/S0
∂C0
< 0.
∂d
and
and
d < K/S0 , C0
C0 , as a function of (u, d),
u > (1 + i) > d ≥ K/S0 .
(b) Show that
values
increases as the
increases.
is constant for the range of
d = 1/u and p = .5. Show that
(
1/2 h
if n is
−n i
P(Sn > S0 ) = P(Sn < S0 ) =
n
(1/2) 1 − n/2 2
if n is
6. Suppose in Example 7.1.1 that
7. Let
S0 = $50.00, r = .12, u = 1.1,
and
d = .9,
odd
even.
and let the length of a
time interval be one day. Find the prices of call and put options that
expire in 90 days with strike price (a)
K = $54.00, (b) K = $47.00. (Use
a spreadsheet with a built-in binomial cdf.)
The Binomial Model
87
8. Find the probability that, in the binomial model, a call option nishes
in the money.
9. A
k -run
of upticks
is a sequence of
k
consecutive stock price increases
not contained in any larger such sequence. Show that if
then the probability of a
k -run
of upticks in
N
N/2 ≤ k < N
time periods is
pk [2q + (N − k − 1)q 2 ].
Cash-or-nothing call option ).
10. (
that the cost
V0
Let
A
be a xed amount of cash. Show
of a claim with payo
AI(K,∞) (SN )
is
(1 + i)−N AΨ(m, N, p∗ ),
where
m
is dened as in Theorem 7.3.1.
Asset-or-nothing call option ).
11. (
payo
SN I(K,∞) (SN )
is
Show that the cost
S0 Ψ(m, N, p̂),
where
m
and
V0 of a claim with
p̂ are dened as in
Theorem 7.3.1.
12. Show that a portfolio long in an asset-or-nothing call option (Exercise 11) and short in a cash-or-nothing call option with cash
K
(Ex-
ercise 10) has the same time-n value as a portfolio with a long position
in a call option with strike price
13. Let
is
K.
1 ≤ M < N . Show that the cost V0 of a claim with payo (SN −SM )+
S0 Ψ(k, L, p̂) − (1 + i)−L Ψ(k, L, p∗ ),
where
L := N − M,
and
p̂
k := (bac + 1)+ ,
V0 =
S02
v
1+i
SN (SN − K)
is
N
− KS0 ,
v = (u + d)(1 + i) − ud.
15. Show that the cost of a claim with payo
V0 =
S02
v
1+i
SN (SN − K)+
is
N
Ψ(m, N, p̃) − KS0 Ψ(m, N, p̂),
m, and p̂ are dened as in Theorem 7.3.1, v = (u + d)(1 + i) − ud,
p̃ = p∗ u2 /v .
where
and
L ln d
,
ln d − ln u
is dened as in Theorem 7.3.1.
14. Show that the cost of a claim with payo
where
a :=
Option Valuation: A First Course in Financial Mathematics
88
16. Use Exercises 14 and 15 and the law of one price to nd the cost
a claim with payo
SN (K − SN )+ .
17. Show that price of a claim with payo
is
V0 =
V0
of
f (Sm , Sn ), where 1 ≤ m < n ≤ N ,
m n−m+j m n − m ∗ k ∗ n−k
1 X X
f (S0 uj dm−j , S0 uk dn−k ),
p q
aN j=0
j
k−j
k=j
where
a := (1 + i).
18. Referring to Example 7.1.1 with
n = 100, p = .5, and d = .8, use a
u that results in P(Sn > S0 ) ≈
spreadsheet to nd the smallest value of
.85.
show that if
N
1
2 (S1 + SN ) − K
is suciently large, specically,
19. Consider a claim with payo
uN −1 >
+
. Use Exercise 17 to
2K − S0 d
,
S0 d
then there exist nonnegative integers
k1
and
k2
less than
N
such that
the price of the claim is
V0 =
where
(S0 d − 2K)q ∗
S0 dq ∗
Ψ(k1 , N − 1, p∗ ) +
Ψ(k1 , N − 1, p̂)
N
2(1 + i)
2
(S0 u − 2K)p∗
S0 up∗
∗
+
Ψ(k
,
N
−
1,
p
)
+
Ψ(k2 , N − 1, p̂),
2
2(1 + i)N
2
p̂
k1 is the smallest
S0 d + S0 uk dN −k > 2K , and k2 is the
k such that S0 u + S0 uk+1 dN −k−1 > 2K .
is dened as in Theorem 7.3.1. Show that
nonnegative integer
k
such that
smallest nonnegative integer
Chapter 8
Conditional Expectation and
Discrete-Time Martingales
Conditional expectation generalizes the notion of expectation by taking into
account the information provided by a given
σ -eld.
The most important
application of this notion is in the denition and construction of martingales.
In this chapter, we develop the theory of conditional expectation and discretetime martingales on a nite probability space. These ideas will be applied in
the next chapter to place the binomial model in a broader context, leading to
the formulation of more general option valuation models.
We assume throughout the chapter that
the
for
σ -eld of all subsets
all ω ∈ Ω. Note that
of
Ω,
and
P
any real function
As usual, the expectation of
X
Ω
X
on
with respect to
F is
P(ω) > 0
is a nite sample space,
is a probability measure with
P
Ω
is an
F -random variable.
E X.
is denoted by
8.1 Denition of Conditional Expectation
In Example 2.3.2, we observed that a partition
consisting of
asserts that
∅
P of Ω generates a σ -eld
P . The following lemma
and all nite unions of members of
every σ-eld of subsets of Ω arises in this way.
Lemma 8.1.1. Let G be a σ-eld of subsets of Ω. Then G is generated by a
partition of Ω.
Proof.
Let B1 , B2 , . . . , Bm be the distinct members of G . For each m-tuple
m
= (1 , 2 , . . . m ) with j = ±1, dene B = B11 B22 · · · Bm
, where
(
Bj if j = 1,
j
Bj =
Bj0 if j = −1.
B are pairwise disjoint since two distinct 's will dier in some
j , and the intersection of the corresponding B 's will be contained
0
in Bj Bj = ∅. Some of the sets B are empty, but every Bj is a union of those
B for which j = 1. Denoting the nonempty B 's by A1 , A2 , . . . , An , we see
that G is generated by the partition P = {A1 , A2 , . . . , An }.
The sets
coordinate
89
Option Valuation: A First Course in Financial Mathematics
90
Remark 8.1.2.
(Fn )N
n=1 is a ltration of σ -elds on Ω such that Fn
is generated by the partition {An,1 , An,2 , . . . , An,mn }. Since Fn ⊆ Fn+1 , each
An,j is a union of some of the sets An+1,1 , An+1,2 , . . ., An+1,mn+1 . Therefore,
we can assign to each outcome ω ∈ Ω a unique sequence (j1 , j2 , . . . , jN ) with
the property that ω ∈ An,jn for each n. This provides a dynamic interpretation
Suppose
of an abstract experiment. (For a coin toss, the sequence is equivalent to a
sequence of heads and tails; see Example 5.1.4.) Figure 8.1 illustrates the idea
for the case
N = 3.
ω
ω
A12
A11
FIGURE 8.1:
A22
A32
A23
A33
ω
A24
A21
A31
ω
described by the sequence
A34
A35
(2, 4, 5)
Lemma 8.1.3. Let G be a σ-eld of subsets of Ω and let X and Y be G random variables on Ω. If {A1 , A2 , . . . , An } is a generating partition for G ,
then
(i)
X ≥ Y ⇐⇒ E(XIAj ) ≥ E(Y IAj )
(ii)
X = Y ⇐⇒ E(XIAj ) = E(Y IAj )
Proof.
for all j ; and
for all j .
The necessity of (i) is clear and (ii) follows from (i). To prove the
Z = X − Y and assume that E(ZIAj ) ≥ 0 for all j . We
A = {Z < 0} is empty.
Suppose to the contrary that A 6= ∅. Since A ∈ G there exists a subset
S
J ⊆ {1, 2, . . . , n} such that A = j∈J Aj . Since the sets Aj are pairwise
suciency of (i), set
show that the set
disjoint,
E(ZIA ) =
X
j∈J
On the other hand, by denition of
E(ZIA ) =
X
E(ZIAj ) ≥ 0.
A,
Z(ω)IA (ω)P(ω) =
ω∈Ω
This contradiction shows that
X
Z(ω)P(ω) < 0.
ω∈A
A
must be empty and hence that
X ≥Y.
We are now in a position to prove the existence and uniqueness of conditional expectation.
Conditional Expectation and Discrete-Time Martingales
91
Theorem 8.1.4. If X is a random variable on (Ω, F, P) and G is a σ-eld
contained in F , then there exists a unique G -random variable Y such that
E (IA Y ) = E (IA X)
Proof.
for all A ∈ G.
{A1 , A2 , . . . , Am } be a partition
G -random variable Y on Ω by
Let
Dene a
m
X
Y =
aj IAj ,
aj :=
j=1
Since
Aj Ak = ∅
for
Ω
of
generating
(8.1)
G
(Lemma 8.1.1).
E(IAj X)
.
P(Aj )
j 6= k ,
IAk Y =
m
X
aj IAj IAk = ak IAk
j=1
so that
E (IAk Y ) = ak P(Ak ) = E(IAk X).
Since any member of
G
is a disjoint union of sets
Ak ,
(8.1) holds. Uniqueness
follows from Lemma 8.1.3.
Denition 8.1.5. The G -random variable Y of Theorem 8.1.4 is called the
conditional expectation of X given G and is denoted by E(X|G). In the special
case that G = σ(X1 , X2 , . . . , Xn ), E(X|G) is called the conditional expectation
of X given X1 , X2 , . . . , Xn and is denoted by E(X|X1 , X2 , . . . , Xn ).
Corollary 8.1.6. If X , X1 , . . ., Xn are random variables on (Ω, F, P), then
there exists a function g (x1 , x2 , . . . , xn ) such that
X
E(X|X1 , X2 , . . . , Xn ) = gX (X1 , X2 , . . . , Xn ).
Proof. σ(X1 , X2 , . . . , Xn )
is generated by the partition consisting of sets of
the form
A(x) = {X1 = x1 , X2 = x2 , . . . , Xn = xn },
Dene

 E(IA(x) X)
P(A(x))
gX (x) =

0
if
x := (x1 , x2 , . . . , xn ).
(8.2)
A(x) 6= ∅,
otherwise.
From the proof of Theorem 8.1.4, we have
E(X|X1 , X2 , . . . , Xn ) =
X
gX (x)IA(x) .
x
If
ω ∈ A(x),
then
x = (X1 (ω), X2 (ω), . . . , Xn (ω))
hence
E(X|X1 , X2 , . . . , Xn )(ω) = gX (x)IA(x) (ω) = gX (X1 (ω), X2 (ω), . . . Xn (ω)).
Since
Ω
is the union of the sets
A(x),
the equation holds for all
ω ∈ Ω.
Option Valuation: A First Course in Financial Mathematics
92
8.2 Examples of Conditional Expectation
The conditional expectation operator averages the values of a random vari-
X
able
taking into account information provided by a
viewed as the best prediction of
X
given
this idea.
G.
σ -eld G .
It may be
The following examples illustrate
Example 8.2.1.
Let G = {∅, Ω}. Since, obviously, E(IA E X) = E(IA X) if
A = ∅ or A = Ω, E(X|G) = E(X). Thus, the best prediction of X given
the information G , which is to say no information at all, is simply the expected
value of X .
either
Example 8.2.2.
E(X|G) = X :
If
G is the σ -eld consisting of all subsets of Ω, then, trivially,
X given all possible information is X
the best prediction of
itself.
Example 8.2.3.
Toss a coin N times and observe the outcome heads H or
T on each toss. Let p be the probability of heads on a single toss and
set q = 1 − p. The sample space for the experiment is Ω = Ω1 × Ω2 · · · × ΩN ,
where Ωn = {H, T } is the set of outcomes of the nth toss. The probability
tails
law for the experiment may be expressed as
P(ω) = pH(ω) q T (ω) , ω = (ω1 , ω2 , . . . , ωN ),
where
H(ω) denotes the number of heads
n < N . For ω ∈ Ω we shall write
ω
in
and
T (ω)
the number of tails.
Fix
ω = (ω 0 , ω 00 ),
Let
Gn
denote the
ω 0 ∈ Ω 1 × Ω 2 · · · × Ωn ,
σ -eld
ω 00 ∈ Ωn+1 × · · · × ΩN .
generated by the sets
Aω0 = {(ω 0 , ω 00 ) | ω 00 ∈ Ωn+1 × · · · × ΩN }.
Gn
represents the information generated by the rst
claim that for any random variable
n
(8.3)
tosses of the coin. We
X,
E(X|Gn )(ω) = E(X|Gn )(ω 0 , ω 00 ) =
X
pH(η) q T (η) X(ω 0 , η),
(8.4)
η ∈ Ωn+1 × · · · × ΩN .
Equation
η
where the sum on the right is taken over all
(8.4) asserts that the best prediction of
by the rst
n
X
given the information provided
tosses (the known) is the average of
X
over the remaining
outcomes (the unknown).
Y (ω) = Y (ω 0 , ω 00 ) and
n tosses, Y is Gn -measurable. It
To verify (8.4), denote the sum on the right by
note that, since
Y
depends only on the rst
Conditional Expectation and Discrete-Time Martingales
93
E(Y IA ) = E(XIA ) for the sets A = Aω0
00
00
pH(ω ) q T (ω ) = 1 we have
therefore suces to show that
in (8.3). Noting that
E(IA Y ) =
P
ω 00
X
dened
Y (ω)P(ω)
ω∈A
=
X
0
0
00
Y (ω 0 , ω 00 )pH(ω ) q T (ω ) pH(ω ) q T (ω
00
)
ω 00
0
0
= pH(ω ) q T (ω )
X
H(ω 0 ) T (ω 0 )
X
00
pH(ω ) q T (ω
00
)
ω 00
=p
q
X
pH(η) q T (η) X(ω 0 , η)
η
H(η) T (η)
p
q
X(ω 0 , η)
η
=
X
p
H(ω) T (ω)
q
X(ω)
ω∈A
= E(IA X),
as required.
Example 8.2.4.
natural ltration
function
f (x),
E
Let
U
Consider the geometric binomial price process
(FnS ). We show that for 0 ≤ n < m ≤ N
f (Sm )|FnS
k X
k j k−j
p q
f uj dk−j Sn ,
=
j
j=0
denote the sum on the right. Since
suces to show that
assume that
A
E (f (Sm )IA ) = E(U IA )
U
S
with its
and any real-valued
k := m − n.
FnS -measurable it
S
Fn . For this, we may
is obviously
for all
is of the form
A∈
A = {η} × Ωn+1 × · · · × ΩN ,
where
η ∈ Ω 1 × · · · × Ωn ,
since these sets generate
FnS .
Noting that
P(A) = P1 (η1 )P2 (η2 ) · · · Pn (ηn )
and
X
(ωm+1 ,...,ωN )
Pm+1 (ωm+1 )Pm+1 (ωm+1 ) · · · PN (ωN ) = 1
we have
E [f (Sm )IA ] =
X
f (Sm (ω)) P(ω)
ω∈A
=
X
ω∈A
f (ωn+1 · · · ωm Sn (ω)) P(ω)
= P(A)
X
f (ωn+1 · · · ωm Sn (η)) Pn+1 (ωn+1 ) · · · Pm (ωm ),
Option Valuation: A First Course in Financial Mathematics
94
where the sum in the last equality is taken over all sequences
Collecting
together
(ωn+1 , . . . , ωm )
all
terms
in
the
last
sum
for
(ωn+1 , . . . , ωm ).
which
the
sequence
j u's, j = 0, 1, . . . , k , we see that
k X
k j k−j
E (f (Sm )IA ) = P(A)
p q
f uj dk−j Sn (η) = P(A)U (η).
j
j=0
has exactly
j ≤ k,
X
E IA f (uj dk−j Sn ) =
f (uj dk−j Sn (ω))P(ω) = P(A)f (uj dk−j Sn (η))
Similarly, for each
ω∈A
hence
E(IA U ) = P(A)
k X
k
j=0
Therefore,
j
pj q k−j f uj dk−j Sn (η) = P(A)U (η).
E (f (Sm )IA ) = E(IA U ),
as required.
8.3 Properties of Conditional Expectation
In the proofs of the following theorems, we rely on the fact that a
variable
Y
is the conditional expectation of
E(Y IA ) = E(XIA )
X
with respect to
for all
G
i
G -random
A ∈ G.
The rst theorem shows that conditional expectation has properties similar
to those of ordinary expectation.
Theorem 8.3.1. Let X and Y be random variables on (Ω, F, P) and let G be
a σ-eld contained in F . Then
(i) (unit property) E(1|G) = 1;
(ii) (linearity) E(αX + βY |G) = αE(X|G) + βE(Y |G), α, β ∈ R;
(iii) (order property) X ≤ Y =⇒ E(X|G) ≤ E(Y |G); and
(iv) (absolute value property) |E(X|G)| ≤ E(|X||G).
Proof. We leave the proof of (i) to the reader. For (ii), let Z denote the
G -random
tation,
variable
αE(X|G) + βE(Y |G)
and let
A ∈ G.
By linearity of expec-
E(IA Z) = αE [IA E(X|G)] + βE [IA E(Y |G)]
= αE(IA X) + βE(IA Y )
= E [IA (αX + βY )] ,
Conditional Expectation and Discrete-Time Martingales
verifying (ii). Property (iii) follows from Lemma 8.1.3 since for
95
A ∈ G,
E [IA E(X|G)] = E(IA X) ≤ E(IA Y ) = E [IA E(Y |G)] .
Part (iv) follows from
±E(X|G) = E(±X|G) ≤ E(|X|G).
The next theorem shows that known factors may be moved outside the
conditional expectation operator.
Theorem 8.3.2 (Factor Property). Let X and Y be random variables on
(Ω, F, P) and let G be a σ -eld contained in F . If X is a G -random variable,
then E(XY |G) = XE(Y |G). In particular, E(X|G) = X .
Proof.
XE(Y |G) is G -measurable it suces to
E [IA XE(Y |G)] = E(IA XY ) for all A ∈ G . Let the range of X
{x1 , x2 , . . . , xn } and set Aj = {X = xj }. Then Aj ∈ G and
Since the random variable
show that
be
IA X =
n
X
xj IAAj .
j=1
By linearity,
E [IA XE(Y |G)] =
n
X
j=1
n
X
xj E IAAj E(Y |G) =
xj E(IAAj Y ) = E(IA XY ),
j=1
which veries the rst assertion of the theorem. The last assertion follows by
taking
Y = 1.
The next theorem shows that if the information provided by
dent of that provided by
as when
no
X
then the best predictor of
X
information is given.
given
G is indepenG is the same
Theorem 8.3.3 (Independence Property). Let X be a random variable on
(Ω, F, P) and G a σ -eld contained in F . If X is independent of G , that is, if
X and IA are independent for all A ∈ G , then E(X|G) = E(X).
Proof. Obviously E X is G -measurable, and by independence
E(IA X) = (E IA )(E X) = E[IA E(X)]
for all
A ∈ G.
The following theorem asserts that successive predictions of
X
based on
nested levels of information produce the same result as a single prediction
using the least information.
Theorem 8.3.4 (Iterated Conditioning Property). Let X be a random variable on (Ω, F, P) and let G and H be σ-elds with H ⊆ G ⊆ F . Then
E [E(X|G)|H] = E(X|H).
Option Valuation: A First Course in Financial Mathematics
96
Proof.
Let
Y = E(X|G).
We need to show that
E [IA E(Y |H)] = E(IA X)
E(Y |H) = E(X|H),
that is,
A ∈ H.
for all
But, by the dening property of conditional expectation with respect to
the last equation is simply
E(IA Y ) = E(IA X),
H,
which holds by the dening
property of conditional expectation with respect to
G.
8.4 Discrete-Time Martingales
Denition 8.4.1. A stochastic process (Mn ) = (Mn )Nn=0 on (Ω, F, P) adapted
to a ltration (Fn )N
n=0 is said to be a P, (Fn ) -martingale if
E(Mn+1 |Fn ) = Mn , n = 0, 1, . . . , N − 1.1
(8.5)
If there is no possibility ofambiguity we will drop one or both of the components of the prex P, (Fn ) . For the special case Fn = σ(M0 , . . . Mn ), we will
omit reference to the ltration.
Remarks 8.4.2.
(a) Since
Mn
is
Fn -measurable
E(Mn+1 − Mn |Fn ) = 0,
we can write (8.5) as
n = 0, 1, . . . , N − 1.
This has the following gambling interpretation: Let
of a gambler on the
nth
Mn
represent the winnings
N plays. A fair game
Mn+1 − Mn on the next play,
the rst n plays, is zero.
play of a game consisting of
requires that the best prediction of the gain
based on the information obtained during
(b) By (8.5) and the iterated conditioning property, martingales satisfy
the following
multistep property :
E(Mm |Fn ) = Mn , 0 ≤ n ≤ m ≤ N.
Example 8.4.3.
variables on
Let
(Ω, F, P)
X0 , X1 , . . . , XN
be a sequence of independent random
with mean 1 and set
and independence properties,
Mn = X0 X1 · · · Xn .
By the factor
E(Mn+1 − Mn |M0 , M1 , . . . , Mn ) = Mn E(Xn+1 − 1|M0 , M1 , . . . , Mn )
= Mn E(Xn+1 − 1)
= 0.
Therefore,
1A
(Mn )
is a martingale.
martingale may begin at indices other than
n = 0.
Conditional Expectation and Discrete-Time Martingales
Example 8.4.4.
ables on
Let
(Ω, F, P)
97
X1 , . . . , XN be a sequence of independent random varip and set Mn = X1 + X2 + · · · + Xn − np. By
with mean
the independence property,
E(Mn+1 − Mn |M1 , . . . , Mn ) = E(Xn+1 − p|M1 , . . . , Mn )
= E(Xn+1 − p)
= 0,
hence
(Mn )
is a martingale.
Example 8.4.5.
a ltration on
Ω.
Let
X
Dene
(Ω, F, P)
Mn = E(X|Fn ), n = 0, 1, . . . , N .
be a random variable on
and let
(Fn )
be
By the iterated
conditioning property,
E(Mn+1 |Fn ) = E [E(X|Fn+1 )|Fn ] = E(X|Fn ) = Mn .
Therefore,
(Mn )
is an
(Fn )-martingale.
The following theorem asserts that reducing the amount of information
provided by a ltration preserves the martingale property. (The same is not
necessarily true if information is
increased.)
Theorem 8.4.6. Let Gn and Fn be ltrations with Gn ⊆ Fn ⊆ F , n =
0, 1, . . . , N . If (Mn ) is adapted to (Gn ) and is an (Fn )-martingale, then it is
also a (Gn )-martingale. In particular, an (Fn )-martingale M = (Mn ) is an
(FnM )-martingale.
Proof.
This follows from the factor and iterated conditioning properties:
E(Mn+1 |Gn ) = E [E(Mn+1 |Fn )|Gn ] = E(Mn |Gn ) = Mn .
The proof of the next theorem is left to the reader.
Theorem 8.4.7. If (Mn ) and (Mn0 ) are (Fn )-martingales and α, α0 ∈ R, then
(αMn + α0 Mn0 ) is a (Fn )-martingale.
Option Valuation: A First Course in Financial Mathematics
98
8.5 Exercises
1. Let
X
and
Y
be discrete random variables. For
g(x) =
X pX,Y (x, y)
pX (x)
y
where the sum is taken over all
g(x)
y
for which
may be dened arbitrarily. Show that
pX (x) > 0,
dene
y,
pX,Y (x, y) > 0. If pX (x) = 0,
E(Y |X) = g(X).
G be a σ -eld contained in F , X a G -random variable, and Y an
F -random variable independent of G . Show that E(XY ) = E(X)E(Y ).
2. Let
Hint: Condition on G .
G be a σ -eld contained in F , X a G -random variable, and Y an
F -random variable with X − Y independent of G . Show that, if either
E X = 0 or E X = E Y , then
3. Let
E(XY ) = E X 2
and
E(Y − X)2 = E Y 2 − E X 2 .
4. Prove Theorem 8.4.7.
5. Verify the multistep property of Remark 8.4.2(b).
6. Show that if
(Mn ) is a martingale then E(Mn ) = E(M0 ), n = 1, 2, . . . , N .
X = (Xn ) be a sequence of independent random variables on
(Ω, F, P) with mean 0 and variance σ 2 , and let Yn := X1 +X2 +· · ·+Xn .
7. Let
Show that
Mn := Yn2 − nσ 2 ,
n = 1, 2, . . . , N
X
denes an (Fn )-martingale.
X = (Xn )
(Ω, F, P) with
8. Let
and set
be a sequence of independent random variables on
P(Xn = 1) = p and P(Xn = −1) = q := 1 − p,
Pn
Yn = j=1 Xj . For a > 0 dene
Mn := eaYn pea + qe−a
Show that
(Mn )
is an
−n
,
(FnX )-martingale.
(Xn ), (Yn ) and (Fn ) be as in Exercise
Y
X
Show that r n is an (Fn )-martingale.
9. Let
n = 1, 2, . . . N.
8,
0 < p < 1,
and
r := qp−1 .
Conditional Expectation and Discrete-Time Martingales
99
(Xn ) and (Yn ) be sequences of independent random variables on
(Ω, F, P), each adapted to a ltration (Fn ), such that, for each n ≥ 1,
Xn and Yn are independent of each other and also of Fn−1 , where F0 =
{∅, Ω}. Suppose also that E(Xn ) = E(Yn ) = 0 for all n. Set
10. Let
An = X1 + X2 + · · · + Xn
Show that
(An Bn )
is an
Bn = Y1 + Y2 + · · · + Yn .
and
(Fn )-martingale.
N
N
(An )N
n=0 and (Bn )n=0 be (Fn )n=0 -martingales on (Ω, F, P) and
2
Cn = An − Bn . Show that
E (Am − An )2 |Fn = E [Cm − Cn |Fn ] , 0 ≤ n ≤ m ≤ N.
11. Let
let
12. Let
(Xn ) be a sequence of independent Bernoulli random variables with
p and set Yk = X1 + X2 + · · · + Xk . For all cases nd
E(Ym |Yn ), (b) E(Xj |Yn ), (c) E(Xk |Xj ), and (d) E(Yn |Xj ).
parameter
(a)
Hint: For (a) and m < n, use Exercises 1, 3.12, and 6.19.
13. Let
(Mn )
be an
(Fn )-martingale
on
(Ω, F, P).
E[(Mn − Mm )Mk ] = 0,
14. (Doob decomposition). Let
(Xn )N
n=0
A0 = 0,
(Fn )N
n=0
Show that
0 ≤ k ≤ m ≤ n.
be a ltration on
an adapted process. Dene
and
and
An = An−1 + E(Xn − Xn−1 |Fn−1 ), n = 1, 2, . . . , N.
Show that, with respect to
a martingale.
(Ω, F, P)
(Fn ), (An )
is predictable and
(Xn − An )
is
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Chapter 9
The Binomial Model Revisited
In this chapter, we give a martingale interpretation of the main results of
Section 7.2. This will suggest an approach to option pricing that can be applied
to general nite models. We also determine the proper price of an American
claim, describe optimal strategies for both the writer and the holder of an
American put, and consider the eect of dividends in the binomial model.
9.1 Martingales in the Binomial Model
Recall that in the binomial model the security
u
to go up by a factor of
probability
q = 1 − p.
with probability
The price of
S
p
at time
S
is assumed each period
or down by a factor of
n
Sn = S0 uYn dn−Yn = S0 dn
d
with
is given by
u Y n
d
,
(9.1)
Yn := X1 + X2 + · · · + Xn , the Xj 's independent Bernoulli random
p on the probability space (Ω, P, F) constructed in
S N
Section 7.1. We model the ow of information by the ltration (Fn )n=0 , where
where
variables with parameter
FnS = σ(S0 , S1 , . . . , Sn ) = σ(X1 , . . . , Xn ) = FnX .
(See Exercise 7.3.) All martingales considered in this chapter are relative to
S
FN
= F , the σ -eld of all subsets of Ω.
throughout that 0 < d < 1 + i < u, where i is
this ltration. Note that
We assume
the interest
rate per period. By Corollary 7.2.4, this is equivalent to the property that the
binomial model is arbitrage-free. As in Section 7.2,
probability measure on
Ω
denotes the risk-free
dened by the probability vector
(p∗ , q ∗ ) :=
1+i−d u−1−i
,
u−d
u−d
Recall that for a stochastic process
dened by
P∗
X̃n = (1 + i)−n Xn .
(Xn ),
.
the discounted process
X̃
is
The following two theorems provide the key
connection between Chapters 7 and 8.
101
Option Valuation: A First Course in Financial Mathematics
102
Theorem 9.1.1. The discounted stock price process
martingale.
Proof.
Let
v = u/d.
Since
Sn+1 = dSn v Xn+1 ,
(S̃n )N
n=0
is a P∗ -
the factor and independence
properties of conditional expectation imply that
E∗ (Sn+1 |FnS ) = dSn E∗ v Xn+1 |FnS
= dSn E∗ v Xn+1
= d (p∗ v + q ∗ ) Sn
= (1 + i)Sn .
(1 + i)n+1 ,
Dividing by
we obtain the martingale property
S̃n .
Recall that the value process
process
(φ, θ)
V = (Vn )N
n=0
E∗ (S̃n+1 |FnS ) =
of a self-nancing portfolio
satises
Vn+1 = θn+1 Sn+1 + (1 + i)[Vn − θn+1 Sn ],
n = 0, 1, . . . , N − 1
(9.2)
(Theorem 5.2.5). Moreover, the binomial model is complete: given any contingent claim
V
H
such that
there exists a (unique) self-nancing portfolio with value process
VN = H
(Corollary 7.2.4).
Theorem 9.1.2. The discounted value process (Ṽn
self-nancing portfolio is a P∗ -martingale. Thus,
of a
:= (1 + i)−n Vn )N
n=0
Vn = (1 + i)n−m E∗ Vm |FnS , 0 ≤ n ≤ m ≤ N.
(9.3)
In particular, the time-n value of a contingent claim H is
Vn = (1 + i)n−N E∗ H|FnS , 0 ≤ n ≤ N.
Proof.
It is clear from the denition of value process that
ltration
(FnS ).
Ṽ
is adapted to the
Moreover, from (9.2), the predictability of the process
θ
and
Theorem 9.1.1, we have
E∗ (Vn+1 |FnS ) = θn+1 E∗ (Sn+1 |FnS ) + (1 + i)(Vn − θn+1 Sn )
= θn+1 (1 + i)Sn + (1 + i)(Vn − θn+1 Sn )
= (1 + i)Vn .
Dividing by
(1 + i)n+1
shows that
Ṽ
is a martingale. Equation (9.3) follows
from the multistep property, and the last assertion of the theorem is a consequence of (9.3) and Theorem 5.3.3.
Combining Theorem 9.1.2 and Example 8.2.4, we have the following result,
which will be needed in Section 9.3.
The Binomial Model Revisited
103
Corollary 9.1.3. Let 0 ≤ n < m ≤ N If Vm = f (Sm ) for some function f ,
then
Vn = (1 + i)−k E∗ f (Sm )|FnS
k X
k ∗ j ∗ k−j
−k
= (1 + i)
f uj dk−j Sn ,
p q
j
j=0
k := m − n.
9.2 Change of Probability
Theorem 9.1.2 expresses
Vn
as a conditional expectation relative to the
risk-neutral probability measure
P∗ .
Vn as a
P. This is
It is also possible to express
conditional expectation relative to the
original
probability measure
the content of the following theorem.
Theorem 9.2.1. There exists a positive random variable Z on (Ω, F, P) with
E(Z) = 1 such that
Vn = (1 + i)n−m Zn−1 E(Zm Vm |FnS ), 0 ≤ n ≤ m ≤ N,
where Zn := E(Z|FnS ).
We give a proof that may be applied to more general settings. The core of
the proof consists of the following three lemmas.
Lemma 9.2.2. There exists a unique positive random variable Z on (Ω, F, P)
with E Z = 1 such that, for any random variable X ,
In particular,
Proof.
Dene
Z
E∗ X = E(XZ).
(9.4)
P∗ (A) = E(IA Z).
(9.5)
by
Z(ω) =
P∗ (ω)
.
P(ω)
(Ω, F, P),
X
X
E(XZ) =
X(ω)Z(ω)P(ω) =
X(ω)P∗ (ω) = E∗ (X).
Then, for any random variable
ω
That
Z
X
on
ω
Z is a random variable
A = {ω} in (9.5), we have P∗ (ω) = Z(ω)P(ω).
is unique follows from the observation that if
satisfying (9.4), then, taking
Option Valuation: A First Course in Financial Mathematics
104
The random variable
Z
in Lemma 9.2.2 is called the
derivative of P∗ with respect to P
Radon-Nikodym
d P∗
d P . It provides a con∗
nection between the expectations E and E. The next lemma shows that Z
and is denoted by
provides an analogous connection between the corresponding conditional expectations.
Lemma 9.2.3. For any σ-elds H ⊆ G ⊆ F and any G -random variable X ,
E∗ (X|H) =
Proof.
A∈H
Let
Y = E(Z|H).
E(XZ|H)
.
E(Z|H)
We show rst that
(9.6)
Y > 0.
A = {Y ≤ 0}.
Let
and
Then
E(IA Y ) = E(IA Z) = E∗ (IA ) = P∗ (A),
where we have used the dening property of conditional expectations in the
rst equality and the dening property of the Radon-Nikodym derivative in
the second. Since
IA Y ≤ 0, P∗ (A) = 0.
A = ∅ and Y > 0.
H-random variable U :=
Therefore,
To verify Equation (9.6), we show that the
Y −1 E(XZ|H) has the
∗
spect to P , namely,
dening property of conditional expectation with re-
E∗ (IA U ) = E∗ (IA X)
for all
A ∈ H.
Using the factor property, we have
E∗ (IA U ) = E(IA U Z) = E [E(IA U Z|H)] = E (IA U Y )
= E [IA E(XZ|H)] = E(IA XZ) = E∗ (IA X),
as required.
The following result is a martingale version of the preceding lemma.
Lemma 9.2.4. Given a ltration (Gn )Nn=0 contained in F set
Zn = E(Z|Gn ), n = 0, 1, . . . , N.
Then (Zn ) is a martingale with respect to (Gn ), and for any Gm -random variable X ,
Proof.
E∗ (X|Gn ) = Zn−1 E(XZm |Gn ),
That
(Zn )
0 ≤ n ≤ m ≤ N.
is a martingale is the content of Example 8.4.5. By the
iterated conditioning and factor properties, we have for
m≥n
E(XZ|Gn ) = E [E(XZ|Gm )|Gn ] = E [XE(Z|Gm )|Gn ] = E [XZm |Gn ] .
Applying Lemma 9.2.3 with
G = Gm
E∗ (X|Gn ) =
and
H = Gn ,
we see that
E(XZ|Gn )
E(XZm |Gn )
=
.
E(Z|Gn )
Zn
The Binomial Model Revisited
For the proof Theorem 9.2.1, take
X = Vm
105
(Gn ) = (FnS )
and
in
Lemma 9.2.4 so that
E∗ (Vm |FnS ) = Zn−1 E(Vm Zm |FnS ),
0 ≤ n ≤ m ≤ N.
The theorem now follows from Equation (9.3).
Remark 9.2.5.
The proofs of Lemmas 9.2.2, 9.2.3, and 9.2.4 are completely
general, valid for any nite sample space and any pair of probability measures
P
and
P∗
that are
equivalent,
that is, satisfy
P(ω) > 0
i
P∗ (ω) > 0.
In this
general setting, one also has the following converse to Lemma 9.2.2: Given
a positive random variable
Z
with
E Z = 1,
the equation
P∗ (A) = E(IA Z)
denes a probability measure such that (9.4) holds for any random variable
X.
9.3 American Claims in the Binomial Model
In Theorem 7.2.1, we constructed a hedge that allows the writer of a European claim to cover her obligation at maturity
N.
In this section we devise
a hedging strategy for the writer of an American claim. Such a hedge is more
n ≤ N.
f (x) is a
nonnegative function. (For example in the case of an American put, f (x) =
(K − x)+ .) At time N , the portfolio needs to cover the amount f (SN ) hence
the value process (Vn ) of the hedge must satisfy
complex, as it must cover the writer's obligation at any time
We assume the payo at time
n
is of the form
f (Sn ),
where
VN = f (SN ).
At time
amount
N − 1, there are two possibilities: If the claim is exercised,
f (SN −1 ) must be covered, so, in this case, we need
then the
VN −1 ≥ f (SN −1 ).
If the claim is not exercised, then the portfolio must have a value sucient to
cover the claim
VN
at time
N.
By risk-neutral pricing, that value is
S
(1 + i)−1 E∗ (VN |FN
−1 ).
Therefore, in this case, we should have
S
VN −1 ≥ (1 + i)−1 E∗ (VN |FN
−1 ).
We can satisfy both cases in an optimal way by requiring that
S
VN −1 = max f (SN −1 ), (1 + i)−1 E∗ (VN |FN
−1 ) .
106
Option Valuation: A First Course in Financial Mathematics
The same argument may be made at each stage of the process. This leads to
the backward recursion formula
VN = f (SN ),
Vn = max{f (Sn ), (1 + i)−1 E∗ (Vn+1 |FnS )}, n = N − 1, N − 2, . . . , 0.
The process
V
(9.7)
so dened may be used to construct a self-nancing trading
strategy exactly as in the proof of Theorem 7.2.1. Thus, we have
Theorem 9.3.1. The process V = (Vn ) dened by (9.7) is the value process for a self-nancing portfolio with trading strategy (φ, θ), where for n =
1, 2, . . . , N and ω = (ω1 , ω2 , . . . , ωn−1 ),
θn (ω) =
Vn (ω, u) − Vn (ω, d)
,
Sn (ω, u) − Sn (ω, d)
and
φn (ω) = (1 + i)1−n (Vn−1 (ω) − θn Sn−1 (ω)).
The portfolio covers the claim at any time n, that is, VN = f (SN ) and Vn ≥
f (Sn ), n = 1, 2, . . . , N − 1. Hence, the proper price of the claim is the initial
cost V0 of setting up the portfolio.
Remark 9.3.2.
value process
Ṽ
In contrast to the case of a European claim, the discounted
is
not
a martingale. However, it follows from (9.7) that
(1 + i)−1 E∗ (Vn+1 |FnS ),
and multiplying this inequality by
(1 + i)−n
Vn ≥
yields
Ṽn ≥ E∗ (Ṽn+1 |FnS ), 0 ≤ n < N.
A process satisfying such an inequality is called a
supermartingale.
We will
return to this notion in the next section.
The following theorem gives an algorithm for constructing
V
based directly
on the backward recursion scheme (9.7).
Theorem 9.3.3. Let (vn ) be the sequence of functions dened by setting
vN (s) = f (s), and
vn (s) = max f (s), avn+1 (us) + bvn+1 (ds) , n = N − 1, . . . , 0,
(9.8)
where a = (1 + i)−1 p∗ and b = (1 + i)−1 q∗ . Then
Vn = vn (Sn ),
n = 1, 2, . . . , N.
In particular, for each ω,
Vn (ω) = max f uk dn−k S0 , avn+1 uk+1 dn−k S0 + bvn+1 uk dn+1−k S0 ,
where k := Yn (ω) is the number of upticks during the rst n time periods.
The Binomial Model Revisited
Proof.
Clearly,
lary 9.1.3 with
VN = vN (SN ).
m = n + 1,
Suppose
107
Vn+1 = vn+1 (Sn+1 ).
By Corol-
(1 + i)−1 E∗ (Vn+1 |FnS ) = avn+1 (uSn ) + bvn+1 (dSn ).
Substituting this expression into (9.7), we see that
Vn = vn (Sn ).
The conclu-
sion of the theorem now follows by backward induction.
For small values of
N,
the algorithm (9.8) may be readily implemented on
a spreadsheet. The following example illustrates the case
Example 9.3.4.
For
v4 (s)
v3 (u3 S0 )
v3 (u2 dS0 )
v3 (ud2 S0 )
v3 (d3 S0 )
v2 (u2 S0 )
v2 (udS0 )
v2 (d2 S0 )
v1 (uS0 )
v1 (dS0 )
v0 (S0 )
The price
=
=
=
=
=
=
=
=
=
=
=
N = 4,
(9.8) may be explicitly rendered as
f (s)
max{ f (u3 S0 ),
max{ f (u2 dS0 ),
max{ f (ud2 S0 ),
max{ f (d3 S0 ),
max{ f (u2 S0 ),
max{ f (udS0 ),
max{ f (d2 S0 ),
max{ f (uS0 ),
max{ f (dS0 ),
max{ f (S0 ),
V0 = v0 (S0 )
av4 (u4 S0 )
av4 (u3 dS0 )
av4 (u2 d2 S0 )
av4 (ud3 S0 )
av3 (u3 S0 )
av3 (u2 dS0 )
av3 (ud2 S0 )
av2 (u2 S0 )
av2 (udS0 )
av1 (uS0 )
+
+
+
+
+
+
+
+
+
+
bv4 (u3 dS0 ) }
bv4 (u2 d2 S0 ) }
bv4 (ud3 dS0 ) }
bv4 (d4 S0 ) }
bv3 (u2 dS0 ) }
bv3 (ud2 S0 ) }
bv3 (d3 S0 ) }
bv2 (udS0 ) }
bv2 (d2 S0 ) }
bv1 (dS0 ) }
of an American put may be calculated using this
scheme. Table 9.1 gives American put prices
for various values of
N = 4.
K , u,
and
P0a
for
S0 = $20.00, i = .10,
and
d.
K
u
d
P0a
K
u
d
P0a
$18.00
1.5
.9
$0.76
$22.00
1.5
.9
$2.43
$18.00
2.0
.9
$1.49
$22.00
2.0
.9
$3.21
$18.00
1.5
.6
$3.18
$22.00
1.5
.6
$5.45
$18.00
2.0
.6
$4.90
$22.00
2.0
.6
$6.79
TABLE 9.1: American put prices for
S0 = $20
Additional material on American claims in the binomial model, including
a version of the hedge that allows consumption at each time
in [16].
n,
may be found
Option Valuation: A First Course in Financial Mathematics
108
9.4 Stopping Times
We have shown how the writer of an American claim may construct a
hedge to cover her obligation at any exercise time
n ≤ N.
The
holder
of the
claim has a dierent concern, namely, when to exercise the claim to obtain
the largest possible payo. In this section, we develop the tools needed to
determine the holder's optimal exercise time.
It is clear that the optimal exercise time of a claim depends on the price
of the underlying asset at that time and therefore must be a random variable.
Moreover, its value must be determined using only present or past information.
This leads to the formal notion of a
stopping time.
Denition 9.4.1. Let (Fn ) = (Fn )Nn=0 be a ltration on Ω. An (Fn )-stopping
time is a random variable τ with values in the set {0, 1, . . . , N } such that
{τ = n} ∈ Fn ,
n = 0, 1, . . . , N.
If there is no possibility of ambiguity, we omit reference to the ltration.
Note that if τ is a stopping time, then the set {τ ≤ n}, as a union of the sets
{τ = j} ∈ Fj , j ≤ n, is a member of Fn . It follows that {τ ≥ n+1} = {τ ≤ n}0
also lies in Fn .
Example 9.4.2.
value
b
The rst time a stock falls below a value
(
min{n | Sn (ω) ∈ A}
τ (ω) =
N
where
A = (∞, a) ∪ (b, ∞). That τ
(Sn ) may be seen from
ltration of
or exceeds a
if
{n | Sn (ω) ∈ A} =
6 ∅
otherwise,
is a stopping time relative to the natural
the calculations
{τ = n} = {Sn ∈ A} ∩
and
{τ = N } =
n−1
\
j=0
{Sj 6∈ A},
j=0
{Sj 6∈ A}.
For a related example, consider a process
(In )
n<N
N\
−1
(In ),
like the S&P 500 or the Nikkei 225, and a ltration
and
a
is described mathematically by the formula
say a stock market index
(Fn )
to which both
are adapted. Dene
(
min{n | Sn (ω) > In (ω)}
τ (ω) =
N
if
{n | Sn (ω) > In (ω)} =
6 ∅
otherwise.
(Sn )
The Binomial Model Revisited
Then
τ
109
is a stopping time, as may be seen from calculations similar to those
above. Such a stopping time could result from an investment decision to sell
the stock the rst time it exceeds the index value.
It is easy to see that the function
(
max{n | Sn (ω) ∈ A}
τ (ω) =
N
is
not
if
{n | Sn (ω) ∈ A} =
6 ∅
otherwise
a stopping time. This is a mathematical formulation of the obvious fact
that without foresight (or insider information) an investor cannot determine
when the stock's value will lie in a set
The constant function
τ = m,
A
where
for the last time.
m
is easily seen to be a stopping time. Also, if
so is
τ ∧ σ,
is a xed integer in
τ
and
σ
{0, 1, . . . , N },
are stopping times, then
where
(τ ∧ σ)(ω) := min(τ (ω), σ(ω)).
This follows immediately from the calculation
{τ ∧ σ = n} = {τ = n, σ ≥ n} ∪ {σ = n, τ ≥ n}.
In particular,
τ ∧ m is a stopping time. This observation leads to the notion of
stopped process, an essential tool in determining the optimal time to exercise
a claim. To dene such a process, we need the notion of an optional stopping
a
of a process.
Denition 9.4.3. Let (Xn )Nn=0 be a stochastic process adapted to a ltration
(Fn )N
n=0 and let τ be a stopping time. The random variable Xτ dened by
Xτ =
N
X
I{τ =j} Xj
j=0
is called an optional stopping of the process X .
From the denition,
immediately that if
Xτ (ω) = Xj (ω) for any ω for which τ (ω) = j . It follows
Xj ≤ Yj for all j then Xτ ≤ Yτ and X̃τ ≤ Ỹτ , where
X̃τ := (1 + i)−τ Xτ .
Denition 9.4.4. For a given stopping time τ , the stochastic process
is called the stopped process for τ .
(Xτ ∧n )N
n=0
For example, if
τ
a,
and the path of the stopped price process for a scenario
then
Sτ = a,
is the rst time the stock's value reaches a specied value
ω
is
S0 (ω), S1 (ω), . . . , Sτ −1 (ω), a.
To nd the optimal exercise time for an American put, we shall need the
following generalization of a martingale:
Option Valuation: A First Course in Financial Mathematics
110
Denition 9.4.5. A stochastic process (Mn ) adapted to a ltration (Fn ) is
said to be a P, (Fn ) -supermartingale, respectively, -submartingale, if
E(Mn+1 |Fn ) ≤ Mn ,
respectively, E(Mn+1 |Fn ) ≥ Mn , n = 0, 1, . . . , N − 1.
When there is no ambiguity, we will drop one or both of the components of the
prex P, (Fn ) . If Fn = σ(M0 , . . . Mn ), we omit reference to the ltration.
Note that
(Mn )
is a supermartingale i
(−Mn )
is a submartingale, and
(Mn )
is a martingale i it is both a supermartingale and a submartingale. Moreover,
a supermartingale has the multistep property
E(Mm |Fn ) ≤ Mn , 0 ≤ n < m ≤ N,
as may be seen by iterated conditioning. Submartingales have the analogous
property with the inequality reversed.
Mn denotes a gambler's acnth game. The submartingale
For a gambling interpretation, suppose that
cumulated winnings at the completion of the
property then asserts that the game favors the player, since the best prediction
of his gain
in the rst
Mn+1 − Mn
n games, is
in the next game, relative to the information accrued
nonnegative. Similarly, the supermartingale property
describes a game that favors the house.
Proposition 9.4.7 below implies that a fair (unfair) game is still fair (unfair)
when stopped according to a rule that does not require prescience. For its
proof, we require the following lemma.
Lemma 9.4.6. Let (Yn ) be a process adapted to a ltration (Fn ) and let τ be
a stopping time. Then the stopped process (Yτ ∧n ) is adapted to (Fn ) and
Proof.
Yτ ∧(n+1) − Yτ ∧n = (Yn+1 − Yn )I{n+1≤τ } .
(9.9)
Note rst that
Yτ ∧(n+1) = Y0 +
n+1
X
j=1
(Yj − Yj−1 )I{j≤τ } , 0 ≤ n < N.
(9.10)
Indeed, because of the indicator functions, the sum on the right in (9.10) may
be written
τ ∧(n+1)
X
j=1
(Yj − Yj−1 ),
which collapses to Yτ ∧(n+1) − Y0 . Since the terms on the right in
Fn+1 -measurable, (Yτ ∧n ) is an adapted process. Subtracting from
analogous equation with n replaced by n − 1 yields (9.9).
(9.10) are
(9.10) the
Proposition 9.4.7. If (Mn ) is a (Fn )-martingale (supermartingale, submartingale) and if τ is an (Fn )-stopping time, then the stopped process (Mτ ∧n )
is a martingale (supermartingale, submartingale).
The Binomial Model Revisited
Proof.
By Lemma 9.4.6,
(Mτ ∧n )
is adapted to
111
(Fn ).
Also, from (9.9),
Mτ ∧(n+1) − Mτ ∧n = (Mn+1 − Mn )I{n+1≤τ } ,
hence, by the
Fn -measurability
of
I{n+1≤τ } ,
E Mτ ∧(n+1) − Mτ ∧n |Fn = I{n+1≤τ } E(Mn+1 − Mn |Fn ).
(9.11)
The conclusion of the proposition is immediate from (9.11). For exam-
(Mn ) is a supermartingale,
E Mτ ∧(n+1) |Fn ≤ Mτ ∧n .
ple, if
then
E(Mn+1 − Mn |Fn ) ≤ 0
hence
9.5 Optimal Exercise of an American Claim
In Section 9.3 we found that the writer of an American claim can cover
her obligations with a hedge whose value process
V
is given by the backward
recursive scheme
VN = HN , Vn = max{Hn , (1 + i)−1 E∗ (Vn+1 |FnS )} = vn (Sn ), 0 ≤ n < N,
where
vn
Hn := f (Sn ) denotes the payo of the claim at time n and the functions
are dened as in Theorem 9.3.3. In terms of the corresponding discounted
processes, we have
ṼN = H̃N , Ṽn = max{H̃n , E∗ (Ṽn+1 |FnS )}, 0 ≤ n < N,
which expresses
(Ṽn )
as the so-called
this section, we use the value process
V
Snell envelope
(9.12)
of the process
(H̃n ).
In
to nd the optimal time for the holder
to exercise the claim. Specically, we show that the holder should exercise the
Hn = V n .
m = 0, 1, . . . , N , let Tm denote the set of all (FnS )-stopping
in the set {m, m + 1, . . . , N }. Dene τm ∈ Tm by
claim the rst time
For
values
times with
τm = min{j ≥ m | Vj = Hj } = min{j ≥ m | Ṽj = H̃j }.
Note that
τm
VN = HN .
(FnS )-supermartingale (Remark 9.3.2), so is the
is well dened since
Recall that, because
stopped process
(Ṽτ ∧n ),
Ṽ
is a
where
Ṽτ ∧n = (1 + i)−τ ∧n Vτ ∧n
(Proposition 9.4.7). The following lemma asserts that, for the special stopping
time
τ = τm ,
much more can be said.
Lemma 9.5.1. For each m = 0, 1, . . . , N −1, the stopped process (Ṽτ
is a (FnS )N
n=m -martingale.
m ∧n
)N
n=m
Option Valuation: A First Course in Financial Mathematics
112
Proof.
(Ṽτm ∧n )
By Lemma 9.4.6,
is adapted to the ltration and
Ṽτm ∧(n+1) − Ṽτm ∧n = (Ṽn+1 − Ṽn )I{n+1≤τm } .
m ≤ n < N,
i
(Ṽn+1 − Ṽn )I{n+1≤τm } |FnS = 0.
Therefore, it suces to show that, for
E∗
Since
I{n+1≤τm }
h
FnS -measurable, the last equation
I{n+1≤τm } E∗ Ṽn+1 |FnS = I{n+1≤τm } Ṽn .
and
To verify (9.13), x
Ṽn
are
ω ∈ Ω
and consider two cases: If
is equivalent to
(9.13)
τm (ω) < n + 1,
then
the indicator functions are zero, hence (9.13) is trivially satised. If, on the
other hand, τm (ω) ≥ n + 1, then Ṽn (ω) 6= H̃n (ω), hence, from (9.12), Ṽn (ω) =
E∗ (Ṽn+1 |FnS )(ω). Therefore, (9.13) holds at each ω .
The following theorem is the main result of the section. It asserts that, after
time
m,
the optimal time to exercise an American claim is
τm ,
in the sense
that the largest expected discounted payo, given the available information,
occurs at that time.
Theorem 9.5.2. For any m ∈ {0, 1, . . . , N },
S
S
) = Ṽm .
E∗ (H̃τm |Fm
) = max E∗ (H̃τ |Fm
τ ∈Tm
In particular,
E∗ H̃τ0 = max E∗ H̃τ = V0 .
τ ∈T0
Proof.
Since
(Ṽτm ∧n )N
n=m
is a martingale (Lemma 9.5.1) and
Ṽτm = H̃τm ,
S
S
S
Ṽm = Ṽτm ∧m = E∗ (Ṽτm ∧N |Fm
) = E∗ (Ṽτm |Fm
) = E∗ (H̃τm |Fm
).
τ ∈ Tm .
Ṽτ ≥ H̃τ ,
Now let
Since
(Ṽτ ∧n )
is a supermartingale (Proposition 9.4.7) and
S
S
S
Ṽm = Ṽτ ∧m ≥ E∗ (Ṽτ ∧N |Fm
) = E∗ (Ṽτ |Fm
) ≥ E∗ (H̃τ |Fm
).
Therefore,
S
Ṽm = maxτ ∈Tm E∗ (H̃τ |Fm
),
Remark 9.5.3.
For the case
m = 0,
completing the proof.
Theorem 9.5.2 asserts that it is optimal
to exercise an American claim the rst time
f (Sn ) = vn (Sn ),
where
vN (s) = f (s),
vn (s) = max [f (s), avn+1 (us) + bvn+1 (ds)] , n = 0, 1 . . . , N − 1
The Binomial Model Revisited
113
(see Theorem 9.3.3). This leads to the following simple algorithm which may
be used to nd
if
else if
else if
τ0 (ω)
for any scenario
ω = (ω1 , ω2 , . . . , ωN ):
f (S0 ) = V0 ,
f (S0 ω1 ) = v1 (S0 ω1 ),
f (S0 ω1 ω2 ) = v2 (S0 ω1 ω2 ),
τ0 (ω) = 0,
τ0 (ω) = 1,
τ0 (ω) = 2,
.
.
.
else if
else
.
.
.
f (S0 ω1 · · · ωN −1 ) = vN −1 (S0 ω1 · · · ωN −1 ),
τ0 (ω) = N − 1,
τ0 (ω) = N.
The algorithm also gives the stopped scenarios for which
N,
and so forth. For small values of
on a spreadsheet by comparing the values of
example illustrates this for the case
Example 9.5.4.
where
τ0 = 1, τ0 = 2,
the algorithm is readily implemented
f
and
v
along paths. The next
N = 4.
Consider an American put that matures in four periods,
S0 = $10.00
and
i = .1.
Table 9.2 gives optimal exercise scenarios
and payos (displayed parenthetically) for various values of
fourth column gives the prices
K
u
d
P0a
20
3
.3
$13.49
20
2
.3
$11.77
12
3
.3
$7.01
12
3
.6
$5.13
12
2
.3
$6.05
12
2
.6
$4.04
8
3
.3
$4.07
8
3
.6
$2.50
P0a
K , u,
and
d.
The
of the put. In row 2, for example, we see
Optimal Stopping Scenarios and Payos
d (17.00); udd (17.30); uudd, or udud (11.90)
d (17.00); ud (14.00); uud (8.00)
d (9.00); udd (9.30); uudd, or udud (3.90)
dd (8.40); dudd, or uddd (5.52)
d (9.00); udd (10.20); uudd, or udud (8.40)
d (6.00); udd (4.80)
dd (7.10); dud, or udd (5.30)
ddd (5.84); uddd, dudd, or ddud (1.52)
TABLE 9.2: Put Prices and Stopping Scenarios
that the claim should be exercised after one time unit if the stock rst goes
down, after two time units if the stock goes up then down, and after three
time units if the stock goes up twice in succession then down.
Scenarios missing from the table are either not optimal or result in zero
payos. For example, in row 1, it is never optimal to exercise at time 2 and
the missing optimal scenarios
uuu, uudu, uduu
all have zero payos. In row
4, it is never optimal to exercise at time 1 and the missing optimal scenarios
uu, udu, duu, uddu, dudu all have zero payos. Note that, if an optimal
ω1 , ω2 , . . . , ωn has a zero payo at time n, then there is no hope of
ever obtaining a nonzero payo; all later scenarios ω1 , ω2 , . . . , ωn , ωn+1 , . . . will
scenario
also have zero payos (see Exercise 5).
Option Valuation: A First Course in Financial Mathematics
114
For additional material regarding stopping times and American claims in
the binomial model, including the case of path-dependent payos, see, for
example, [16].
9.6 Dividends in the Binomial Model
So far in this chapter, we have assumed that our stock
In this case, the binomial price process
Sn+1 = Zn+1 Sn ,
where
V
(Xn )
(Sn )
S
pays no dividends.
satises the recursion equation
Zn+1 := d
u Xn+1
d
,
is the Bernoulli process dened in Section 7.1. The value process
of a self-nancing portfolio based on the stock may then be expressed as
Vn+1 = θn+1 Zn+1 Sn + (1 + i)(Vn − θn+1 Sn ), n = 0, 1, . . . , N − 1.
Now suppose that at each of the times
dividend that is a fraction
assume that
δn
δn ∈ (0, 1)
n = 1, 2, . . . , N
S
of the value of
is a random variable and that the process
the stock pays a
at that time. We
(δn )N
n=1
is adapted
to the price process ltration. An arbitrage argument shows that after the
dividend is paid the value of the stock is reduced by exactly the amount of
the dividend (see Section 4.8). Thus, at time
dividend
δn+1 Zn+1 Sn ,
n + 1,
just after payment of the
the value of the stock becomes
Sn+1 = (1 − δn+1 )Zn+1 Sn ,
n = 0, 1, . . . , N − 1.
(9.14)
Since dividends contribute to the portfolio, the value process must satisfy
Vn+1 = θn+1 Sn+1 + (1 + i)(Vn − θn+1 Sn ) + θn+1 δn+1 Zn+1 Sn
= θn+1 (1 − δn+1 )Zn+1 Sn + (1 + i)(Vn − θn+1 Sn ) + θn+1 δn+1 Zn+1 Sn
= θn+1 Zn+1 Sn + (1 + i)(Vn − θn+1 Sn ).
(9.15)
Thus, the value process in the dividend-paying case satises the same recursion equation as in the non-dividend-paying case. Since the proof of Theorem 7.2.1 relies only on this equation, the conclusion of that theorem holds
in the dividend-paying case as well. In particular, given a European claim
there exists a unique self-nancing trading strategy
V
such that
(φ, θ)
H,
with value process
VN = H .
One easily checks that in the dividend-paying case the discounted price
process is no longer a martingale. (In this connection, see Exercise 6.) Nevertheless, as in the non-dividend-paying case, we have
The Binomial Model Revisited
115
Theorem 9.6.1. The discounted value process (Ṽn := (1 + i)−n Vn )Nn=0 of a
self-nancing portfolio
(φ, θ) based on a risk-free bond and the dividend-paying
stock is a P∗ , (FnS ) -martingale. In particular,
Vn = (1 + i)n−m E∗ Vm |FnS , 0 ≤ n ≤ m ≤ N,
(9.16)
and the time-n value of a contingent claim H is
Vn = (1 + i)n−N E∗ H|FnS , 0 ≤ n ≤ N.
Proof.
Using (9.15), noting that
θn+1 , Sn , and Vn
are
FnS -measurable, we have
E∗ (Vn+1 |FnS ) = θn+1 Sn E∗ (Zn+1 |FnS ) + (1 + i)(Vn − θn+1 Sn ).
Since
Zn+1
is independent of
FnS ,
E∗ (Zn+1 |FnS ) = E∗ (Zn+1 ) = up∗ + dq ∗ = 1 + i.
Therefore,
Ṽ
E∗ (Vn+1 |FnS ) = (1 + i)Vn ,
and dividing by
(1 + i)n+1
(9.17)
shows that
is a martingale.
9.7 The General Finite Market Model
Many of the ideas in the preceding sections carry over to the case of a
general stock price process
space
(Ω, F, P),
S = (Sn )N
n=0
on an arbitrary nite probability
where, as in the binomial model,
F
is the set of all subsets
of
Ω.
n.
Martingales may be used eectively to describe option valuation in this
Here, the stock is no longer restricted to only two movements at time
general setting, as illustrated by the following theorem. For its statement,
recall that probability measures
P∗
and
positive at exactly the same outcomes
ω
P
are
equivalent
if the measures are
(see Remark 9.2.5).
Theorem 9.7.1. If the discounted general price process (S̃n ) is a P∗ martingale for some probability measure P∗ equivalent to P, then the discounted
value process (Ṽn ) for any self-nancing portfolio is also a P∗ -martingale. In
particular, the time-n value of a European claim H with VN = H is
Vn = (1 + i)n−N E∗ H|FnS , 0 ≤ n ≤ N,
and the fair price of the claim is V0 = (1 + i)−N E∗ (H).
Proof.
The proof of the rst assertion of the theorem is the same as that of
Theorem 9.1.2, as it depends only on the characterization of self-nancing
portfolio given in (9.2).
To establish the second assertion it suces by Theorem 5.3.3 to show
Option Valuation: A First Course in Financial Mathematics
116
that the existence of
P∗
implies that the market is arbitrage-free. To this
V satisfying V0 = 0
P(VN ≥ 0) = 1. From the martingale property for Ṽ and the fact that
P∗ (ω) = 0 whenever VN (ω) < 0, we have
X
VN (ω)P∗ (ω) = E∗ (VN ) = (1 + i)N E∗ (V0 ) = 0.
end, consider any trading strategy with value process
and
VN (ω)>0
Since the terms in the sum are nonnegative,
and hence also
P (ω) = 0.
VN (ω) > 0 implies that P ∗ (ω) = 0
But then,
X
P(VN > 0) =
P(ω) = 0.
VN (ω)>0
Thus, the market is arbitrage-free.
The proof of Theorem 9.7.1 showed that the existence of a probability
measure
P∗
with the stated properties implies that the market is arbitrage-
free. The converse of this result is also true, although not as easy to prove.
The following theorem summarizes these results, providing a solution to part
of the general option-pricing problem.
Theorem 9.7.2 (First Fundamental Theorem of Asset Pricing). The market
is arbitrage-free i the discounted price process of the stock is a P∗ -martingale
for some probability measure P∗ equivalent to P.
The remaining part of the option-pricing problem is to determine conditions under which an arbitrage-free market is complete. The solution is given
by the following theorem.
Theorem 9.7.3 (Second Fundamental Theorem of Asset Pricing). A market
is arbitrage-free and complete i the discounted price process of the stock is a
P∗ -martingale for a unique probability measure P∗ equivalent to P.
Proofs of these theorems and a detailed discussion of nite market models
may be found in [2, 4].
The Binomial Model Revisited
117
9.8 Exercises
The following exercises refer to a stock S with geometric binomial
price process S .
1. Show that the following are stopping times:
(a)
τa :=
the rst time the stock price exceeds the average of all of its
previous values;
(b)
(c)
τb :=
the rst time the stock price exceeds all of its previous values;
τc := the rst time the stock price exceeds at least one of its previous
values.
2. Show that the rst time the stock price increases twice in succession is
a stopping time.
3. Following the format of Example 9.5.4, nd the prices and optimal exercise time scenarios of an American claim with payos given by the
f (x) = x(K − x)+ .
i = .1, u = 1.9, and d = .3.
function
Use the data
S0 = $10.00, K = $12.00,
4. Referring to Section 9.2, dene
Zn =
p∗
p
Yn q∗
q
n−Yn
so that, in the notation of Lemma 9.2.2,
(FnS ) = (FnX )
E(Zm |FnS ) = Zn ,
5. Let
ZN = Z .
Use the fact that
to show directly that
ω ∈ Ω and n = τ0 (ω).
k ≥ n.
1 ≤ n ≤ m.
Show that if
f (Sn (ω)) = 0
then
f (Sk (ω)) = 0
for all
6. Referring to Section 9.6, assume that the process
(δn ) is predictable with
respect to the price process ltration. Dene
ξn :=
n
Y
(1 − δj ),
n = 1, 2, . . . , N.
j=1
Show that the process
Ŝn := (1 + i)−n ξn−1 Sn ,
is a
P∗ -martingale.
0 ≤ n ≤ N,
118
Option Valuation: A First Course in Financial Mathematics
7. Referring to Section 9.6, assume that
δn = δ
for each
n, where 0 < δ < 1
is a constant.
(a) Prove the dividend-paying analog of Corollary 9.1.3, namely,
Vn = an−m
m−n
X
j=0
where
m − n ∗ j ∗ m−n−j
f bm−n uj dm−n−j Sn ,
p q
j
a := (1 + i), b := 1 − δ ,
(b) Use (a) to show that the cost
and
C0
0 ≤ n ≤ m ≤ N.
of a call option in the dividend-
paying case is
C0 = S0 (1 − δ)N Ψ(m, N, p̂) − (1 + i)−N KΨ(m, N, p∗ ),
p̂ = (1 + i)−1 p∗ u and m is the
N m N −m
which S0 (1 − δ) u d
> K.
where
for
smallest nonnegative integer
Chapter 10
Stochastic Calculus
Stochastic calculus extends classical dierential and integral calculus to functions with a random component arising from indeterminacy or system noise.
The fundamental construct is the Ito integral, whose description and analysis,
as well as an explication of it's role in solving stochastic dierential equations
(SDEs), are the main goals of this chapter. The principal tool used in determining the solution of an SDE is the Ito-Doeblin formula, a generalization of
the chain rule of Newtonian calculus. To put the theory in perspective, we
begin with a brief discussion of classical dierential equations.
10.1 Dierential Equations
An
ordinary dierential equation
(ODE) is an equation involving an un-
known function and its derivatives. A
rst order
ODE
with initial condition
is of the form
x0 = f (t, x),
where
t
f
(10.1)
(t, x) and x0 is a given value. The variable
x the space variable. A solution
function x = x(t) satisfying (10.1) on some open
is a continuous function of
may be thought of as the
of (10.1) is a dierentiable
interval
x(0) = x0 ,
I
containing
0.
time
variable and
Equation (10.1) is frequently written in dierential
form as
dx = f (t, x) dt,
x(0) = x0 .
Explicit solutions of (10.1) are possible only in special cases. For example,
if
f (x, t) =
h(t)
,
g(x)
then (10.1) may be written
g(x)x0 (t) = h(t),
x(0) = x0
x(t) must satisfy G(x(t)) = H(t)+c, where G(x) and H(t) are
g(x) and h(t), respectively, and c is an arbitrary constant.
condition x(0) = x0 may then be used to determine c. The result
hence a solution
antiderivatives of
The initial
119
Option Valuation: A First Course in Financial Mathematics
120
may be obtained formally by writing the dierential equation in separated
form as
g(x)dx = h(t)dt
Example 10.1.1.
and then integrating.
x0 (t) = 2t sec(x(t), x(0) = x0 , has
2
separated form cos x dx = 2t dt, which integrates to sin x = t + c, c = sin x0 .
−1 2
The solution may be written x(t) = sin
(t + sin x0 ), which is valid for
x0 ∈ (−π/2, π/2) and for t suciently near 0.
Example 10.1.2.
The dierential equation
Let
x(t) be
r(t).
the value at time
t
of an account earning
∆x = x(t + ∆t) −
x(t) earned over the time interval [t, t + ∆t] is approximately x(t)r(t)∆t. This
interest at the variable rate
For small
∆t,
the amount
leads to the initial value problem
dx(t) = x(t)r(t)dt,
x0 is
r(τ ) dτ .
where
Rt
0
the deposit. The solution is
An ODE is inherently
x0
and the rate
for all
t
near
0.
f (t, x)
deterministic
x(0) = x0 ,
x(t) = x0 eR(t) ,
where
R(t) =
in the sense that the initial condition
uniquely and completely determine the solution
There are circumstances, however, under which
f (t, x)
x(t)
is not
completely determined but rather is subject to random uctuations that are
the result of noise in the system. For example, if
x(t) is size of an investment
at time t, a model that incorporates random uctuation of the interest rate is
given by
x0 (t) = [r(t) + ξ(t)]x(t),
where
ξ(t)
(10.2)
is a random variable. The same dierential equation arises if
x(t)
is the size of a population whose relative growth rate is subject to random
uctuations from environmental changes. Because (10.2) has a random component, one would expect its solution to be a random variable. Equations like
this are called
stochastic dierential equations. In this chapter, we attempt to
make these ideas precise and show how to solve such equations.
A
partial dierential equation (PDE) is an equation involving an unknown
function of several variables and its partial derivatives. As we shall see in
Chapter 11, a stochastic dierential equation can give rise to a PDE whose
solution may be used to construct the solution of the original SDE. This
method will be used in Chapter 11 to obtain the Black-Scholes option pricing
formula.
10.2 Continuous-Time Stochastic Processes
Recall that a discrete-time stochastic process is a sequence of random variables that models an experiment involving a sequence of trials. While useful
Stochastic Calculus
121
and eective in many contexts, discrete-time models are not always suciently
rich to capture all the important features of an experiment. Furthermore,
discrete-time processes have inherent mathematical limitations, notably the
unavailability of calculus techniques. Continuous-time processes oer a more
realistic way to model the dynamics of experiments unfolding in time and
allow the introduction of powerful tools from stochastic calculus.
Denition 10.2.1. A (continuous-time) stochastic or random process on
a probability space (Ω, F, P) is a real-valued function X on D × Ω, where
D is an interval of real numbers, such that the function X(t, · ) is an F random variable for each t ∈ D. The set D is called the index set for the
process, and the functions X( · , ω), ω ∈ Ω, are the paths of the process. If
X does not depend on ω the process is said to be deterministic. If X 1 , X 2 ,
. . ., X d are stochastic processes with the same index set, then the d-tuple
X = (X 1 , X 2 , . . . , X d ) is called a d-dimensional stochastic process.
Xt or X(t) for the random
X by (Xt )t∈D or just (Xt ),
Depending on context, we will use the notations
variable
X(t, · ).
We will also denote the process
this notation reecting the interpretation of a stochastic process as a collection
of random variables indexed by
D
and hence as a mathematical description
of a random system changing in time. The interval
[0, T ]
or
D
is usually of the form
[0, ∞).
An example of a continuous-time process is the price of a stock, a path
being the price history for a particular market scenario. The position at time
t
of a particle randomly bombarded by the molecules of a liquid in which it is
suspended is a three- dimensional stochastic process. Surprisingly, there is an
important connection between these seemingly unrelated examples, one that
we shall examine later.
As noted above, a continuous-time stochastic process may be viewed as a
mathematical description of an evolving experiment. Related to this notion
is the ow of information revealed by the experiment. As in the discrete-time
case, this evolution of information may be modeled by a ltration.
Denition 10.2.2. A (continuous-time) ltration on (Ω, F, P ) is a collection
(Ft )t∈D of σ -elds Ft indexed by members t of an interval D ⊆ R such that
for s, t ∈ D and s < t.
Fs ⊆ Ft ⊆ F
A stochastic process (Xt )t∈D is said to be adapted to a ltration (Ft )t∈D if Xt
is a Ft -random variable for each t ∈ D. A d-dimensional process is adapted
to a ltration if each coordinate process is adapted.
As with stochastic processes, we frequently omit reference to the symbol
and denote the ltration by
(Ft ).
Associated with each stochastic process
the smallest ltration to which
information related solely to
X
X.
X
is its
natural ltration.
D
It is
is adapted and consists of time-dependent
Option Valuation: A First Course in Financial Mathematics
122
Denition 10.2.3. Let (Xt ) be a stochastic process. For each index t let
denote the intersection of the collection of σ-elds
containing all events of the form {Xs ∈ J}, where J is an arbitrary interval
and s ≤ t. (FtX ) is called the natural ltration for (Xt ) or the ltration
generated by the process (Xt ).
FtX = σ(Xs : s ≤ t)
An important example of a natural ltration is the ltration generated by
a Brownian motion, the fundamental process used in the construction of the
stochastic integral. Brownian motion and its natural ltration, described in the
next section, form the basis of the continuous-time pricing models discussed
in later chapters.
10.3 Brownian Motion
In 1827, Robert Brown observed that pollen particles suspended in a liquid
exhibited highly irregular motion. Later, it was determined that this motion
resulted from random collisions of the particles with molecules in the ambient liquid. In 1900, L. Bachelier noted the same irregular variation in stock
prices and attempted to describe this behavior mathematically. In 1905, Albert Einstein used Brownian motion, as the phenomenon came to be called, in
his work on measuring Avogadro's number. A rigorous mathematical model of
Brownian motion was developed in the 1920s by Norbert Wiener. Since then,
the mathematics of Brownian motion and its generalizations has become one
of the most active and important areas of probability theory.
To gain a better appreciation of its denition, it is instructive to view (onedimensional) Brownian motion as a limit of random walks in the following
sense. Suppose that every
∆t
(n)
1/2. Let Zt denote
the position of the particle at time t = n∆t, that is, after n moves. We assume
2
1
that ∆t and ∆x are related by the equation (∆x) = ∆t. Let Xj = 1 if the
j th move is to the right and Xj = 0 otherwise. The X
Pjn's are independent
Bernoulli variables with parameter p = 1/2, and Yn :=
j=1 Xj ∼ B(p, n) is
the number of moves to the right during the time interval [0, t]. Thus
!
Yn − n/2 √
(n)
p
Zt = Yn ∆x + (n − Yn )(−∆x) = (2Yn − n)∆x =
t,
n/4
right or left a distance
1 This
(n)
Zt is
∆x,
seconds a particle starting at the origin moves
each move with probability
is needed to produce the desired result that the limit
N (0, t).
Zt
of the random variables
Stochastic Calculus
the last equality because
lim
n→∞
(n)
P(Zt
t = n(∆x)2 .
≤ z) = lim P
n→∞
123
By the Central Limit Theorem,
Yn − n/2
z
p
≤√
t
n/4
!
=Φ
z
√
t
.
Thus, as the step length and step time tend to zero via the relation
√
(n)
∆t, Zt
tends in distribution to a random variable
∆x =
A similar
Zt − Zs is the limit in distribution of random walks over
[s, t] and that Zt − Zs ∼ N (0, t − s).
argument shows that
the time interval
Zt ∼ N (0, t).
With these ideas in mind we make the following denition:
Denition 10.3.1. Let (Ω, F, P) be a probability space. A (standard) Brownian motion or Wiener process is a stochastic process W on (Ω, F) that satises
the following conditions:
(a)
W0 = 0;
(b)
W (t) − W (s) ∼ N (0, t − s), 0 ≤ s < t;
(c)
the paths of W are continuous; and
(d)
W (t)
has independent increments; that is, if 0 < t1 < t2 < · · · < tn
then the random variables W (t1 ), W (t2 ) − W (t1 ), . . . , W (tn ) − W (tn−1 )
are independent.
The Brownian ltration on (Ω, F, P) is the natural ltration (FtW )t≥0 .
Rigorous proofs of the existence of Brownian motion may be found in
advanced texts on probability theory. The most common of these proofs uses
Kolmogorov's Extension Theorem; another constructs Brownian motion from
wavelets. The interested reader is referred to [18, 19].
Brownian motion has the unusual property that, while the paths are continuous, they are nowhere dierentiable. This corresponds to Brown's observation that the paths of the pollen particles seemed to have no tangents.
fractal. These properties partially account for the usefulness of Brownian motion
Moreover, Brownian motion looks the same at any scale; that is, it is a
in modeling systems with noise.
10.4 Variation of Brownian Paths
A useful way to measure the seemingly erratic behavior of Brownian motion
is by the
variation
of a function.
of its paths. For this we need the notion of
mth
variation
Option Valuation: A First Course in Financial Mathematics
124
Denition 10.4.1. Let P = {a = t0 < t1 < · · · < tn = b} be a partition of
the interval [a, b] and let m be a positive integer. For a real-valued function f
on [a, b] dene
(m)
VP
(f ) =
n−1
X
j=0
|∆fj |m ,
∆fj := f (tj+1 ) − f (tj ).
The function f is said to have bounded (unbounded) mth variation on [a, b]
if the quantities VP(m) (f ), taken over all partitions P , form a bounded (unbounded) set of real numbers.
Example 10.4.2.
The continuous function

t sin (1/t)
w(t) :=
0
if
if
w
on
[0, 2/π]
0<t≤
dened by
2
,
π
t=0
has unbounded rst variation. This can be seen by considering partitions of
the form
2
2
2
Pn = 0,
,
,...,
nπ (n − 1)π
π
Pn−1
and noting that the corresponding sums
j=0 |∆wj | are the partial
a divergent series. By contrast, for each > 0 the related function

t1+ sin (1/t) if 0 < t ≤ 2 ,
u(t) :=
π
0
if t = 0,
has bounded rst variation on every interval
[a, b].
sums of
(Exercise 2(d).)
If f is a stochastic process, Denition 10.4.1 may be applied to the paths
(m)
f ( · , ω). In this case, VP (f ) is a random variable. In particular, for Brownian
motion, we have the denition
(m)
VP
(W ) :=
n−1
X
j=0
|∆Wj |m .
It may be shown that the paths of Brownian motion have unbounded rst
variation in every interval. (See, for example, [6].)
The following theorem describes the key property of Brownian motion that
accounts for the primary dierence between stochastic calculus and classical
calculus.
Theorem 10.4.3. Let P = {a = t0 < t1 < · · · < tn = b} be a partition of the
interval [a, b] and set ||P|| = maxj ∆tj , where ∆tj := tj+1 − tj . Then
(2)
lim VP (W ) = b − a
||P||→0
Stochastic Calculus
125
in the mean square sense, that is,
2
(2)
lim E VP (W ) − (b − a) = 0.
||P||→0
Proof.
Let
n−1
X
(2)
AP = VP (W ) − (b − a) =
j=0
(∆Wj )2 − ∆tj ,
so that
E(A2P ) =
n−1
X n−1
X
j=0 k=0
E
(∆Wj )2 − ∆tj (∆Wk )2 − ∆tk .
(10.3)
By independent increments, the terms in the double sum for which
reduce to zero hence
E(A2P ) =
n−1
X
j=0
X
2 n−1
E (∆Wj )2 − ∆tj =
E(Zj2 − 1)2 (∆tj )2 ,
j 6= k
(10.4)
j=0
where
∆Wj
∼ N (0, 1).
Zj := p
∆tj
The quantity
c := E(Zj2 − 1)2
is nite and does not depend on
j
(see Exam-
ple 6.3.2) hence
E(A2P ) ≤ c||P||
||P|| → 0
Letting
forces
Remarks 10.4.4.
quadratic variation
n−1
X
j=0
∆tj = c||P||(b − a).
E(A2P ) → 0.
(2)
lim||P||→0 VP (W ) is called the
the interval [a, b]. That Brownian
The mean square limit
of Brownian motion on
motion has nonzero quadratic variation on any interval is in stark contrast to
the functions one encounters in Newtonian calculus. (See Exercise 2 in this
regard.)
For
m ≥ 3,
the mean square limit of
the continuity of
(m)
VP
W (t)
(W ) =
(m)
VP
(W )
is zero. This follows from
and the inequality
n−1
X
j=0
(2)
|∆Wj |m−2 |∆Wj |2 ≤ max |∆Wj |m−2 VP (W ).
j
Option Valuation: A First Course in Financial Mathematics
126
10.5 Riemann-Stieltjes Integrals
To motivate the construction of the Ito integral, we rst give a brief
overview of the Riemann-Stieltjes integral. For details, the reader is referred
to [15].
Let
f
and
w be bounded functions on an interval [a, b]. A Riemann-Stieltjes
sum of f with respect to w is a sum of the form
RP =
n−1
X
f (t∗j )∆wj ,
∆wj := w(tj+1 ) − w(tj ),
j=0
P = {a = t0 < t1 < · · · < tn = b} is a partition of [a, b] and t∗j ∈
[tj , tj+1 ], j = 0, 1, . . . , n − 1. The Riemann-Stieltjes integral of f with respect
to w is dened as the limit
Z b
f (t) dw(t) = lim RP ,
where
||P||→0
a
||P|| = maxj (tj+1 − tj ). The limit is required to be independent of
t∗j 's. The integral may be shown to exist if f is continuous
and w has bounded rst variation on [a, b]. The Riemann integral is obtained
as a special case by taking w(t) = t. More generally, if w is continuously
where
the choice of the
dierentiable then
b
Z
b
Z
f (t)w0 (t) dt.
f dw =
a
a
The Riemann-Stieltjes integral has many of the familiar properties of the
Riemann integral, notably
Z
b
b
Z
(αf + βg) dw = α
Z
a
a
Z
b
Z
g dw
and
a
c
f dw =
a
b
f dw + β
Z
f dw +
a
b
f dw,
a < c < b.
c
10.6 Stochastic Integrals
For the remainder of the chapter,
probability space
(Ω, F, P).
W
denotes a Brownian motion on a
In this section, we construct the Ito integral
Z
I(F ) =
b
F (t) dW (t),
a
(10.5)
Stochastic Calculus
where
F (t)
is a stochastic process on
such a process
F,
for each
ω∈Ω
Z
(Ω, F, P)
127
with continuous paths. Given
we may form the ordinary Riemann integral
b
F (t, ω)2 dt.
a
For technical reasons, we shall assume that the resulting random variable
Rb
a
F (t)2 dt
has nite expectation:
Z
!
b
F (t)2 dt
E
a
< ∞.
To construct the Ito integral, consider sums of the form
n−1
X
F (t∗j , ω)∆Wj (ω),
j=0
∆Wj := W (tj+1 ) − W (tj ),
(10.6)
P := {a = t0 < t1 < t2 < · · · tn = b} is a partition of [a, b] and t∗j ∈
[tj , tj+1 ]. In light of the above discussion on the Riemann-Stieltjes integral,
it might seem reasonable to dene I(F )(ω) as the limit of these sums as
||P|| → 0. However, this fails for several reasons.
First, the paths of W do not have bounded rst variation, so we can't
where
expect the sums in (10.6) to converge in the usual sense. What is needed instead is mean square convergence. Second, even with the appropriate mode of
convergence, the limit of the sums in (10.6) generally depends on the choice of
∗
the intermediate points tj . To eliminate this problem, we shall always take the
∗
point tj to be the
endpoint tj of the interval [tj , tj+1 ]. These restrictions,
left
however, are not sucient to ensure a useful theory. We shall also require that
the random variable
0≤s<t
F (s)
be independent of the increment
W (t) − W (s) for
W (r) for r ≤ s.
F be adapted to
and depend only on the information provided by
Both conditions are realized by requiring that the process
the Brownian ltration. Under these conditions we dene the
F
in (10.5) to be the limit of the
Ito sums
IP (F ) :=
n−1
X
Ito integral
of
F (tj )∆Wj ,
j=0
where the convergence is in the mean square sense:
lim E|IP (F ) − I(F )|2 = 0.
||P||→0
It may be shown that this limit exists for all continuous processes
F
satisfying
the conditions described above, hereafter referred to as the usual conditions.
In the discussions that follow, we shall assume, usually without explicit mention, that these conditions are met.
Option Valuation: A First Course in Financial Mathematics
128
Example 10.6.1.
if
F
If
F (t) is deterministic, that is, does not depend on ω , and
has bounded variation, then the following integration by parts formula is
valid:
b
Z
a
b
Z
F (t) dW (t) = F (b)W (b) − F (a)W (a) −
W (t) dF (t).
(10.7)
a
ω , is interpreted as a Riemann-
Here the integral on the right, evaluated at any
Stieltjes integral.
To verify (10.7), let
n−1
X
F (tj )∆Wj (t) =
j=0
P
be a partition of
n
X
j=1
F (tj−1 )W (tj ) −
[a, b]
n−1
X
and write
F (tj )W (tj )
j=0
n−1
X
= F (tn−1 )W (b) − F (a)W (a) −
As
kPk → 0,
W (tj )∆Fj−1 .
(10.8)
j=1
the sum in (10.8) converges to the Riemann-Stieltjes integral
(both pointwise in
ω
and in the mean square sense) and
F (tn−1 )
converges to
F (b).
If
F0
Z
is continuous, then (10.7) takes the form
b
a
F (t) dW (t) = F (b)W (b) − F (a)W (a) −
b
Z
W (t)F 0 (t) dt,
(10.9)
a
as may be seen by applying the Mean Value Theorem to the increments
Because
F
is deterministic, the random variable
mean zero and variance
Rb
a
F 2 (t) dt
Rb
a
∆Fj .
F (t) dW (t) is normal with
(Corollary 10.6.4, below). It follows from
(10.9) that the random variable
b
Z
a
W (t)F 0 (t) dt + F (a)W (a) − F (b)W (b)
is also normal with mean zero and variance
F (t) = t − b,
Rb
a
F 2 (t) dt.
In particular, taking
we see that
Z
a
b
[W (t) − W (a)] dt
is normal with mean 0 and variance
Example 10.6.2.
Z
a
(t − b)2 dt = (b − a)3 /3.
Theorem 10.4.3 may be used to derive the formula
b
W (t) dW (t) =
a
Rb
W 2 (b) − W 2 (a) b − a
−
.
2
2
(10.10)
Stochastic Calculus
129
To verify this, note rst that, for any sequence of real numbers
2
n−1
X
j=0
xj (xj+1 − xj ) = x2n − x20 −
n−1
X
j=0
(xj+1 − xj )2 ,
as may be seen by direct expansion. In particular, setting
where
have
P = {a = t0 < t1 < · · · < tn = b}
2
n−1
X
j=0
xj ,
xj = W (tj , ω),
[a, b], we
is an arbitrary partition of
(2)
W (tj )∆Wj = W 2 (b) − W 2 (a) − VP (W ).
Equation (10.10) is now an immediate consequence of Theorem 10.4.3.
Remarks.
1
The term (b−a) in (10.10) arises because of the particular choice
2
∗
of tj in the denition of (10.5) as the left endpoint of the interval [tj , tj+1 ]. If we
had instead used
(thus producing what is called the
1
), then the term
2 (b − a) would not appear and the result would
midpoints
integral
Stratonovich
conform to the familiar one of classical calculus. The choice of left endpoints is
dictated by technical considerations, including the fact that this choice makes
the Ito integral a martingale, a result of fundamental importance in both the
theory and applications of stochastic calculus. One can also explain the choice
of the left endpoint heuristically: Consider the parameter
to represent time. If
value
F (tj )
tj
t
in
F (t)
and
W (t)
represents the present, then we should use the known
j th term of the approximation (10.6) of the integral rather
F (t∗j ), t∗j > tj , which may be viewed as anticipating the future.
in the
than a value
The crucial step in Example 10.6.2 is the result proved in Theorem 10.4.3
that the sums
Pn−1
2
j=0 (∆Wj ) converge in the mean square sense to
fact, which is sometimes written symbolically as
b − a.
This
(dW )2 = dt,
is largely responsible for the dierence between the Ito calculus and Newtonian
calculus.
The following theorem summarizes the main properties of the Ito integral.
The processes
F
and
G
are assumed to satisfy the usual conditions described
in the preceding section.
Theorem 10.6.3. Let α, β ∈ R and 0 ≤ a < c < b. Then
(iii)
Rb
Rb
[αF (t) + βG(t)] dW (t) = α a F (t) dW (t) + β a G(t) dW (t);
Rc
Rb
Rb
F (t) dW (t) = a F (t) dW (t) + c F (t) dW (t);
a
R
b
E a F (t) dW (t) = 0;
(iv)
E
(i)
(ii)
Rb
a
R
b
a
2 R
b
F (t) dW (t) = a E F 2 (t) dt;
Option Valuation: A First Course in Financial Mathematics
130
(v)
E
(vi)
E
Proof.
R
b
a
R
b
a
F (t) dW (t)
Rb
a
R
b
G(t) dW (t) = a E (F (t)G(t)) dt;
and
R
b
F (t) dt = a E (F (t)) dt.
For part (i), observe rst that, for any real numbers
x
and
y,
(x + y)2 = 2(x2 + y 2 ) − (x − y)2 ≤ 2(x2 + y 2 ).
Now set
H = αF +βG and X = αI(F )+βI(G). Applying the above inequality
IP (H) = αIP (F ) + βIP (G), we have
twice and using the fact that
2
[I(H) − X] = |I(H) − IP (H) + IP (H) − X|
2
2
2
≤ 2 |I(H) − IP (H)| + 2 |IP (H) − X|
2
≤ 2 |I(H) − IP (H)| + 4α2 |IP (F ) − I(F )|
2
2
+ 4β 2 |IP (G) − I(G)| .
Letting
||P|| → 0
veries part (i).
P of [a, b] containing the intermediate point
c is the union of partitions P1 of [a, c] and P2 of [c, b] hence IP (F ) = IP1 (F ) +
IP2 (F ). For partitions that do not contain c, a relation of this sort holds
approximately, the approximation improving as ||P|| → 0, so that in the limit
For (ii), note that a partition
one obtains (ii).
Part (iii) follows from
E IP (F ) =
n−1
X
E(F (tj )∆Wj ) =
j=0
n−1
X
E F (tj ) E ∆Wj = 0,
j=0
where we have used the independence of
F (tj )
and
∆Wj .
To prove part (iv), note that the terms in the double sum
2
E [IP (F )] =
n−1
X n−1
X
E [F (tj )∆Wj F (tk )∆Wk ]
j=0 k=0
j 6= k
FtW
-measurable,
k
for which
evaluate to zero. Indeed, if
and since
∆Wk
j < k , then F (tj )∆Wj F (tk )
2 it follows that
FtW
k
is
is independent of
E [F (tj )∆Wj F (tk )∆Wk ] = E [F (tj )∆Wj F (tk )] E (∆Wk ) = 0.
Also, because
F (tj )
and
∆Wj
are independent,
2
2
E (F (tj )∆Wj ) = E F 2 (tj ) E (∆Wj ) = E F 2 (tj ) (tj+1 − tj ).
2 This
assertion requires a conditioning argument based on results from Section 12.1. For
a discrete version of the calculation, see Exercise 8.2.
Stochastic Calculus
131
Therefore,
n−1
X
2
E [IP (F )] =
E F 2 (tj ) (∆tj ),
j=0
Rb
which is a Riemann sum for the integral
a
E F 2 (t) dt.
Letting
yields (iv).
||P|| → 0
For (v), dene
F, G = E
b
Z
Z
!
b
G(t) dW (t)
F (t) dW (t)
a
a
and
Z
F, G =
b
E (F (t)G(t)) dt.
a
The bracket functions are linear in each argument separately, and by (iv) yield
F = G. Since
4 F, G = F + G, F + G − F − G, F − G ,
equality holding for F, G , we see that F, G = F, G
the same value when
with a similar
Part (vi) is a consequence of
Fubini's Theorem,
.
which gives general con-
ditions under which integral and expectation may be interchanged. A proof
may be found in standard texts on real analysis.
Corollary 10.6.4. If F (t) is a deterministic process,R then the Ito integral
b
F (t) dW (t) is normal with mean zero and variance a F 2 (t) dt.
a
Rb
Proof.
That
I(F )
(iv) of the theorem. To see that
Rb
F 2 (t) dt follows from (iii) and
a
is normal, note that IP (F ), as sum
has mean 0 and variance
I(F )
F (tj )∆Wj ,
of independent normal random variables
is itself normal (Exam-
ple 3.6.2). A standard result in probability theory implies that
I(F ), as a mean
square limit of normal random variables, is also normal.
10.7 The Ito-Doeblin Formula
Denition 10.7.1. An Ito process is a stochastic process X of the form
t
Z
Xt = Xa +
t
Z
F (s) dW (s) +
a
a ≤ t ≤ b,
G(s) ds,
a
where F and G are continuous processes adapted to (FtW ) and
Z
!
b
2
F (t) dt
E
a
Z
+E
a
b
!
|G(t)| dt
< +∞.
(10.11)
Option Valuation: A First Course in Financial Mathematics
132
Equation (10.11) is usually written in dierential notation as
dX = F dW + G dt.
For example, if we take
b = t
in Equation (10.9) and rewrite the resulting
equation as
Z
t
a
a
FW
W (s)F 0 (s) ds,
F (s) dW (s) +
F (t)W (t) = F (a)W (a) +
then
t
Z
is seen to be an Ito process with dierential
d(F W ) = F dW + W F 0 dt.
Similarly, we can rewrite Equation (10.10) as
which shows that
Z
Wa2
t
t
Z
Wt2
=
W2
is an Ito process with dierential
+2
W (s) dW (s) +
a
1 ds,
a
dW 2 = 2W dW + dt.
Note that if
RXt
is a deterministic function with continuous derivative then
X 0 (s) ds. Thus, by the above convention, dX = X 0 (t)dt, in
a
agreement with the classical denition of dierential.
Xt = Xa +
The Ito-Doeblin formula, described in various forms below, is useful in
generating stochastic dierential equations, the subject of the next section.
The following theorem gives the simplest version of the formula.
Theorem 10.7.2 (Ito-Doeblin Formula, Version 1). Let f (x) have continuous
rst and second derivatives. Then the process f (W ) has dierential
1
df (W ) = f 0 (W ) dW + f 00 (W ) dt.
2
In integral form,
Z
f (W (t)) = f (W (a)) +
a
Proof.
t
f 0 (W (s)) dW (s) +
1
2
Z
t
f 00 (W (s)) ds.
a
We give a plausible argument under the assumption that
series expansions
f (r) − f (s) =
∞
X
f (n) (s)
(r − s)n , r, s ∈ [a, b].
n!
n=1
Detailed proofs may be found, for example, in [9, 18].
f
has Taylor
Stochastic Calculus
P = {a = t0 < t1 < · · · < tn = b}
Let
f (W (t)) − f (W (a)) =
and for each
n−1
X
j=0
133
be a partition of
[a, b].
Then
f (W (tj+1 )) − f (W (tj )),
j
f (W (tj+1 )) − f (W (tj )) =
∞
X
f (n) (W (tj ))
(∆Wj )n
n!
n=1
= f 0 (W (tj ))∆Wj + 12 f 00 (W (tj ))(∆Wj )2 + (∆Wj )3 Rj
Rj .
for a suitable remainder term
Thus,
f (W (t)) − f (W (a)) = AP + BP + CP ,
where
AP =
n−1
X
f 0 (W (tj ))∆Wj ,
j=0
BP =
CP =
n−1
1 X 00
f (W (tj ))(∆Wj )2 ,
2 j=0
and
n−1
X
(∆Wj )3 Rj .
j=0
AP , BP , and CP as ||P|| → 0.
Pn−1
(∆Wj )2 → b − a =
Rj=0
b 00
that BP →
f (W (t)) dt. This is
a
Now consider the mean square limits of
Clearly,
Rb
a
1 dt,
AP →
Rt
f 0 (W (s)) dW (s).
a
Recalling that
it is not unreasonable to expect
indeed the case and may be proved by methods similar to those used in the
m variation of
m ≥ 3 (Remarks 10.4.4), one shows that CP → 0,
proof of Theorem 10.4.3. Finally, using the fact that the order
Brownian motion is zero for
completing the argument.
Remark 10.7.3.
The integral equation in Theorem 10.7.2 may be expressed
as
Z
a
t
1
f (W (s)) dW (s) = f (W (t)) − f (W (a)) −
2
0
Z
t
f 00 (W (s)) ds,
a
which is Ito's version of the fundamental theorem of calculus. The presence of
the integral on the right is a consequence of the nonzero quadratic variation
of
W.
Option Valuation: A First Course in Financial Mathematics
134
Example 10.7.4.
we have
Applying Theorem 10.7.2 to the function
f (x) = xk , k ≥ 2,
1
dW k = kW k−1 dW + k(k − 1)W k−2 dt,
2
which has integral form
W k (t) = W k (a) + k
Z
t
a
t
Z
W k−1 (s) dW (s) + 21 k(k − 1)
W k−2 (s) ds.
a
Rearranging we have
Z
t
W k−1 (s) dW (s) =
a
W k (t) − W k (a) (k − 1)
−
k
2
Z
t
W k−2 (s) ds,
a
which may be viewed as an evaluation of the Ito integral on the left. The
special case
k=2
is the content of Example 10.6.2.
We state without proof three additional versions of the Ito-Doeblin Formula, each of which considers dierentials of functions of several variables. A
proof of the rst may be given along the lines of that of Theorem 10.7.2, using
multivariable Taylor series.
Theorem 10.7.5 (Ito-Doeblin Formula, Version 2). Suppose f (t, x) is continuous with continuous partial derivatives ft , fx , and fxx . Then, suppressing
the variable t in the notation W (t),
df (t, W ) = fx (t, W ) dW + ft (t, W ) dt + 21 fxx (t, W ) dt.
In integral form,
Z
f (t, W (t)) = f (a, W (a)) +
a
t
fx (s, W (s)) dW (s)
Z t
+
ft (s, W (s)) + 21 fxx (s, W (s)) ds.
a
Versions 1 and 2 of the Ito-Doeblin Formula deal only with functions of
the process
W . The following version treats functions of a general Ito process.
Theorem 10.7.6 (Ito-Doeblin Formula, Version 3). Suppose f (t, x) is continuous with continuous partial derivatives ft , fx , and fxx . Let X be an Ito
process with dierential
dX = F dW + G dt.
Then
2
df (t, X) = ft (t, X) dt + fx (t, X) dX + 21 fxx (t, X) (dX)
= fx (t, X) F dW + ft (t, X) + fx (t, X) G + 21 fxx (t, X) F 2 dt.
Stochastic Calculus
·
dt
dW
dt
0
0
135
dW
0
dt
TABLE 10.1: Symbol Table One
Remark 10.7.7.
The second equality in the formula may be obtained from
the rst by substituting
F dW + G dt
for
dX
and using the formal multiplica-
tion rules summarized in the above symbol table. The rules reect the limit
properties
n−1
X
j=0
as
(∆Wj )2 → b − a,
||P|| → 0.
n−1
X
j=0
(∆Wj )∆tj → 0,
n−1
X
and
j=0
(∆tj )2 → 0
Using the table, we have
(dX)2 = (F dW + G dt)2 = F 2 dt.
Example 10.7.8.
Let h(t) be a dierentiable function and X an Ito process.
d(hX) by applying the above formula to f (t, x) = h(t)x. Since
ft (t, x) = h0 (t)x, fx (t, x) = h(t) and fxx (t, x) = 0, we have
We calculate
d(hX) = h dX + h0 X dt = h dX + X dh,
which conforms to the product rule in classical calculus.
The general version of the Ito-Doeblin Formula allows functions of nitely
many Ito processes. We state the formula for the case
n = 2.
Theorem 10.7.9 (Ito-Doeblin Formula, Version 4). Suppose f (t, x, y) is continuous with continuous partial derivatives ft , fx , fy , fxx , fxy , and fyy . Let
Xj be an Ito process with dierential
dXj = Fj dWj + Gj dt,
j = 1, 2,
where W1 and W2 are Brownian motions. Then
df (t, X1 , X2 ) = ft (t, X1 , X2 ) dt + fx (t, X1 , X2 ) dX1 + fy (t, X1 , X2 ) dX2
+ 21 fxx (t, X1 , X2 ) (dX1 )2 + 21 fyy (t, X1 , X2 ) (dX2 )2
+ fxy (t, X1 , X2 ) dX1 · dX2 .
Remark 10.7.10.
of
dt, dW1 ,
and
The dierential
dW2
df (t, X1 , X2 ) may be described in terms
Fj dWj + Gj dt for dXj and using the
by substituting
formal multiplication rules given in Table 10.2. From the table, we see that
(dX1 )2 = F12 dt, (dX2 )2 = F22 dt,
and
dX1 · dX2 = F1 F2 dW1 · dW2 ,
Option Valuation: A First Course in Financial Mathematics
136
·
dt
dW1
dW2
dt
dW1
0
0
0
dt
0 dW1 · dW2
dW2
0
dW1 · dW2
dt
TABLE 10.2: Symbol Table Two
hence
df (t, X1 (t), X2 (t)) = ft dt + fx · (F1 dW1 + G1 dt) + fy · (F2 dW2 + G2 dt)
+ 21 fxx F12 dt + 12 fyy F22 dt + fxy F1 F2 dW1 · dW2
= fx F1 dW1 + fy F2 dW2 + fxy F1 F2 dW1 · dW2
+ ft + fx G1 + fy G2 + 12 fxx F12 + 21 fyy F22 dt,
f are evaluated at (t, X1 (t), X2 (t)). The evaldW1 · dW2 depends on how dW1 and dW2 are related. For
example, if W1 and W2 are independent, then dW1 · dW2 = 0. On the other
hand, if W1 and W2 are correlated, that is,
p
W1 = %W2 + 1 − %2 W3 ,
where the partial derivatives of
uation of the term
where
W2
and
W3
are independent Brownian motions and
0 < |%| ≤ 1,
then
dW1 · dW2 = % dt.
(See, for example, [17].)
Example 10.7.11.
We use Theorem 10.7.9 to obtain Ito's product rule for
dXj = Fj dWj + Gj dt, j = 1, 2. Taking f (x, y) = xy we have
fx = y , fy = x, fxy = 1, and ft = fxx = fyy = 0 hence
(
X2 dX1 + X1 dX2
if W1 and W2 are independent
d(X1 X2 ) =
X2 dX1 + X1 dX2 + %F1 F2 dt if W1 and W2 are correlated.
the dierentials
Thus, in the independent case, we obtain the familiar product rule of classical
calculus.
10.8 Stochastic Dierential Equations
Denition 10.8.1. A stochastic dierential equation (SDE) is an equation
of the form
dX(t) = α(t, X(t)) dW (t) + β(t, X(t)) dt,
Stochastic Calculus
137
where α(t, x) and β(t, x) are continuous functions. A solution of the SDE is
a stochastic process X adapted to (FtW ) and satisfying
t
Z
t
Z
β(t, X(t)) dt,
α(t, X(t)) dW (t) +
X(t) = X(0) +
(10.12)
0
0
where X(0) is a specied random variable.
In certain cases the Ito-Doeblin formula, which generates SDEs from Ito
processes, may be used to nd an explicit form of the solution (10.12). We
illustrate with two general procedures, each based on an Ito process
Z
t
Z
F (s) dW (s) +
Y (t) = Y (0) +
t
G(s) ds.
0
The rst procedure applies Version 3 of the formula to the process
eY (t) .
With
f (t, y) = ey
(10.13)
0
X(t) =
we have
dX = df (t, Y ) = ft (t, Y ) dt + fy (t, Y ) dY + 21 fyy (t, Y ) (dY )2
= X dY + 12 X (dY )2 ,
and since
dY = F dW + G dt
and
(dY )2 = F 2 dt
dX = F X dW + G + 21 F
we obtain
2
X dt.
(10.14)
Equation 10.14 therefore provides a class of SDEs with solutions
t
Z
X(t) = X(0) exp
Z
t
F (s) dW (s) +
0
G(s) ds .
Example 10.8.2.
F =σ
and
(10.15)
0
Let σ and µ be continuous stochastic
G = µ − 21 σ 2 in (10.14), we obtain the SDE
processes. Taking
dX = σX dW + µX dt,
which, by (10.15), has solution
Z
X(t) = X(0) exp
t
Z
σ(s) dW (s) +
0
In case
µ
and
σ
0
t
1 2
µ(s) − 2 σ (s) ds .
are constant, the solution reduces to
X(t) = X(0) exp σW (t) + (µ − 21 σ 2 )t ,
a process known as
geometric Brownian motion. This example will form the
basis of discussion in the next chapter.
Option Valuation: A First Course in Financial Mathematics
138
The second procedure applies Version 3 of the Ito-Doeblin formula to the
process
Y
X(t) = h(t)Y (t),
h(t)
where
is a nonzero dierentiable function and
is given by (10.13). By Example 10.7.8,
dX = h dY + h0 Y dt = h(F dW + G dt) + h0 Y dt.
Rearranging, we obtain the SDE
h0
dX = hF dW + hG + X dt.
h
The solution
X = hY
may be written
X(t) = h(t)
Taking
F = f /h
and
X(0)
+
h(0)
G = g/h,
Z
t
t
Z
F (s) dW (s) +
0
G(s) ds .
0
f and g are continuous
h0
dX = f dW + g + X dt
h
obtain the SDE
where
functions, we
(10.16)
with solution
X(t) = h(t)
X(0)
+
h(0)
Z
0
t
f (s)
dW (s) +
h(s)
0
Example 10.8.3.
g = α,
and
t
Z
Let α, β , and σ be constants
h(t) = exp(−βt) in (10.16). Then
g(s)
ds .
h(s)
with
β>0
and take
dX = σ dW + (α − βX) dt,
(10.17)
f = σ,
(10.18)
which, by (10.17), has solution
X(t) = e
−βt
α
X(0) + (eβt − 1) + σ
β
Z
t
e
βs
dW (s) ,
(10.19)
0
Ornstein-Uhlenbeck process. In nance, the SDE in (10.18) is known
Vasicek equation and is used to describe the evolution of stochastic in-
called an
as the
terest rates (see, for example, [17]). With
is called a
ics.
α=0
and
σ > 0,
Equation (10.18)
Langevin equation, which plays a central role in statistical mechan-
Stochastic Calculus
139
10.9 Exercises
1. Find the solution of each of the following ODEs and the largest open
interval on which it is dened.
(a)
x0 = x2 sin t, x(0) = 1/3;
(b)
x0 = x2 sin t, x(0) = 2;
2t + cos t
x0 =
, x(0) = 1;
2x
x+1
, x(π/6) = 1/2, 0 < t < π/2.
x0 =
tan t
(c)
(d)
variation of order m of a (deterministic) function f
2. The
[a, b]
on an interval
is dened as the limit
(m)
lim VP
||P||→0
(f ).
Prove the following:
(a) If
f
is a bounded function with zero variation of order
then
(b) If
f
f
has zero variation of order
is continuous with bounded
zero variation of order
(c) If
f
k>m
on
m+1
has a bounded rst derivative on
on
[a, b],
[a, b],
f
has
[a, b], then it has bounded rst
m ≥ 2 on [a, b].
3. Show that the Riemann-Stieltjes integral
w
on
then
f (t) = t1+ sin (1/t), f (0) = 0, has bounded
zero variation of order m ≥ 2 on [0, 1].
(d) The function
the function
m
[a, b].
mth variation
[a, b].
variation and zero variation of order
ation and
on
R 2/π
0
1 dw
rst vari-
does not exist for
dened in Example 10.4.2.
4. Show that for any nonzero constant
c, W1 (t) := cW (t/c2 )
denes a
Brownian motion.
5. Show that for
r ≤ s ≤ t,
W (s)+W (t) ∼ N (0, t+3s)
and
W (r)+W (s)+W (t) ∼ N (0, t+3s+5r).
Generalize.
6. Show that for
a > 1/2,
lim sa W (1/s) = 0,
s→0
where the limit is taken in the mean square sense.
Option Valuation: A First Course in Financial Mathematics
140
7. Use Theorem 10.6.3 to nd
(a)
(b)
(c)
V Xt
if
Xt =
Rt√
s W (s) dW (s);
exp W 2 (s) dW (s);
0
Rtp
|W (s)| dW (s).
0
0
Rt
8. Show that
Z
a
b
[W (t) − W (b)] dt
is normal with mean 0 and variance
(b − a)3 /3.
(See Example 10.6.1.)
9. Use the Ito-Doeblin formulas to show that
(b)
(c)
t
Z
1 t W (s)
e
ds;
2 a
a
Z t
Z t
t2
2
2
W 2 (s) ds; and
sW (s) dW (s) = tW (t) − −
2
0
0
"
#
2
X
X dX
dY
dY
dX dY
d
=
−
+
−
, where X
Y
Y
X
Y
Y
X Y
Z
(a)
eW (s) dW (s) = eW (t) − eW (a) −
and
Y
are Ito processes.
10. Let
X
be an Ito process with
in terms of
dW
and
dt,
of (a)
dX = F dW + Gdt. Find
X 2 ; (b) ln X ; (c) tX 2 .
11. Show that, for the process given in Equation (10.19),
the dierentials,
limt→∞ E Xt =
α
β.
Chapter 11
The Black-Scholes-Merton Model
With the methods of Chapter 10 at our disposal, we are now able to derive the
celebrated Black-Scholes formula for the price of a call option. The formula is
based on the solution of a partial dierential equation arising from an SDE
that governs the price of the underlying stock
S.
We assume throughout that
there are no arbitrage opportunities in the market.
11.1 The Stock Price SDE
Let
W
be a Brownian motion on a probability space
of a single share of
where
µ
and
σ
S
is assumed to satisfy the SDE
dS
= σ dW + µ dt,
S
(Ω, F, P).
The price
(11.1)
drift
are constants called, respectively, the
and
volatility
of
the stock. Equation (11.1) asserts that the relative change in the stock price
has two components: a deterministic part
µ dt, which accounts for the general
σ dW , which reects the unpre-
trend of the stock, and a random component
dictable nature of
S.
The volatility is a measure of the riskiness of the stock
and its sensitivity to changes in the market. If
with solution
σ = 0,
then (11.1) is an ODE
St = S0 eµt .
Equation (11.1) may be written in standard form as
dS = σS dW + µS dt,
(11.2)
which is the SDE of Example 10.8.2. The solution there was found to be
St = S0 exp σWt + µ − 21 σ 2 t .
(11.3)
The integral version of (11.2) is
Z
St = S0 +
t
Z
σS(s) dW (s) +
0
t
µS(s) ds.
(11.4)
0
Taking expectations in (11.4) and using Theorem 10.6.3, we that
Z
E St = S0 + µ
t
E S(s) ds.
0
141
Option Valuation: A First Course in Financial Mathematics
142
The function
S0 e
µt
x(t) := E St
. This is the solution of (11.1) for the case
x0 = µx,
hence E St =
σ = 0 and represents the return
therefore satises the ODE
on a risk-free investment. Thus, taking expectations in (11.4) removes the
random component of (11.1).
Although we won't consider such a general setting, both the drift
the volatility
σ
µ
and
may be stochastic processes. In this case, the solution to (11.1)
is given by
Z
t
t
Z
σ(s) dW (s) +
St = S0 exp
0
0
µ(s) − 21 σ 2 (s) ds ,
as was shown in Example 10.8.2.
11.2 Continuous-Time Portfolios
As in the binomial model, the basic construct in determining the value of a
claim is a self-nancing, replicating portfolio based on
B,
S
and a risk-free bond
r. The value
t is denoted by Bt , where we take the initial value B0 to
Bt = ert , which is the solution of the ODE
the bond account earning interest at a constant annual rate
of the bond at time
be one unit. Thus,
dB = rB dt, B0 = 1.
We assume that the market allows unlimited trading in shares of
of
B.
S
and units
The following is the continuous-time analog of Denition 5.2.1.
Denition 11.2.1. A portfolio or trading strategy for (B, S) is a twodimensional stochastic process (φ, θ) = (φt , θt )0≤t≤T adapted to the price ltration (FtS )0≤t≤T . The random variables φt and θt are, respectively, the number
of units of B and shares of S held at time t. The value of the portfolio at time
t is dened as
Vt = φt Bt + θt St ,
0 ≤ t ≤ T.
The process V = (Vt )0≤t≤T is the value or wealth process of the trading
strategy, and V0 is the initial investment or initial wealth.
Denition 11.2.2. A portfolio (φ, θ) is self-nancing if
dV = φ dB + θ dS.
(11.5)
To understand the implication of 11.5, consider a discrete version of the
portfolio process dened at times
tj ,
t0 = 0 < t1 < t2 < · · · < tn = T .
Sj is known is
the value of the portfolio before the price
φj Bj−1 + θj Sj−1 ,
At time
The Black-Scholes-Merton Model
where, for ease of notation, we have written
becomes known and the new bond value
Sj
Bj
for
143
Stj ,
and so forth. After
Sj
is noted, the portfolio has value
Vj = φj Bj + θj Sj .
At this time, the number of stocks and bonds may be adjusted (based on the
information provided by
Fj ),
but for the portfolio to be self-nancing, this
restructuring must be accomplished without changing the current value of the
φj+1
portfolio. Thus, the new values
θj+1
and
must satisfy
φj+1 Bj + θj+1 Sj = φj Bj + θj Sj .
It follows that
∆Vj = φj+1 Bj+1 + θj+1 Sj+1 − (φj+1 Bj + θj+1 Sj )
= φj+1 ∆Bj + θj+1 ∆Sj ,
which is the discrete version of (11.5), in agreement with Theorem 5.2.5.
As in the discrete case, a portfolio may be used as a hedging strategy, that
is, an investment in shares of
of the writer at maturity
T.
S
and units of
the claim, the latter formally dened as a
complete
B
devised to cover the obligation
In this case, the portfolio is said to
if every claim can be replicated.
FTS -random
replicate
variable. A market is
As with discrete time portfolios, the importance of continuous time portfolios derives from the law of one price, which implies that in an arbitrage-free
market the value of a claim is that of a replicating, self-nancing trading strategy. We use this observation in the next section to obtain a formula for the
value of a claim with underlying
S.
11.3 The Black-Scholes-Merton PDE
To derive the Black-Scholes formula, we begin by assuming the existence
VT
of a self-nancing portfolio whose value
of a European claim, where
f
at time
T
is the payo
conditions. For such a portfolio, the value of the claim at any time
is
Vt .
We seek a function
v(t, s)
such that
Vt = v(t, St ), 0 ≤ t ≤ T,
Note that if
S0 = 0
f (ST )
is a continuous function with suitable growth
and
v(T, ST ) = f (ST ).
then (11.3) implies that the process
In this case, the claim is worthless and
t ∈ [0, T ]
Vt = 0
for all
S
t.
is identically zero.
Therefore,
v
must
satisfy the boundary conditions
v(T, s) = f (s), s ≥ 0,
and
v(t, 0) = 0, 0 ≤ t ≤ T.
(11.6)
144
Option Valuation: A First Course in Financial Mathematics
To determine
v , we begin by applying Version 3 of the Ito-Doeblin formula
Vt = v(t, St ). Using (11.2), we have
(Theorem 10.7.6) to the process
dV = vt dt + vs dS + 21 vss (dS)2
= σvs S dW + vt + µvs S + 21 σ 2 vss S 2 dt,
where the partial derivatives of
v
are evaluated at
(t, St ).
(11.7)
Additionally, from
(11.5), we have
dV = θ dS + φ dB
= θS(µ dt + σ dW ) + rφB dt
= σθS dW + [µθS + r(V − θ)S] dt.
Equating the respective coecients of
dt
and
dW
(11.8)
in (11.7) and (11.8) leads
to the equations
µθS + r(V − θS) = vt + µvs S + 21 σ 2 vss S 2
and
θ = vs .
Substituting the second equation into the rst and simplifying yields the partial dierential equation
vt + rsvs + 21 σ 2 s2 vss − rv = 0, s > 0, 0 ≤ t < T.
(11.9)
Equation (11.9) together with the boundary conditions in (11.6) is called the
Black-Scholes-Merton
(BSM) PDE.
The following theorem gives the solution
v(t, s)
to (11.9). The assertion
of the theorem may be veried directly, but it is instructive to see how the
solution may be obtained from that of a simpler PDE. The latter approach is
carried out in Appendix B.
Theorem 11.3.1 (General Solution of the BSM PDE). The solution of
with the boundary conditions (11.6) is given by
(11.9)
v(t, s) = e−r(T −t) G(t, s), 0 ≤ t < T, where
Z ∞ n √
o
G(t, s) :=
f s exp σ T − t y + (r − 12 σ 2 )(T − t) ϕ(y) dy.
−∞
v of the BSM PDE, to complete the circle of
v( · , S) is indeed the value process of a self-nancing,
Having obtained the solution
ideas we must show that
replicating trading strategy. This is carried out in the following theorem, whose
proof uses
v( · , S)
to construct the strategy. A martingale proof is given in
Chapter 13.
Theorem 11.3.2. Given a European claim with payo f (ST ), there exists a
self-nancing replicating strategy for the claim with value process
Vt = v(t, St ),
0 ≤ t ≤ T,
where v(t, s) is the solution of the BSM PDE.
(11.10)
The Black-Scholes-Merton Model
Proof.
Dene
V
by (11.10) and dene adapted processes
θ(t) = vs (t, St )
V
Then
145
and
φ=B
is the value process of the strategy
−1
θ
and
φ
by
(V − θS).
(φ, θ),
and from (11.7) and (11.9)
dV = σθS dW + [r(V − θS) + µθS] dt
= θS(µ dt + σ dW ) + r(V − θS) dt
= θ dS + φ dB.
Therefore,
(φ, θ)
is self-nancing. Since
v(T, s) = f (s),
the strategy replicates
the claim.
From Theorem 11.3.2, we obtain the celebrated Black-Scholes option pricing formula :
Corollary 11.3.3. The value at time t ∈ [0, T ) of a standard call option with
strike price K and maturity T is given by
Ct = St Φ d1 (T − t, St , K, σ, r) − Ke−r(T −t) Φ d2 (T − t, St , K, σ, r) ,
(11.11)
where the functions d1 and d2 are dened by
ln (s/K) + (r + 21 σ 2 )τ
√
and
σ τ
√
ln (s/K) + (r − 21 σ 2 )τ
√
d2 (τ, s, K, σ, r) =
= d1 − σ τ .
σ τ
d1 (τ, s, K, σ, r) =
In particular, the cost of the option is
C0 = S0 Φ d1 (T, S0 , K, σ, r) − Ke−rT Φ d2 (T, S0 , K, σ, r) .
Proof.
Taking
f (x) = (x − K)+
in Theorem 11.3.1 and applying Theo-
rem 11.3.2, we see that the value of the call option at time
−r(T −t)
Ct = e
(11.12)
t ∈ [0, T )
is
G(t, St ),
where
Z
∞
G(t, s) =
−∞
√
+
s exp σ τ y + (r − 12 σ 2 )τ − K ϕ(y) dy, τ := T − t.
To evaluate the integral, note that the integrand is increasing in
y < −d2 , where dj := dj (τ, s, K, σ, r). Thus,
Z ∞
Z
√
1 2
G(t, s) = s
exp σ τ y + r − 2 σ τ ϕ(y) dy − K
y
and equals
zero when
−d2
∞
ϕ(y) dy
−d2
Z
2
√
se(r−σ /2)τ ∞
√
exp − 12 y 2 + σ τ y dy − K [1 − Φ(−d2 )]
2π
−d2
rτ
= se Φ(d1 ) − KΦ(d2 ),
(11.13)
=
the last equality by Exercise 12.
Option Valuation: A First Course in Financial Mathematics
146
Example 11.3.4.
stock with price
Table 11.1 gives prices
S0 = $20.00. C0
C0
and
P0
for options based on a
is calculated using (11.12) and
from the put-call parity formula. The table suggests that
C0
P0
is obtained
is increasing in
T
K
r
σ
C0
P0
T
K
r
σ
C0
P0
.5
18
.06
.1
$2.55
$0.01
2
18
.06
.1
$4.09
$0.06
.5
18
.06
.2
$2.77
$0.24
2
18
.06
.2
$4.64
$0.61
.5
18
.12
.1
$3.05
$0.00
2
18
.12
.1
$5.85
$0.01
.5
18
.12
.2
$3.20
$0.16
2
18
.12
.2
$6.09
$0.25
.5
22
.06
.1
$0.14
$1.49
2
22
.06
.1
$1.37
$0.89
.5
22
.06
.2
$0.61
$1.96
2
22
.06
.2
$2.47
$1.99
.5
22
.12
.1
$0.28
$1.00
2
22
.12
.1
$2.90
$0.21
.5
22
.12
.2
$0.82
$1.54
2
22
.12
.2
$3.71
$1.02
and
P0
with
TABLE 11.1: Variation of
the variables
σ, r,
and
T
C0
and decreasing in
K.
T , K, r
and
σ
These and other relations will
be examined in the next section.
11.4 Properties of the BSM Call Function
The
Black-Scholes-Merton (BSM) call function
is dened by
C = C(τ, s, K, σ, r) = sΦ(d1 ) − Ke−rτ Φ(d2 ),
where
d1,2 = d1,2 (τ, s, K, σ, r) =
τ, s, K, σ, r > 0,
ln (s/K) + (r ± σ 2 /2)τ
√
.
σ τ
For the sake of brevity, we shall occasionally suppress one or more arguments
C and d1,2 . By Corollary 11.3.3, C(T, S0 , K, σ, r) is the price
C0 of a call option with strike price K , maturity T , and underlying stock price
S0 . In the notation of (11.11), C(T − t, St , K, σ, r) = Ct , the value of the call
at time t.
in the functions
The analogous
BSM put function
is dened as
P = P (τ, s, K, σ, r) = C(τ, s, K, σ, r) − s + Ke−rτ , τ, s, K, σ, r > 0.
P (T, S0 , K, σ, r) is the price of the correspondPt := P (T − t, St , K, σ, r) is its value at time t.
By the put-call parity relation,
ing put option and
(11.14)
The Black-Scholes-Merton Model
147
We state below two theorems that summarize the analytical properties of
the BSM call function. The rst expresses various measures of sensitivity of
an option price to market parameters in terms of the standard normal cdf and
density functions. The second describes the limiting behaviors of the price with
respect to these parameters. The proofs are given in Appendix C. Analogous
properties of the BSM put function may be derived from these theorems using
(11.14).
Theorem 11.4.1 (Growth Rates of C).
(i)
(ii)
(iii)
∂C
= Φ(d1 )
∂s
2
∂ C
1
= √ ϕ(d1 )
2
∂s
sσ τ
∂C
σs
= √ ϕ(d1 ) + Kre−rτ Φ(d2 )
∂τ
2 τ
Remarks.
(iv)
(v)
(vi)
√
∂C
= s τ ϕ(d1 )
∂σ
∂C
= Kτ e−rτ Φ(d2 )
∂r
∂C
= −e−rτ Φ(d2 )
∂K
(a) The quantities
∂C ∂ 2 C
∂C ∂C
,
, −
,
,
∂s
∂s2
∂τ
∂σ
and
∂C
∂r
delta, gamma, theta, vega, and rho, and are known
Greeks. A detailed analysis with concrete examples may be
are called, respectively,
collectively as the
found in [7].
(b) Theorem 11.4.1 shows that
τ , σ,
and
r,
and decreasing in
K.
C
is increasing in each of the variables
s,
These analytical facts have simple nancial
S0 and/or decrease in K will likely
(ST − K)+ and therefore should require a higher call price.
T or r decreases the discounted strike price Ke−rT , reducing
explanations. For example, an increase in
increase the payo
An increase in
the initial cash needed to cover the strike price at maturity, making the option
more attractive.
(c) Since
v(t, s) = C(T − t, s, K, σ, r),
property (i) implies that
vs (t, s) = Φ (d1 (T − t, s, K, σ, r)) .
Recalling that
vs (t, St ) = θt represents the stock
t, we see that the expression
holdings in the replicating
portfolio at time
St Φ d1 (T − t, St , K, σ, r)
in the Black-Scholes formula gives the time-t value of the stock holdings, and
the dierence
v(t, St ) − St Φ d1 (T − t, St , K, σ, r) = −Ke−r(T −t) Φ d2 (T − t, St , K, σ, r)
the time-t value of the bond holdings. In other words, the portfolio should
always be long
Ke−r(T −t) Φ(d2 ).
Φ(d1 )
shares of the stock and short the cash amount
Option Valuation: A First Course in Financial Mathematics
148
Theorem 11.4.2 (Limiting Behavior of C ).
(i)
(ii)
(iii)
(iv)
(v)
lim (C − s) = −Ke−rτ
(vi)
lim C = 0
(vii)
lim C = s
(viii)
s→∞
s→0+
τ →∞
lim C = (s − K)+
(ix)
τ →0+
lim C = s
K→0+
lim C = s
σ→∞
lim C = (s − e−rτ K)+
σ→0+
lim C = s
r→∞
lim C = 0
K→∞
Remarks.
As with Theorem 11.4.1, the analytical assertions of Theo-
rem 11.4.2 have simple nancial interpretations. For example, part (i) implies
S0 , C0 ≈ S0 − Ke−rT , which is the initial value of a portfolio
with payo ST − K . This is to be expected, as a larger S0 makes it more likely
that for large
that the option will be exercised, resulting in precisely this payo.
Part (iii) asserts that for large
T
the cost of the option is roughly the same
as the initial value of the stock. This can be understood by noting that if
is large the discounted strike price
Ke−rT
T
is negligible. If the option nishes
in the money, a portfolio consisting of cash in the (small) amount
Ke−rT
and
a long call will have the same maturity value as a portfolio that is long in
the stock. The law of one price then dictates that the two portfolios have the
same start-up cost, which is roughly that of the call. A similar explanation
may be given for (ix).
Part (iv) implies that for
S0 > K
and small
T
the price of the option is
the dierence between the initial value of the stock and the strike price. This
is to be expected, as the holder would likely receive an immediate payo of
S0 − K .
Part (vi) conrms the following argument: For small a strike price (in com-
parison to
S0 )
the option will almost certainly nish in the money. Therefore,
a portfolio long in the option will have about the same payo as one long in
the stock. By the law of one price, the portfolios should have the same initial
cost.
S0 > e−rT K then the option price
rT
is roughly the cost of a bond with face value S0 e
− K . As the stock has little
Part (viii) asserts that if
σ
is small and
volatility, this is also the expected payo of the option. Therefore, the option
and the bond should have the same price.
The Black-Scholes-Merton Model
149
11.5 Exercises
In the following exercises, all derivatives are assumed to have underlying S and maturity T . The price process of S is given by
(11.3).
1. Suppose
S
S0 = $50.00.
sells for
If
r = .10
and
σ = .2,
use the Black-
Scholes formula and the put-call parity relation to nd the prices of call
and put options that expire in 90 days if
K =
(a) $47.00; (b) $53.00.
(Use a spreadsheet with a built in normal cdf.)
2. Show that the function
where
v(t, s) = αs+βert
satises the BSM PDE (11.9),
α and β are constants. What portfolio does the function represent?
3. Show that the BSM put function
lim P
s→∞
P
is decreasing is
and
s.
Calculate
lim P.
s→0+
4. Show that
Pt (s) = Ke−r(T −t) Φ − d2 (T − t, s) − sΦ − d1 (T − t, s) .
5. A
cash-or-nothing call option
pays a constant amount
A
if
ST > K
and
pays nothing otherwise. Use Theorem 11.3.2 to show that the value of
the option at time
t
is
Vt = Ae−r(T −t) Φ d1 (T − t, St , K, σ, r) .
6. An
asset-or-nothing call option pays the amount ST
if
ST > K
and zero
otherwise. Use Theorem 11.3.2 to show that the value of the option at
time
t
is
Vt = St Φ d1 (T − t, St , K, σ, r) .
Use this result together with that of Exercise 5 to show that in the BSM
model a portfolio long in an asset-or-nothing call and short in a cash-ornothing call with cash
K
has the same time-t value as a standard call
option.
V0
0 < K1 < K2 .
7. Use Exercise 6 to nd the cost
ST I(K1 ,K2 ) (ST ),
8. A
collar option
where
of a claim with payo
has payo
VT = min max(ST , K1 ), K2 ,
where
Show that the value of the option at time
t
0 < K1 < K2 .
is
Vt = K1 e−r(T −t) + C(T − t, St , K1 ) − C(T − t, St , K2 ).
VT =
Option Valuation: A First Course in Financial Mathematics
150
9. A
break forward
is a derivative with payo
VT = max(ST , F ) − K = (ST − F )+ + F − K,
where
F = S0 erT
is the forward value of the stock and
K
is initially set
at a value that makes the cost of the contract zero. Determine the value
Vt
t
of the derivative at time
10. Find
dS k
in terms of
dW
and nd
K.
dt.
and
11. Find the probability that a call option with underlying
money.
12. Let
p and q
be constants with
p > 0, and let x1
and
x2
S
nishes in the
be extended real
numbers. Verify that
Z
x2
e−px
2
+qx
dx = eq
2
/4p
r
x1
where
Φ(∞) := 1
and
q − 2px2
π
q − 2px1
√
√
−Φ
,
Φ
p
2p
2p
Φ(−∞) := 0.
elasticity of the call price C0 = C(T, s, K, σ, r) with respect to the
stock price s is dened as
13. The
EC =
which is the percent increase in
EC =
C0
s ∂C0
,
C0 ∂s
due to a 1% increase in
sΦ(d1 )
,
sΦ(d1 ) − Ke−rT Φ(d2 )
Conclude that
EC > 1,
d1,2 := d1,2 (T, s, K, σ, r).
implying that the option is more sensitive to
change than the underlying stock. Show also that (a)
and (b)
s. Show that
lims→0+ EC = +∞.
lims→+∞ EC = 1
Interpret nancially.
elasticity of the put price P0 = P (T, s, K, σ, r) with respect to the
stock price s is dened as
14. The
EP = −
which is the percent decrease in
EP =
and that (a)
P0
s ∂P0
,
P0 ∂s
due to a 1% increase in
sΦ(−d1 )
Ke−rT Φ(−d2 ) − sΦ(−d1 )
lims→+∞ EP = +∞
and (b)
lims→0+ EP = 0.
nancially and compare with Exercise 13.
15. Referring to Theorem 11.3.1 show that
G(t, s) =
where
1
σ
s. Show that
p
2π(T − t)
d2 = d22 (T − t, s, z, σ, r).
Z
0
∞
2
f (z)e−d2 /2
dz
,
z
Interpret
Chapter 12
Continuous-Time Martingales
In Chapter 9, the main results of option pricing in the binomial model were
interpreted in the context of discrete-time martingales. In Chapter 13, we carry
out a similar program for the Black-Scholes-Merton model, using continuoustime martingales to nd the fair price of a derivative. The current chapter
provides the necessary tools to implement this program. The main result is
Girsanov's Theorem, which guarantees the existence of risk-neutral probability
measures, a fundamental construct in the theory of option valuation.
(Ω, F, P) denotes a xed probability space
W is a Brownian motion on (Ω, F, P).
Throughout the chapter,
expectation operator
E
and
with
12.1 Conditional Expectation
Let
X
be an
F -random
variable and G a σ -eld contained in F . In TheoΩ is nite and P(ω) > 0 for all ω then there exists a
variable E(X|G), called the conditional expectation
rem 8.1.4 we showed that if
unique positive
G -random
of X given G , such that
E [IA E(X|G)] = E(IA X)
for all
A ∈ G.
(12.1)
Conditional expectation, as dened by (12.1), may be shown to exist in the
E X exists); however, E(X|G)
G -random variable Y that diers from
current general setting as well (provided that
may no longer be unique. Indeed, any
E(X|G) on a set
of probability zero also satises (12.1). For this reason, prop-
erties involving conditional expectation hold only
almost surely, that is, on a
set of probability one. For ease of exposition, we will usually omit the qualier
that a given property holds only almost surely.
The following theorem summarizes the properties of conditional expectation that we shall need in the sequel. Most of the proofs are the same as in the
nite case (Section 8.3), since they rely essentially on the dening property
(12.1) and the (almost sure) uniqueness of conditional expectation.
Theorem 12.1.1 (Properties of Conditional Expectation). Let X and Y be
variables with nite expectation. Then
F -random
151
Option Valuation: A First Course in Financial Mathematics
152
(i) (unit property)
(ii) (linearity)
E(1|G) = 1;
E(αX + βY |G) = αE(X|G) + βE(Y |G), α, β ∈ R;
X ≤ Y ⇒ E(X|G) ≤ E(Y |G);
(iii) (order property)
(iv) (absolute value property)
(v) (factor property)
|E(X|G)| ≤ E(|X||G);
if X is G -measurable then E(XY |G) = XE(Y |G);
if X and G are independent, that is, if X and
are independent for all A ∈ G , then E(X|G) = E(X);
(vi) (independence property)
IA
if H, G are σ-elds of events such that
then E [E(X|G)|H] = E(X|H).
(vii) (iterated conditioning property)
H⊆G⊆F
12.2 Martingales: Denition and Examples
Denition 12.2.1. A stochastic process
(Mt )t≥0 on (Ω, F, P) adapted to a
ltration (Ft )t≥0 is said to be a P, (Ft ) -martingale (or, simply, a martingale)
if
E(Mt |Fs ) = Ms ,
0 ≤ s ≤ t.
(12.2)
A (FtW )-martingale with continuous paths is called a Brownian martingale.
Note that, by the factor property, (12.2) is equivalent to
E(Mt − Ms |Fs ) = 0,
0 ≤ s ≤ t.
The following processes are examples of Brownian martingales.
Example 12.2.2.
Wt
t≥0
Theorem 12.1.1(vi),
Example 12.2.3.
: The independent increment property of Brow-
Wt − Ws is independent of FsW
E(Wt − Ws |FsW ) = E(Wt − Ws ) = 0.
Wt2 − t t≥0 : For 0 ≤ s ≤ t,
nian motion implies that
for all
s ≤ t.
By
Wt2 = [(Wt − Ws ) + Ws ]2 = (Wt − Ws )2 + 2Ws (Wt − Ws ) + Ws2 .
Taking conditional expectations and using linearity and the factor and independence properties yields
E(Wt2 |FsW ) = E(Wt − Ws )2 + 2Ws E(Wt − Ws ) + Ws2 = t − s + Ws2 .
Continuous-Time Martingales
Example 12.2.4.
eWt −t/2
t≥0
: For
properties imply that
0 ≤ s ≤ t,
153
the factor and independence
E(eWt |FsW ) = eWs E(eWt −Ws |FsW ) = eWs E(eWt −Ws ) = eWs +(t−s)/2 ,
the last equality by Exercise 6.14. Therefore,
E eWt −t/2 |FsW = eWs −s/2 .
The above examples are special cases of the following theorem.
Theorem 12.2.5. Every Ito process of the form
Z
t
Xt = X0 +
F (s) dW (s)
0
is a Brownian martingale.
Proof.
For
0 ≤ s < t,
Xt − Xs = I(F ) = lim IP (F ),
||P||→0
where
P = {s = t0 < t1 < · · · < tn = t}
Z
is a partition of
t
I(F ) =
F (u) dW (u),
IP (F ) =
s
n−1
X
[s, t],
F (tj )∆j W,
j=0
and convergence is in the mean square sense:
lim E|IP (F ) − I(F )|2 = 0.
||P||→0
Now let
A ∈ FsW .
Since
E (IA IP (F )) − E (IA I(F )) ≤ E IA (IP (F ) − I(F )) ≤ E IP (F ) − I(F )
and
E2 |IP (F ) − I(F )| ≤ E|IP (F ) − I(F )|2
(Exercise 6.17), we see that
E [IA (Xt − Xs )] = E [IA I(F )] = lim E [IA IP (F )] .
||P||→0
Furthermore, since
s ≤ tj
for all
j,
(12.3)
linearity, independence, and iterated con-
Option Valuation: A First Course in Financial Mathematics
154
ditioning imply that
X n−1
E IP (F )|FsW =
E F (tj )∆j W |FsW
j=0
=
n−1
X
j=0
=
n−1
X
j=0
=
n−1
X
h i
W
|F
E E F (tj )∆j W |FtW
s
j
h
i
W
E F (tj )E ∆j W |FtW
|F
s
j
E F (tj )E(∆j W )|FsW
j=0
= 0.
Therefore,
E [IA IP (F )] = E IA E(IP (F )|FsW ) = 0,
which, by (12.3), implies that
E(Xt − Xs |FsW ) =
is referred to [9].
0.
For the
E [IA (Xt − Xs )] = 0 for all A ∈ FsW . Therefore,
proof that Xt has continuous paths, the reader
Theorem 12.2.5 asserts that Ito processes
are Brownian martingales. This is
dX = F dW + G dt,
Example 12.2.6.
Let
Yt
0
By (10.14), the process
t
1
F (s) dW (s) −
2
Xt = eYt
F
Z
dX = F dW
t
α,
to be a constant
(Xt )
F 2 (s) ds.
0
satises the SDE
term, Theorem 12.2.5 implies that
particular, taking
with dierential
be the Ito process
Yt = Y0 +
dt
X
true for general Ito processes given by
as the reader may readily verify.
Z
is no
not
dX = F X dW . Since there
is a Brownian martingale. In
we see that the process
exp αWt − 12 α2 t , t ≥ 0,
is a Brownian martingale. Example 12.2.4 is the special case
12.3 Martingale Representation Theorem
A martingale of the form
Z
Xt = X0 +
t
F (s) dW (s),
0
α = 1.
Continuous-Time Martingales
where
X0
square integrable,
is constant, is
that is,
155
E Xt2 < ∞
for all
t.
This
may be seen by applying Theorem 10.6.3 to the integral terms in
Xt2 = X02 + 2X0
t
Z
Z
2
t
F (s) dW (s)
F (s) dW (s) +
.
0
0
It is a remarkable fact that all square integrable Brownian martingales are
Ito processes of the above form. We state this result formally in the following
theorem. For a proof, see, for example, [18].
Theorem 12.3.1 (Martingale Representation Theorem). If (Mt )t≥0 is a
square-integrable Brownian martingale, then there exists a square-integrable
process (ψt ) adapted to (FtW ) such that
Z
Mt = M0 +
0
Example 12.3.2.
Let
H
be an
t
ψ(s) dW (s), t ≥ 0.
FTW -random
Mt = E(H|FtW ),
variable with
(Mt )
Dene
0 ≤ t ≤ T.
The iterated conditioning property shows that
that
E H 2 < ∞.
(Mt )
is a martingale. To see
is square-integrable, note rst that
H 2 ≥ Mt2 + 2Mt (H − Mt ),
(H − Mt )2 . For each positive integer n
2
W
that An ∈ Ft . Since IAn Mt is bounded
as may be seen by expanding
dene
An = {Mt ≤ n}
it has
and note
nite expectation, hence we may condition on the inequality
IAn H 2 ≥ IAn Mt2 + 2IAn Mt (H − Mt )
to obtain
IAn E(H 2 |FtW ) ≥ IAn Mt2 + 2IAn Mt E(H − Mt |FtW ) = IAn Mt2 .
Letting
n→∞
and noting that for each
ω
the sequence
IAn (ω)
eventually
equals 1, we see that
E(H 2 |FtW ) ≥ Mt2 .
Taking expectations yields
It may be shown that
E(H 2 ) ≥ E(Mt2 )
each Mt may be
hence
(Mt )
is square-integrable.
modied on a set of probability
zero so that the resulting process has continuous paths (see [18]). Thus,
has the representation described in Theorem 12.3.1.
(Mt )
156
Option Valuation: A First Course in Financial Mathematics
12.4 Moment Generating Functions
The proof of Girsanov's Theorem given in the next section is based on the
following important notion from probability theory.
Denition 12.4.1. The moment generating function (mgf ) φX of a random
variable X is dened by
φX (λ) = E eλX
for all real numbers λ for which the expectation is nite.
To see how
φX
gets its name, expand
expectations to obtain
Dierentiating we have
Example 12.4.2.
eλX
in a power series and take
∞
X
λn
E X n.
φX (λ) =
n!
n=0
(n)
φX
(0) = E X n ,
the
nth
moment
of
X.
X ∼ N (0, 1). Then
Z ∞
2
1
φX (λ) = √
eλx−x /2 dx
2π −∞
λ2 /2 Z ∞
2
e
= √
e−(x−λ) /2 dx
2π −∞
Let
2
= eλ
To nd the moments of
X
/2
write
eλ
2
/2
and compare power series to obtain
E X 2n =
.
=
∞
X
λ2n
n!2n
n=0
E X 2n+1 = 0
and
(2n)!
= (2n − 1)(2n − 3) · · · 3 · 1.
n!2n
(See Example 6.3.2, where these moments were found by direct integration.)
More generally, if
X ∼ N (µ, σ 2 )
then
Y := (X − µ)/σ ∼ N (0, 1)
φX (λ) = E eλ(σY +µ) = eµλ φY (σλ) = eµλ+σ
2
λ2 /2
hence
.
Moment generating functions derive their importance from the following
theorem, which asserts that the distribution of a random variable is completely
determined by its mgf. A proof may be found in standard texts on probability.
Theorem 12.4.3. If random variables
they have the same cdf.
X
and Y have the same mgf, then
Continuous-Time Martingales
Example 12.4.4.
Xj ∼ N (µj , σj2 ), j = 1, 2.
Let
If
157
X1
and
X2
are indepen-
dent, then, by Example 12.4.2,
2
2
φX2 +X2 (λ) = E eλX1 eλX2 = φX1 (λ)φX2 (λ) = eµ1 λ+(σ1 λ) /2 eµ2 λ+(σ2 λ) /2
2
= eµλ+(σλ)
where
µ = µ1 +µ2
and
/2
,
σ 2 = σ12 +σ22 . By Theorem 12.4.3, X1 +X2 ∼ N (µ, σ 2 ),
a result obtained in Example 3.6.2 with considerably more eort.
Denition 12.4.5. The moment generating function φ of a random vector
is dened (whenever the expectation exists) by
X
X = (X1 , X2 , . . . , Xn )
φX (λ) = E eλ·X ,
where λ · X :=
Pn
j=1
λ = (λ1 , λ2 , . . . , λn ),
λj Xj .
The following result generalizes Theorem 12.4.3 to random vectors.
Theorem 12.4.6. If X = (X1 , X2 , . . . , Xn ) and
the same mgf, then they have the same joint cdf.
Corollary 12.4.7. Let
dependent i for all λ
X = (X1 , X2 , . . . , Xn ).
Y = (Y1 , Y2 , . . . , Yn )
Then X1 , X2 , . . . , Xn are in-
φX (λ) = φX1 (λ1 )φX2 (λ2 ) · · · φXn (λn ).
Proof.
The necessity is clear. To prove the suciency, let
independent random variables with
(Y1 , Y2 , . . . , Yn ).1
Then
φY j = φXj
have
FYj = FXj
for all
(12.4)
Y1 , Y2 , . . . , Yn be
j and set Y =
so, by independence and (12.4),
φY (λ) = φY1 (λ1 )φY2 (λ2 ) · · · φYn (λn ) = φX (λ).
By Theorem 12.4.6,
FX = FY .
Therefore,
FX (x1 , x2 , . . . , xn ) = FY (x1 , x2 , . . . , xn )
= FY1 (x1 )FY2 (x2 ) · · · FYn (xn )
= FX1 (x1 )FX2 (x2 ) · · · FXn (xn ),
which shows that
1 The
X1 , X2 , . . . , Xn
are independent.
Yj is generally taken to be the j th coordinate function on a new
Rn , where the probability measure is dened so that the sets (−∞, x1 ] ×
(−∞, x2 ] × · · · × (−∞, xn ] have probability FX1 (x1 )FX2 (x2 ) · · · FXn (xn ).
random variable
probability space
Option Valuation: A First Course in Financial Mathematics
158
12.5 Change of Probability and Girsanov's Theorem
In Remark 9.2.5 we observed that if
random variable
Z
with
E(Z) = 1,
Ω
is nite then, given a nonnegative
the equation
P∗ (A) = E(IA Z),
denes a probability measure
∗
on
P
Conversely, any probability measure
A ∈ F,
(12.5)
∗
(Ω, F) such that P (ω) > 0 i P(ω) > 0.
P∗ with this positivity property satises
(12.5). The positivity property is a special case of the notion of equivalent
probability measures, that is, measures having precisely the same sets of probability zero. These ideas carry over to the general setting as follows:
Theorem 12.5.1 (Change of Probability). Let Z be a positive random variable on (Ω, F) with E Z = 1. Then (12.5) denes a probability measure P∗ on
(Ω, F) equivalent to P. Moreover, all probability measures P∗ equivalent to P
arise in this manner and satisfy
E∗ (X) = E(XZ)
(12.6)
for all F -random variables X for which E(XZ) is dened, where E is the
expectation operator corresponding to P∗ .
∗
The proof of Theorem 12.5.1 may be found in advanced texts on prob-
Z is called the
∗
P∗ with respect to P and is denoted by dP
dP .
P and P∗ described in (12.5) and (12.6) is frequently
ability theory. As in the nite case, the random variable
Radon-Nikodym derivative
The connection between
of
expressed as
dP∗ = ZdP.
Replacing
X
in (12.6) by
XZ −1 ,
we obtain the companion formulas
E(X) = E∗ (XZ −1 )
that is,
P(A) = E∗ (IA Z −1 ),
and
dP = Z −1 dP∗ .
Y := X + α of a
α. The random variable
∗
probability measure P equivalent
For an illuminating example, consider the translation
standard normal random variable
X
by a real number
Y
is normal so one might ask if there is a
to
P
under which
Y
is
standard
normal. To answer this question, suppose
∗
Z = dP
dP .
Z = g(X) for some
that such a probability measure exists and set
X,
somehow depend on
we assume that
Since
Z
function
should
g(x)
to
be determined. By (12.5),
P∗ (Y ≤ y) = E I{Y ≤y} Z = E I(−∞,y] (X + α)g(X) ,
and, since
that
X ∼ N (0, 1) under P, the law of the unconscious statistician implies
P∗ (Y ≤ y) =
Z
∞
Z
y−α
I(∞,y] (x + α)g(x)ϕ(x) dx =
−∞
g(x)ϕ(x) dx.
−∞
Continuous-Time Martingales
If
Y
is to be standard normal with respect to
Z
P∗ ,
159
we must therefore have
y−α
g(x)ϕ(x) dx = Φ(y).
−∞
Dierentiating yields
g(y − α)ϕ(y − α) = ϕ(y)
so that
2
ϕ(y + α)
= e−αy−α /2 .
ϕ(y)
∗
to the probability measure P dened by
dP∗ = g(X) dP = exp −αX − 12 α2 dP.
g(y) =
Thus, we are led
One easily checks that
X +α
is indeed standard normal under
P∗ .
Girsanov's Theorem generalizes this result from a single random variable
to an entire process.
Theorem 12.5.2 (Girsanov's Theorem). Let (Wt )0≤t≤T be a Brownian motion on the probability space (Ω, F, P) and let α be a constant. Dene
Zt = exp −αWt − 21 α2 t , 0 ≤ t ≤ T.
Then the process W ∗ dened by
Wt∗ := Wt + αt,
0 ≤ t ≤ T,
is a Brownian motion on the probability space (Ω, F, P∗ ), where dP∗ = ZT dP.
Proof. Note rst that (Zt )0≤t≤T is a P-martingale (Example 12.2.6). In particular,
E ZT = E Z0 = 1
hence
P∗
at 0 and has continuous paths, so it remains to show that, under
independent increments and
Let
Wt∗ − Ws∗
t − s, 0 ≤ s < t.
0 ≤ t0 < t1 < · · · < tn ≤ T
and variance
X = (X1 , X2 , . . . , Xn )
W ∗ starts
P , W ∗ has
is well-dened. It is clear that
∗
is normally distributed with mean 0
and dene random vectors
and
X ∗ = (X1∗ , X2∗ , . . . , Xn∗ ),
where
Xj := Wtj − Wtj−1
and
Xj∗ := Wt∗j − Wt∗j−1 , j = 1, 2, . . . , n.
The core of the proof is determining the mgf of
Let
λ = (λ1 , λ2 , . . . , λn ).
∗ λ·X
E e
X∗
with respect to
P∗ .
By the factor and independence properties,
ZT = E eλ·X E (ZT |Ftn ) = E eλ·X Ztn
= E exp λ · X − αWtn − 21 α2 tn


n
X
= E exp  (λj − α)Xj − αWt0 − 21 α2 tn ,
=E e
λ·X
j=1
= E exp −αWt0 − 21 α2 tn
n
Y
j=1
E exp (λj − α)Xj .
Option Valuation: A First Course in Financial Mathematics
160
The factors in the last expression are mgfs of normal random variables hence,
by Example 12.4.2,
 

n
X
1
= exp  α2 (t0 − tn ) +
(λj − α)2 (tj − tj−1 ).
2
j=1
E∗ eλ·X
Since
λ · X∗ = λ · X + α
we see that
∗
1
E∗ eλ·X = e 2 A ,
A = α2 (t0 − tn ) +
=
n
X
j=1
n
X
j=1
n
X
j=1
λj (tj − tj−1 ),
where
(λj − α)2 (tj − tj−1 ) + 2α
n
X
j=1
λj (tj − tj−1 )
λ2j (tj − tj−1 ).
Thus,

∗
E∗ eλ·X = exp 
∗
Since
2
j=1

λ2j (tj − tj−1 ).
2
E∗ eλj Xj = e(tj −tj−1 )λj /2 so Wt∗j − Wt∗j−1 is normally distributed
zero and variance tj − tj−1 (Example 12.4.2 and Theorem 12.4.3).
In particular,
with mean
n
1X
n
Y
∗
E∗ eλ·X =
∗
E∗ eλj Xj ,
j=1
∗
∗
the increments Wt − Wt
are independent (Corollary 12.4.7). Therefore,
j
j−1
∗
∗
W is a P -Brownian motion.
Remark 12.5.3.
The general version of Girsanov's Theorem allows more
than one Brownian motion. It asserts that for
Wj
on
(Ω, F, P)
ability measure
and constants
P∗
independent
αj , j = 1, 2, . . . , d,
Brownian motions
there exists a
relative to which the processes
single
prob-
Wj∗ (t) := Wj (t) + αj t,
0 ≤ t ≤ T , are Brownian motions with respect to ltration generated by the
processes Wj . The αj 's may even be stochastic processes provided they satisfy
the
Novikov condition
"
E exp
In this case,
Wj∗ (t)
1
2
Z
!#
T
β(s) ds
0
is dened as
Wj (t) +
< ∞,
Rt
0
β :=
αj (s) ds.
d
X
αj2 .
j=1
(See, for example, [17].)
Continuous-Time Martingales
161
12.6 Exercises
1. Find the mgfs of (a) a binomial random variable
(n, p);
(b) a geometric random variable
2. Find the mgf of a random variable
terval
X
X
X
with parameters
with parameter
p.
uniformly distributed on the in-
[0, 1].
3. Show that, for
X and Y
fX (x) > 0 for
4. Let
0 ≤ s ≤ t,
(a)
E(Ws Wt ) = s
and (b)
be jointly distributed continuous random variables with
all
x.
E(Y |X) = g(X),
Show that
Z
∞
g(x) =
−∞
where
fX,Y (x, y)
y dy.
fX (x)
fX,Y (x, y)
is called the conditional
fX (x)
g(x) if X and Y are independent?
The function
What is
E(Wt |Ws ) = Ws .
(Wt , Ws )
1
x−y
y
p
√
√
ft,s (x, y) :=
ϕ
ϕ
.
s
t−s
s(t − s)
5. Show that, for
0 < s < t,
the joint density
6. Use Exercises 4 and 5 to nd
7. Show that
M := eαWt
E(Ws |Wt )
+h(t)
t≥0
ft,s
density of Y given X .
for
of
is given by
0 < s < t.
is a martingale i
h(t) = −α2 t/2+h(0).
8. Show that
Conclude
E (Wt3 |FsW ) = Ws3 + 3(t − s)Ws ,
3
that Wt − 3tWt is a martingale.
0 ≤ s ≤ t.
Hint: Expand [(Wt − Ws ) + Ws ]3 .
9. Find
10. The
E(Wt2 |Ws )
and
E(Wt3 |Ws )
for
Hermite polynomials Hn (x, t)
0 < s < t.
are dened by
where
fx,t (λ) := f (λ, x, t) = exp (λx − 21 λ2 t).
(a) Show that
∞
X
λn
exp λx − 21 λ2 t =
Hn (x, t) .
n!
n=0
(n)
Hn (x, t) = fx,t (0),
Option Valuation: A First Course in Financial Mathematics
162
(b) Use (a) and the fact that
tingale to show that the
tingale for each
cases
H2 (Wt , t)
n.
exp (λWt − 21 λ2 t) t≥0 is a Brownian marprocess (Hn (Wt , t))t≥0 is a Brownian mar-
(Example 12.2.3 and Exercise 8 are the special
and
(n+1)
(c) Show that fx,t
(λ)
H3 (Wt , t),
respectively.)
(n)
(n−1)
= (x − λt)fx,t (λ) − ntfx,t
(λ) and hence that
Hn+1 (x, t) = xHn (x, t) − ntHn−1 (x, t).
(d) Use (c) to nd explicit representations of the martingales
and
H4 (Wt , t)
H5 (Wt , t).
(e) Use the Ito-Doeblin formula to show that
t
Z
Hn (Wt , t) =
nHn−1 (Ws , s) dWs .
0
This gives another proof that
(Hn (Wt , t))t≥0
is a martingale.
(Wt )0≤t≤T be a Brownian motion on the probability space (Ω, F, P)
α be a constant. Suppose that X is a random variable inde∗
pendent of WT . Show that X has the same cdf under P as under P ,
11. Let
and let
where
dP∗ = exp −αWT − 12 α2 T dP.
12. Let
W1
and
W2
be independent Brownian motions and
Show that the process
W = %W1 +
p
1 − %2 W2
0 < |%| < 1.
is a Brownian motion.
Chapter 13
The BSM Model Revisited
The continuous-time martingale theory developed in Chapter 12 is used in
the present chapter as an alternative method of determining the fair price of
a derivative in the Black-Scholes-Merton model. The last section of the chapter provides the connection, in the form of the Feynman-Kac Representation
Theorem, between the martingale approach to option pricing and the PDE
approach of Chapter 11.
(Ω, F, P) denotes a xed probability space with
E, and W is a Brownian motion on (Ω, F, P). As in Chapconsists of a risk-free bond B with price process governed
Throughout the chapter,
expectation operator
ter 11, our market
by the ODE
dB = rB dt, B0 = 1.
and a stock
S
with price process
S
following the SDE
dS = σS dW + µS dt,
where
µ
σ
and
are constants. As shown in Chapter 10, the solution to the
SDE is
St = S0 exp σWt + (µ − 12 σ 2 )t , 0 ≤ t ≤ T.
All martingales in this chapter are relative to the ltration
that, because
a continuous
St
(FtS )Tt=0 .
Note
Wt may each be expressed in terms of the other by
W
S
for all t. As in Chapter 11, we assume
function, Ft = Ft
and
throughout that the market is arbitrage-free.
13.1 Risk-Neutral Valuation of a Derivative
Denition 13.1.1. The probability measure P∗ on (Ω, F) dened by
dP∗ = ZT dP,
1
2
ZT := e−αWT − 2 α
T
,
α :=
µ−r
,
σ
is called the risk-neutral probability measure for the price process S . The
corresponding expectation operator is denoted by E∗ .
163
Option Valuation: A First Course in Financial Mathematics
164
The following theorem and its corollary are the main results of the chapter.
Martingale proofs are given in the next section. Note that the conclusion of
the corollary is in agreement with Theorem 11.3.2, obtained by PDE methods.
Theorem 13.1.2. Let H be a claim, that is, an FT -random variable, with
E H 2 < ∞. Then there exists a unique self-nancing, replicating strategy for
H with value process V such that
Vt = e−r(T −t) E∗ (H|Ft ) ,
0 ≤ t ≤ T.
(13.1)
Corollary 13.1.3. If H is a European claim of the form H = f (ST ), where
f is continuous and E H 2 < ∞, then
where
Vt = e−r(T −t) E∗ (f (ST )|Ft ) = e−r(T −t) G(t, St ), 0 ≤ t ≤ T,
Z
∞
G(t, s) :=
−∞
(13.2)
o
n √
f s exp σ T − t y + (r − 12 σ 2 )(T − t) ϕ(y) dy.
As noted in Chapter 11, the no-arbitrage assumption implies that
(13.3)
Vt
must
be the time-t value of the claim.
Example 13.1.4.
By Corollary 13.1.3, the time-t value of a forward contract
with forward price
K
is
Ft = e−r(T −t) E∗ (ST − K|Ft ) .
Because there is no cost in entering a forward contract,
(13.4)
F0 = 0,
and therefore
K = E∗ ST = erT E∗ S̃T = erT S0 .
Here we have used the fact that
S̃
is a
P∗ -martingale
(13.5)
(Lemma 13.2.3, be-
low). (Recall that Equation 13.5 was obtained in Section 4.3 using a general
arbitrage argument.) By the martingale property again,
E∗ (ST |Ft ) = erT E∗ (S̃T |Ft ) = erT S̃t = er(T −t) St .
(13.6)
Substituting (13.5) and (13.6) into (13.4), we see that
Ft = e−r(T −t) er(T −t) St − erT S0 = St − ert S0 ,
in agreement with Equation (4.4) of Section 4.3.
Example 13.1.5.
maturity
T
The time-t value of a call option with strike price
K
and
is
Ct = e−r(T −t) E∗ (ST − K)+ |Ft = e−r(T −t) G(t, St ), 0 ≤ t ≤ T,
where
this
f
G(t, s)
is given by (13.3) with
f (x) = (x − K)+ .
Evaluating (13.3) for
produces the BSM formula, exactly as in Corollary 11.3.3.
The BSM Model Revisited
165
13.2 Proofs of the Valuation Formulas
Lemmas 13.2.113.2.4 in this section are used in the proof of Theorem 13.1.2.
Lemma 13.2.1. Under the risk-neutral probability P∗ , the process
Wt∗ := Wt + αt,
0 ≤ t ≤ T,
µ−r
,
σ
α :=
is a Brownian motion with Brownian ltration (FtS ).
Proof.
Since
Wt
and
Wt∗
dier by a constant,
clusion now follows from Girsanov's Theorem.
∗
FtW = FtW = FtS .
The con-
We omit the straightforward verication of the next lemma.
Lemma 13.2.2. In terms of W ∗ , the price process S is given by
St = S0 exp σWt∗ + r − 21 σ 2 t ,
0 ≤ t ≤ T.
From Lemma 13.2.2 and Example 12.2.6, we have
Lemma 13.2.3. The discounted price process S̃ , given by
S̃t := e−rt St = S0 exp σWt∗ − 21 σ 2 t ,
0 ≤ t ≤ T,
˜ ∗.
is a P∗ -martingale with S̃ = σdW
Lemma 13.2.4. Let (φ, θ) be a self-nancing portfolio adapted to (FtS ) with
value process
Vt = φt Bt + θt St ,
0 ≤ t ≤ T.
Then the discounted value process Ṽ , given by Ṽt := e−rt Vt , is a P∗ -martingale.
Proof.
By Ito's product rule and the self-nancing condition
dV = φ dB+θ dS ,
dṼt = −re−rt Vt dt + e−rt dVt
= −re−rt [φt Bt + θt St ] dt + e−rt [rφt Bt dt + θt dSt ]
= −re−rt θt St dt + e−rt θt dSt
= θt dS̃t
= σθt S̃t dWt∗ ,
the last equality from the Ito-Doeblin formula and Lemma 13.2.3. It follows
from Theorem 12.2.5 that
Ṽ
is a
P∗ -martingale.
The proof of Corollary 13.1.3 uses the following lemma.
Option Valuation: A First Course in Financial Mathematics
166
Lemma 13.2.5. Let G be a σ-eld contained in F , let X be a G -random variable and Y an F -random variable independent of G . If g(x, y) is a continuous
function with E∗ |g(X, Y )| < ∞, then
E∗ [g(X, Y )|G] = G(X),
where G(x) := E∗ g(x, Y ).
Proof. We give an outline
R denote
R = J × K,
of the proof. Let
R2 containing
g = IR , then
of subsets of
intervals. If
(13.7)
all rectangles
the smallest
where
J
and
σ -eld
K are
E∗ [g(X, Y )|G] = E∗ [IJ (X)IK (Y )|G] = IJ (X)E∗ [IK (Y )|G] = IJ (X)E∗ IK (Y ),
where we have used the
IK (Y )
and
G.
Since
G -measurability
of
IJ (X)
and the independence of
G(x) = E∗ [IJ (x)IK (Y )] = IJ (x)E∗ IK (Y ),
we see that (13.7) holds for indicator functions of rectangles. From this it
may be shown that (13.7) holds for indicator functions of
R
all
members of
and hence for linear combinations of these indicator functions. Because a
continuous function is the limit of a sequence of such linear combinations,
(13.7) holds for any function
Remark 13.2.6.
g
satisfying the conditions of the lemma.
For future reference, we note that Lemma 13.2.5 extends to
G -measurable random variable X . For example, if X1 and X2
G -measurable and if g(x1 , x2 , y) is continuous with E∗ |g(X1 , X2 , Y )| < ∞
more than one
are
then
E∗ [g(X1 , X2 , Y )|G] = G(X1 , X1 ),
where
G(x1 , x2 )
E∗ g(x1 , x2 , Y ).
=
The
proof
is
similar
to
that
of
Lemma 13.2.5.
Proof of Theorem
Dene a process
V
13.1.2
by Equation (13.1). Then
ṼT = e−rT H
and
Ṽt = e−rt Vt = E∗ ṼT |Ft , 0 ≤ t ≤ T.
By Example 12.3.2,
Ṽ
is a square-integrable
P∗ -martingale
Martingale Representation Theorem, (Theorem 12.3.1),
t
Z
ψ(s) dW ∗ (s),
Ṽt = Ṽ0 +
0
for some process
ψ
adapted to
θ=
ψ
σ S̃
0 ≤ t ≤ T,
(FtS ).
Now set
and
φ = B −1 (V − θS).
so that, by the
The BSM Model Revisited
Then
(φ, θ)
is adapted to
(Ft )
and
V = θS + φB .
167
Furthermore,
dVt = ert dṼt + rVt dt
= ert ψt dWt∗ + rVt dt
ert ψt
dS̃t + rVt dt
(by Lemma 13.2.3)
σ S̃t
= ert θt −re−rt St dt + e−rt dSt + rVt dt
=
= θt dSt + r [Vt − θt St ] dt
= θt dSt + φt dBt .
Therefore,
process
(φ, θ) a self-nancing, replicating trading strategy for H
with value
V.
(φ0 , θ0 ) is a self-nancing, replicating
trading strategy for H based on S and B . By Lemma 13.2.4, the value process
V 0 of the strategy is a martingale hence
To show uniqueness, suppose that
Ṽt0 = E∗ (ṼT0 |FtS ) = E∗ (e−rT H|FtS ) = Ṽt .
Therefore,
V0 =V.
Proof of Corollary
13.1.3
By Lemma 13.2.2, we may write
√
ST = St exp σ T − t Yt + (r − 12 σ 2 )(T − t) ,
where
(13.8)
W ∗ − Wt∗
.
Yt := √T
T −t
Now dene
n √
o
g(t, x, y) = f x exp σ T − t y + (r − 21 σ 2 )(T − t) .
Yt ∼ N (0, 1) under P∗ , the law of
∗
that E g(t, x, Yt ) = G(t, x). Moreover,
Since
the unconscious statistician im-
plies
from (13.8),
g (t, St , Yt ) = f (ST ).
Therefore, by Lemma 13.2.5,
E∗ [f (ST )|FtS ] = E∗ [g(t, St , Yt )|FtS ] = G(t, St ).
13.3 Valuation under P
(Vt ) in terms of the origP. It is the continuous-time analog of Theorem 9.2.1.
The following theorem expresses the value process
inal probability measure
Option Valuation: A First Course in Financial Mathematics
168
Theorem 13.3.1. The time-t value of a claim H with E H 2 < ∞ is given by
Vt = e−r(T −t)
where
Proof.
2
E(HZT |FtS )
= e−(r+α /2)(T −t) E(e−α(WT −Wt ) H|FtS ),
S
E(ZT |Ft )
1
ZT := e−αWT − 2 α
2
T
and α :=
(13.9)
µ−r
.
σ
E |HZT | = E∗ |H| is nite, the conditional
expectation
S
E(HZT |Ft ) is dened. Since Vt = e−r(T −t) E∗ H|FtS , the rst equality in
(13.9) is equivalent to
Since
E(HZT |FtS ) = E∗ (H|FtS )E(ZT |FtS ).
Xt = E∗ (IA H|FtS ). Then,
E(IA HZT ) = E∗ (IA H) = E∗ Xt = E [Xt ZT ] = E E(Xt ZT |FtS )
= E IA E∗ (H|FtS )E(ZT |FtS ) ,
To verify (13.10), let
A ∈ FtS
(13.10)
and set
establishing (13.10) and hence the rst equality in (13.9).
For the second equality we have
E(e−αWT H|FtS )
E(HZT |FtS )
=
,
E(ZT |FtS )
E(e−αWT |FtS )
and by the factor and independence properties,
E(e−αWT |FtS ) = E(e−α(WT −Wt ) e−αWt |FtS )
= e−αWt E e−α(WT −Wt )
2
= e−αWt eα
(T −t)/2
,
the last equality by Exercise 6.14. Therefore
2
E(HZT |FtS )
= e−α (T −t)/2 E e−α(WT −Wt ) H|FtS .
S
E(ZT |Ft )
13.4 The Feynman-Kac Representation Theorem
We now have two ways of deriving the Black-Scholes pricing formula, one
using PDE techniques and the other using martingale methods. The connection between the two methods is given by the Feynman-Kac Representation
Theorem, which gives a probabilistic solution to a class of PDEs. The following
version of the theorem is sucient for our purposes.
The BSM Model Revisited
169
Theorem 13.4.1 (Feynman-Kac Representation Theorem). Let µ(t, x),
σ(t, x) and f (x) be continuous functions. Suppose that, for 0 ≤ t ≤ T , Xt
is the solution of the SDE
dXt = µ(t, Xt ) dt + σ(t, Xt ) dWt
(13.11)
and w(t, x) is the solution of the boundary value problem
wt (t, x) + µ(t, x)wx (t, x) + 21 σ 2 (t, x)wxx (t, x) = 0, w(T, x) = f (x).
If
T
Z
2
E [σ(t, Xt )wx (t, Xt )] dt < ∞,
0
(13.12)
(13.13)
then w(t, Xt ) = E f (XT )|FtW , 0 ≤ t ≤ T .
Proof.
f (XT ) = w(T, XT ), the conclusion of the theorem will follow if
T
that (w(t, Xt ))t=0 is a martingale. By Version 3 of the Ito-Doeblin
Since
we show
formula,
dw(t, X) = wt (t, X) dt + wx (t, X) dX + 21 wxx (t, X)(dX)2
= wt (t, X) dt + wx (t, X) µ(t, X) dt + σ(t, X) dW
+ 21 σ 2 (t, X)wxx (t, X) dt
= wt (t, X) + µ(t, X)wx (t, X) + 12 σ 2 (t, X)wxx (t, X) dt
+ σ(t, X)wx (t, X) dW
= σ(t, X)wx (t, X) dW,
the last equality by (13.12). It follows from (13.13) and Theorem 10.6.3(vi)
that
w(t, Xt )
is an Ito process. An application of Theorem 12.2.5 completes
the proof.
Corollary 13.4.2. Suppose that Xt saties
solution of the boundary value problem
(13.11)
and that v(t, x) is the
vt (t, x) + µ(t, x)vx (t, x) + 12 σ 2 (t, x)vxx (t, x) − rv(t, x) = 0, v(T, x) = f (x),
where r is a constant and
Z
0
T
2
E [σ(t, Xt )vx (t, Xt )] dt < ∞.
Then v(t, Xt ) = e−r(T −t) E f (XT )|FtW , 0 ≤ t ≤ T .
Proof.
(13.13)
One easily checks that
w(t, x) := er(T −t) v(t, x)
satises (13.12) and
170
Option Valuation: A First Course in Financial Mathematics
Remark.
In the derivation of the Black-Scholes formula in Chapter 11, -
nancial considerations led to the PDE
vt + rxvx + 21 σ 2 x2 vxx − rv = 0,
0 ≤ t < T.
The corollary therefore provides the desired connection between the PDE and
martingale methods of option valuation.
The BSM Model Revisited
171
13.5 Exercises
1. Show that the risk-neutral probability of a call nishing in the money is
Φ d2 (T, S0 , K, σ, r) .
P∗∗ under which
Φ d1 (T, S0 , K, σ, r) .
2. Use Girsanov's Theorem to nd a probability measure
the probability of a call nishing in the money is
What is
dP∗∗
dP∗ ?
Hint: For the rst part, set
Wt∗∗ := Wt + βt,
β := σ −1 µ − r − σ 2 .
For the second part, use
dP∗∗ dP
dP∗∗
=
.
dP∗
dP dP∗
3. Show that the process
e−(r+σ
2
)t
St
is a
P∗∗ -martingale.
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Chapter 14
Other Options
In Chapter 13, the price of a standard European call option was obtained using
the risk-neutral probability measure given by Girsanov's Theorem. There are
a variety of other options that may be similarly valued. In this chapter we
consider the most common of these.
As in previous chapters, we assume that the markets under consideration
are arbitrage-free, so that the value of a claim is that of a self-nancing,
(Ω, F, P)
replicating portfolio. Throughout,
with expectation operator
E.
as usual, by the generic notation
operator. As before,
S
of the underlying
denotes a xed probability space
Risk-neutral measures on
P∗ ,
(FtS ) denotes
asset S .
with
E∗
Ω
will be denoted,
the corresponding expectation
the natural ltration for the price process
14.1 Currency Options
In this section, we consider derivatives whose underlying is a euro bond.
Let
Dt = erd t
and
Et = ere t
denote the price processes of a US dollar bond
and a euro bond, respectively, where
rates, and let
Q
rd
and
re
are the dollar and euro interest
denote the exchange rate process in dollars per euro (see
Section 4.4). To model the volatility of the exchange rate, we take
Q
to be a
geometric Brownian motion give by
Qt = Q0 exp σWt + (µ − 12 σ 2 )t , 0 ≤ t ≤ T,
where
σ
and
µ
(14.1)
are constants. Dene
St = Qt Et = S0 exp[σWt + (µ + re − 12 σ 2 )t],
which is the dollar value of the euro bond at time
t.
(14.2)
Because of the volatility
of the exchange rate, from the point of view of the domestic investor, the euro
bond is a risky asset. The form of the price process
S
clearly reects that
view.
E H 2 < ∞, we can apply the methods of Chapter 13
self-nancing replicating trading strategy (φ, θ) for H , where
Given a claim
to construct a
H
with
173
Option Valuation: A First Course in Financial Mathematics
174
φt
and
θt
are, respectively, the number of units of the dollar bond and the
euro bond held at time
t.
Set
r := rd − re , α :=
By Girsanov's Theorem,
W∗
µ−r
,
σ
is a Brownian measure under
dP∗ := e
Let
V
Wt∗ := Wt + αt.
and
denote the value process of
−αWT − 21 α2 T
P∗ ,
(14.3)
where
dP
(φ, θ), and set S̃ := D−1 S
and
Ṽ := D−1 V .
P∗ -martingales.
Risk-neutral
Since
S̃
the process
St = S0 exp[σWt∗ + (rd − σ 2 /2)t],
and hence also
Ṽ := D−1 V
are
pricing and the no-arbitrage assumption therefore imply that the time-t dollar
value of
H
is given by
Vt = e−rd (T −t) E∗ (H|Ft ) , 0 ≤ t ≤ T.
Example 14.1.1.
(Currency Forward). Consider a forward contract for the
purchase in dollars of one euro at time
T,
the euro. At time
t
forward at time
the euro costs
QT
T.
Let
K
denote the forward price of
dollars hence the dollar value of the
is
Vt = e−rd (T −t) E∗ (QT − K|Ft ) .
Because there is no cost to enter a forward contract,
K = E∗ QT .
Since
Q = DS̃E −1 ,
(14.4)
V0 = 0
and therefore
E∗ (QT |Ft ) = DT ET−1 E∗ (S̃T |Ft ) = DT ET−1 S̃t = er(T −t) Qt
(14.5)
and in particular
K = E∗ QT = erT Q0 .
(14.6)
Substituting (14.5) and (14.6) into (14.4), we obtain
Vt = e−rd (T −t) er(T −t) Qt − erT Q0 = e−re T ere t Qt − erd t Q0 .
H = f (QT ), where f (x) is continuous. Set
H = f1 (ST ). From (14.2), S is the price process
with µ replaced by µ + re . By Theorem 11.3.2 or
More generally, suppose that
f1 (x) = f e−re T x
so that
of the stock in Chapter 13
Corollary 13.1.3,
Vt = e−rd (T −t) G1 (t, St ),
where
Z
∞
G1 (t, s) :=
−∞
Now dene
n √
o
f1 s exp σ T − t y + rd − 12 σ 2 (T − t) ϕ(y) dy.
G(t, s) = G1 (t, ere t s), so that G1 (t, St ) = G(t, Qt ). Replacing s
G1 (t, s) by ere t s, we arrive at the following result:
the denition of
in
Other Options
Theorem 14.1.2. Let H = f (QT ), where
Then the value of H at time t is
175
f
is continuous and E H 2 < ∞.
Vt = e−rd (T −t) G(t, Qt ),
where
Z
∞
G(t, s) :=
−∞
Example 14.1.3.
n √
o
f exp σ T − t y + (rd − re − 12 σ 2 )(T − t) ϕ(y) dy.
(Currency Call Option). Taking
f (x) = (x − K)+
in The-
orem 14.1.2, we see that the time-t dollar value of an option to buy one euro
for
K
dollars at time
Corollary 11.3.3 with
T is Ct = e−rd (T −t) G(t, Qt ),
r replaced by rd − re ,
where, as in the proof of
G(t, s) = se(rd −re )(T −t) Φ (d1 (s, T − t)) − KΦ (d2 (s, T − t)) ,
d1,2 (s, τ ) :=
ln (s/K) + (rd − re ± σ 2 /2)τ
√
.
σ τ
Thus,
Ct = e−re (T −t) Qt Φ d1 (Qt , T − t) − e−rd (T −t) KΦ d2 (Qt , T − t) .
Kt = e(rd −re )(T −t) Qt
(re −rd )(T −t)
e
Kt we have
This may also be expressed in terms of the forward price
of a euro (see Section 4.4). Indeed, since
Qt =
Ct = e−rd (T −t) Kt Φ dˆ1 (Kt , T − t) − e−rd (T −t) KΦ dˆ2 (Kt , T − t) ,
where
ln (s/K) ± τ σ 2 /2
√
.
dˆ1,2 (s, τ ) =
σ τ
14.2 Forward Start Options
A
forward start option
Consider, for example, a
T0 , for
K = ST0 .
is a contract that gives the holder at time
no extra cost, an option with maturity
T > T0
and strike price
forward start call option, whose underlying call has
(ST − ST0 )+ . Let Vt denote the value of the option at time t. At time
T0 the strike price ST0 is known; hence, for all later times, the forward start
payo
option has the value of the call. Thus, in the notation of Section 11.4,
Vt = C(T − t, St , ST0 , σ, r),
To nd
Vt
for
t ≤ T0 ,
T0 ≤ t ≤ T.
note that
VT0 = C(T − T0 , ST0 , ST0 , σ, r) = C(T − T0 , 1, 1, σ, r)ST0 ,
Option Valuation: A First Course in Financial Mathematics
176
which is the value at time
T0
of a portfolio consisting of
C(T − T0 , 1, 1, σ, r)
units of the underlying security. Since the values of the forward start option
T0 , they must agree at all times t ≤ T0 hence
h
i
Vt = C(T − T0 , 1, 1, σ, r)St = Φ(d1 ) − er(T −T0 ) Φ(d2 ) St , 0 ≤ t ≤ T0 ,
and the portfolio agree at time
where
d1,2 = d1,2 (T − T0 , 1, 1, σ, r) =
r
σ
±
σp
T − T0 .
2
In particular, the initial cost of the forward start call option is
h
i
V0 = Φ(d1 ) − er(T −T0 ) Φ(d2 ) S0 .
14.3 Chooser Options
A
T0
chooser option
gives the holder the right to select at some future date
K,
T > T0 , and underlying S . Let Vt , Ct , and Pt denote, respectively,
time-t values of the chooser option, the call, and the put. In the notation
whether the option is to be a call or a put with common exercise price
maturity
the
of Section 11.4,
Ct = C(T − t, St , K, σ, r)
Since at time
T0
and
Pt = P (T − t, St , K, σ, r).
the holder will choose the option with the higher value,
VT0 = max(CT0 , PT0 )
= max(CT0 , CT0 − ST0 + Ke−r(T −T0 ) )
+
= CT0 + Ke−r(T −T0 ) − ST0 ,
where we have used the put-call parity relation in the second equality. The last
expression is the value at time
K
T0
of a portfolio consisting of a long call option
T and a long put option with strike price
T0 . Since the values of the chooser option
and the portfolio are the same at time T0 , they must be the same for all times
t ≤ T0 . Thus, using put-call parity again and noting that K1 e−r(T0 −t) =
Ke−r(T −t) , we have for 0 ≤ t ≤ T0
with strike price
K1 := Ke−r(T −T0 )
and maturity
and maturity
Vt = C(T − t, St , K, σ, r) + P (T0 − t, St , K1 , σ, r)
= C(T − t, St , K, σ, r) + C(T0 − t, St , K1 , σ, r) − St + Ke−r(T −t) .
In particular,
V0 = C(T, S0 , K, σ, r) + C(T0 , S0 , K1 , σ, r) − S0 + Ke−rT .
(14.7)
Other Options
177
To evaluate (14.7) we apply Black-Scholes:
C(T, S0 , K, σ, r) = S0 Φ(d1 ) − Ke−rT Φ(d2 )
C(T0 , S0 , K1 , σ, r) = S0 Φ(dˆ1 ) − K1 e−rT0 Φ(dˆ2 ),
and
where
ln (S0 /K) + (r ± σ 2 /2)T
√
and
σ T
ln (S0 /K) + rT ± σ 2 T0 /2
√
= d1,2 (T0 , S0 , K1 , σ, r) =
.
σ T0
d1,2 = d1,2 (T, S0 , K, σ, r) =
dˆ1,2
Substituting into (14.7), we obtain the formula
V0 = S0 Φ(d1 ) − Ke−rT Φ(d2 ) + S0 Φ(dˆ1 ) − Ke−rT Φ(dˆ2 ) − S0 + Ke−rT
= S0 Φ(d1 ) + Φ(dˆ1 ) − 1 − Ke−rT Φ(d2 ) + Φ(dˆ2 ) − 1
= S0 Φ(d1 ) − Φ(−dˆ1 ) − Ke−rT Φ(d2 ) − Φ(−dˆ2 ) .
The value of the option for
T0 ≤ t ≤ T is either Ct or Pt , depending on
T0 . To distinguish between the two
whether the call or put was chosen at time
scenarios let
A = {CT0 > PT0 }.
Since
IA = 1 i the call was chosen and IA0 = 1 i the put was chosen we have
Vt = Ct IA + Pt IA0 , T0 ≤ t ≤ T.
In particular, the payo of the chooser option is
VT = ST − K
+
IA + K − ST
+
IA0 .
14.4 Compound Options
compound option is a call or put option whose underlying is another call
S . Consider the case of a call-on-call
option. Suppose that the underlying call has strike price K and maturity T ,
A
or put option, the latter with underlying
and that the compound option has strike price
seek the fair price
V0cc
K0
and maturity
of the compound option at time
The value of the underlying call at time
T0
is
T0 < T .
We
0.
C(ST0 ), where, by the Black-
Scholes formula,
C(s) = C(T − T0 , s, K, σ, r) = sΦ d1 (s) − Ke−r(T −T0 ) Φ d2 (s) ,
Option Valuation: A First Course in Financial Mathematics
178
ln (s/K) + (r ± σ 2 /2)(T − T0 )
√
.
σ T − T0
+
f (s) = C(s) − K0 , the cost of
d1,2 (s) =
By Corollary 13.1.3 with
option is
V0cc = e−rT0
where
Since
Z
∞
−∞
the compound
+
C (g(y)) − K0 ϕ(y) dy,
(14.8)
n p
o
g(y) = S0 exp σ T0 y + r − 12 σ 2 T0 .
+
C (g(y)) − K0
is increasing in
y,
+ C (g(y)) − K0 = C (g(y)) − K0 I(y0 ,∞) ,
where
y0 := inf{y | C (g(y)) > K0 }.
Therefore,
V0cc
−rT0
Z
∞
=e
y0
h
i
g(y)Φ dˆ1 (y) − Ke−r(T −T0 ) Φ dˆ2 (y) ϕ(y) dy
− e−rT0 K0 Φ(−y0 ),
where
√
ln (S0 /K) + σ T0 y + rT + σ 2 (T − 2T0 )/2
√
dˆ1 (y) := d1 g(y) =
σ T − T0
√
ln (S0 /K) + σ T0 y + (r − σ 2 /2)T
ˆ
√
d2 (y) := d2 g(y) =
.
σ T − T0
and
14.5 Path-Dependent Derivatives
Recall that a path-dependent derivative is a contract whose payo depends
not just on the value of the underlying at maturity but on the entire history
of the asset over the duration of the contract. Because of this dependency, the
valuation of path-dependent derivatives is more complex than that of pathindependent derivatives.
In this section we consider the most common path-dependent derivatives:
barrier options, lookback options, and Asian options.
Other Options
14.5.1
179
Barrier Options
The payo for a
barrier option
depends on whether the value of the asset
has crossed a predetermined level, called a
barrier. Because of this added con-
dition, barrier options are generally cheaper than standard options. They are
useful because they allow the holder to forego paying a premium for scenarios
deemed unlikely, while still retaining the essential features of a standard option. For example, if an investor believes that a stock will not fall below $20,
he could buy a barrier call option on the stock with payo
stock remains above $20, and zero otherwise.
(ST − K)+
if the
(ST − K)+ IA if the option
put, where A is a barrier event.
The payo for a barrier option has the form
is a call and
+
(K − ST ) IA
if the option is a
The indicator function acts as a switch, activating or deactivating the option
if the barrier is breached. Barrier events are typically described in terms of
the random variables
MS
and
mS ,
where for a process
M X := max{Xt | 0 ≤ t ≤ T }
X
mX := min{Xt | 0 ≤ t ≤ T }.
and
1
The most common barrier events are
up-and-out option;
down-and-out option;
up-and-in option;
down-and-in option;
{M S ≤ c} :
{mS ≥ c} :
{M S ≥ c} :
{mS ≤ c} :
For the rst two cases, the so-called
deactivated if asset rises above
deactivated if asset falls below
activated if asset rises above
activated if asset falls below
c;
c;
c;
c.
knock-out cases, the barrier is set so
knock-in cases, the option is
that the option is initially active, while for the
initially inactive. For example, in the case of an up-and-out option,
while for a down-and-in option
S0 > c.
In this section, we show that the price
C0do
S0 < c,
of a down-and-out call option
is given by the formula
"
C0do
= S0 Φ(d1 ) −
c
S0
2r2 +1
σ
#
"
−rT
Φ(δ1 ) − Ke
Φ(d2 ) −
c
S0
2r2 −1
#
σ
Φ(δ2 ) ,
(14.9)
where, with
d1,2 =
M := max(K, c),
ln
S0
M
+ (r ±
√
σ T
1 2
2 σ )T
ln
and
δ1,2 =
c2
S0 M
+ (r ± 12 σ 2 )T
√
.
σ T
(14.10)
To establish (14.9), note rst that the payo for a down-and-out call is
CTdo = (ST − K)+ IA ,
1 Strict
where
inequalities may be used here, as well.
A = {mS ≥ c}.
Option Valuation: A First Course in Financial Mathematics
180
Since the option is out of the money if
CTdo = (ST − K)IB ,
ST < K ,
the payo may be written
B = {ST ≥ K, mS ≥ c}.
where
By risk-neutral pricing, the cost of the option is therefore
C0do = e−rT E∗ (ST − K)IB .
(14.11)
t
By Lemma 13.2.2, the value of the underlying at time
∗
St = S0 eσ(Wt +βt) , β :=
where
Wt∗
is a
P∗ -Brownian
r
σ
− , 0 ≤ t ≤ T,
σ
2
∗
1
ZT := e−βWT − 2 β
Since
ST = S0 eσŴT
and
(14.12)
Ŵt := Wt∗ + βt
∗
by dP̂ = ZT dP ,
motion. By Girsanov's Theorem,
is a Brownian motion under the probability measure
where
may be expressed as
2
T
Ŵ
mS = S0 eσm
1
given
P̂
= e−β ŴT + 2 β
2
T
.
, we may express
B
as
B = {ŴT ≥ a, mŴ ≥ b}, a := σ −1 ln (K/S0 ), b := σ −1 ln (c/S0 ).
Note that the barrier is set so that
ST ZT−1 = S0 e
b < 0.
γ ŴT − 21 β 2 T
,
(14.13)
Since
γ := σ + β =
r
σ
+ ,
σ
2
Equation (14.11) and the change of measure formula yield
C0do = e−rT Ê (ST − K)IB ZT−1
h
i
2
= e−(r+β /2)T S0 Ê eγ ŴT IB − K Ê eβ ŴT IB .
It remains then to evaluate
Ê eλŴT IB
for
λ=γ
and
β.
(14.14)
For this we shall
2
need the following lemma, which we state without proof.
Lemma 14.5.1. The joint density fˆm (x, y) of (ŴT , mŴ ) under P̂ is given by
fˆm (x, y) = ĝm (x, y)IE (x, y), where
−(x − 2y)2
2(x − 2y)
ĝm (x, y) = √
exp
2T
T 2πT
E = {(x, y) | y ≤ 0, y ≤ x}.
2 The
derivation of the density formula (14.15) is based on the
(14.15)
and
reection principle
of
Brownian motion, which asserts that the rst time the process hits a specied nonzero level
l
it starts anew, and its probabilistic behavior thereafter is invariant under reection in the
horizontal line through l. For a detailed account, the reader is referred to [3] or [17].
Other Options
181
From the lemma and (14.13), we see that for any real number
Ê eλŴT IB = Ê eλŴT I[a,∞)×[b,∞) (ŴT , mŴ )
ZZ
=
eλx ĝm (x, y) dA,
λ,
(14.16)
D
where
D = E ∩ [a, ∞) × [b, ∞) = {(x, y) | b ≤ y ≤ 0, x ≥ a, x ≥ y}.
The integral in (14.16) depends on the relative values of
K
and
S0 .
K
and
c
and also of
To facilitate its evaluation, we prepare the following lemma.
Lemma 14.5.2. For any real number
yj ,
Z
Z
x2
y2
x1
y1
λ
and extended real numbers xj and
eλx ĝm (x, y) dy dx = Iλ (y2 ; x1 , x2 ) − Iλ (y1 ; x1 , x2 ),
where, for y = y1 or y2 and integration variable x1 < x < x2 ,

o
n 2y−x
2yλ+λ2 T /2

√1 +λT − Φ 2y−x
√2 +λT

e
,
Φ

T
T
Iλ (y; x1 , x2 ) =
Proof.
Iλ (0; x1 , x2 ),


0,
y 6= x
y=x
y = ±∞.
A simple substitution yields
Z
y2
y1
ĝm (x, y) dy = √
where
u(x, y) =
i
1 h u(x,y2 )
e
− eu(x,y1 ) ,
2πT
 2

√
− x−2y
,

2T
u(x, 0),



−∞,
y 6= x
real
y=x
y = ±∞.
Thus,
Z
x2
x1
Z
y2
y1
y = y1
eλx ĝm (x, y) dy dx = Jλ (y2 ; x1 , x2 ) − Jλ (y1 ; x1 , x2 ),
x1
 1 R x2
λx+u(x,y)
dx,

 √2πT x1 e
Jλ (y; x1 , x2 ) = Jλ (0; x1 , x2 ),


0,
where, for
or
y2
and integration variable
< x < x2 ,
y
real and
y=x
y = ±∞.
real
y 6= x
Option Valuation: A First Course in Financial Mathematics
182
It remains to show that
Jλ (y; x1 , x2 ) = Iλ (y; x1 , x2 )
if
y
is real and
Since
x2 − 4xy + 4y 2
x2
y2
2y
=−
−
+ λ+
x,
2T
2T
T
T
λx + u(x, y) = λx −
2
e−2y /T
Jλ (y; x1 , x2 ) = √
2πT
y 6= x.
Z
x2
2
e−x /(2T )+(λ+2y/T )x dx
x1
2y − x1 + λT
2y − x2 + λT
2yλ+λ2 T /2
√
√
=e
Φ
−Φ
,
T
T
where for the last equality we used Exercise 11.12 with
λ + 2y/T .
Thus,
Jλ = Iλ ,
p = 1/(2T )
and
q=
completing the proof.
It is now a straightforward matter to evaluate (14.16). Suppose rst that
K > c,
so that
a > b.
If
K ≥ S0 ,
Z
0
Lemma 14.5.2,
Z
Ê eλŴT IB =
∞
a
then
a≥0
and
D = [a, ∞) × [b, 0]
hence, by
eλx ĝm (x, y) dy dx
b
= Iλ (0; , a, ∞) − Iλ (b; a, ∞)
2b − a + λT
−a + λT
2bλ
λ2 T /2
√
√
−e Φ
.
=e
Φ
T
T
On the other hand, if
K < S0
then
a<0
and
D = {(x, y) | a ≤ x ≤ 0, b ≤ y ≤ x} ∪ [0, ∞) × [b, 0]
(Figure 14.1) so, again by Lemma 14.5.2,
y
x
a
D
b
FIGURE 14.1:
D
for the case
c < K < S0 .
(14.17)
Other Options
0
Z
Ê eλŴT IB =
a
Z
x
eλx ĝm (x, y) dy dx +
183
Z
b
∞
0
Z
0
eλx ĝm (x, y) dy dx
b
= Iλ (0; a, 0) − Iλ (b; a, 0) + Iλ (0; 0, ∞) − Iλ (b; 0, ∞)
−a + λT
λT
λ2 T /2
√
=e
Φ
−Φ √
T
T
2
2b
−
a + λT
2b + λT
2bλ+λ T /2
√
√
−e
Φ
−Φ
T
T
2
2
λT
2b
+
λT
√
+ eλ T /2 Φ √
− e2bλ+λ T /2 Φ
.
T
T
The last expression reduces to (14.17), which therefore holds for all values of
S0
and
K
with
K > c.
We are now ready to evaluate
β
C0do
for the case
K > c.
Taking
λ=γ
and
in (14.17), we see from (14.14) that
C0do = e(γ
2
2b − a + γT
−a + γT
√
√
− e2bγ Φ
S0 Φ
T
T
−a + βT
2b − a + βT
√
√
Φ
− e2bβ Φ
.
(14.18)
T
T
−β 2 −2r)T /2
− Ke−rT
Recalling that
a = σ −1 ln (K/S0 ), b = σ −1 ln (c/S0 ), β =
r
σ
− ,
σ
2
and
γ=
r
σ
+ ,
σ
2
we have
γ 2 − β 2 = 2r, e2bγ =
c
S0
2r2 +1
σ
,
and
e2bβ =
c
S0
2r2 −1
σ
.
Furthermore, one readily checks that
−a + γT
√
T
−a + βT
√
T
2b − a + γT
√
T
2b − a + βT
√
T
ln(S0 /K) + (r + σ 2 /2)T
√
σ T
ln(S0 /K) + (r − σ 2 /2)T
√
=
σ T
2
ln(c /S0 K) + (r + σ 2 /2)T
√
=
σ T
2
ln(c /S0 K) + (r − σ 2 /2)T
√
=
σ T
=
= d1 ,
= d2 ,
= δ1 ,
= δ2 .
Inserting these expressions into (14.18) establishes (14.9) for the case
(M
K>c
= K ).
Now suppose
K ≤ c.
(Figure 14.2) and
Then
a≤b
hence
D = {(x, y) | b ≤ y ≤ 0, x ≥ y}
Option Valuation: A First Course in Financial Mathematics
184
y
x
b
D
b
FIGURE 14.2:
Ê e
λŴT
Z
0
Z
IB =
D
for the case
x
Z
λx
K ≤ c.
∞
0
Z
eλx ĝm (x, y) dy dx
e ĝm (x, y) dy dx +
b
b
0
b
= Iλ (0; b, 0) − Iλ (b; b, 0) + Iλ (0; 0, ∞) − Iλ (b; 0, ∞)
−b + λT
b + λT
λ2 T /2
2bλ
√
√
=e
Φ
−e Φ
.
T
T
(14.19)
Thus, from (14.14),
C0do
Since
=e
−b + γT
b + γT
2bγ
√
√
S0 Φ
−e Φ
T
T
−b
+
βT
b
+
βT
−rT
2bβ
√
√
−e
K Φ
−e Φ
.
T
T
(γ 2 −β 2 −2r)T /2
ln(S0 /c) + (r + σ 2 /2)T
√
σ T
ln(c/S0 ) + (r + σ 2 /2)T
√
=
σ T
ln(S0 /c) + (r − σ 2 /2)T
√
=
σ T
ln(c/S0 ) + (r − σ 2 /2)T
√
=
σ T
−b + γT
√
T
b + γT
√
T
−b + βT
√
T
b + βT
√
T
=
we see that (14.9) holds for the case
the derivation of (14.9).
We remark that the barrier level
to a value less than
K;
K ≤ c (M = c),
c
(14.20)
= d1 ,
= δ1 ,
= d2 ,
= δ2 ,
as well. This completes
for a down-and-out call is usually set
otherwise, the option could be knocked out even if it
expires in the money.
Example 14.5.3.
Table 14.1 gives prices
based on a stock that sells for
C0do
of down-and-out call options
S0 = $50.00. The parameters are T = .5, r = .10
Other Options
and
σ = .20.
and $1.87 for
185
The cost of the corresponding standard call is $7.64 for
K = 55.
c
C0do
K = 45
Notice that the price of the barrier option decreases as
39
42
45
47
49
49.99
$7.64
$7.54
$6.74
$5.15
$2.16
$0.02
K = 45
c
C0do
42
43
45
47
49
49.99
$1.87
$1.86
$1.81
$1.57
$0.78
$0.01
K = 55
TABLE 14.1: Variation of
C0do
with the barrier level
c.
the barrier level increases. This is to be expected, since the higher the barrier
the more likely the option will be knocked out hence the less attractive the
option.
14.5.2
Lookback Options
lookback option
A
is another example of a path-dependent option, the
payo in this case depending on the maximum or minimum value of the asset
over the contract period. There are two main categories of lookback options,
oating strike
and
xed strike.
The holder of a oating strike lookback call
option has the right at maturity to buy the stock for its lowest value over
the duration of the contract, while the holder of a oating strike lookback
put option may sell the stock at its high. The payos of lookback call and
put options are, respectively,
ST − mS
and
M S − ST ,
where
mS
and
MS
are
dened as in Subsection 14.5.1. In the present subsection, we determine the
value
Vt
of a oating strike lookback call option. Fixed strike lookback options
are examined in Exercise 8.
By risk-neutral pricing,
Vt = e−r(T −t) E∗ (ST −mS |Ft ) = St −e−r(T −t) E∗ (mS |Ft ), 0 ≤ t ≤ T,
where we have used the fact that the discounted asset price is a
As in Subsection 14.5.1, the value of the underlying at time
as
St = S0 eσŴt ,
β :=
P∗ -Brownian
motion.
∗
To evaluate E
mS |Ft , we introduce
For a process X and for t ∈ [0, T ], set
where
Wt∗
Ŵt := Wt∗ + βt,
σ
r
− ,
σ
2
(14.21)
P∗ -martingale.
t may be expressed
0 ≤ t ≤ T,
is a
mX
t := min{Xu | 0 ≤ u ≤ t}
and
the following additional notation:
mX
t,T := min{Xu | t ≤ u ≤ T }.
Option Valuation: A First Course in Financial Mathematics
186
Thus, in our earlier notation,
mX = mX
T .
Now let
t<T
and set
Yt = eσ min{Ŵu −Ŵt |t≤u≤T } .
Su = St eσ(Ŵu −Ŵt ) ,
Since
mSt,T = min{St eσ(Ŵu −Ŵt ) | t ≤ u ≤ T } = St Yt ,
and therefore
mS = min(mSt , mSt,T ) = min(mSt , St Yt ).
mSt
Since
is
Ft -measurable and Yt is independent of Ft , we
g(x1 , x2 , y) = min(x1 , x2 y) to conclude that
can apply Re-
mark 13.2.6 with
E∗ (mS |Ft ) = E∗ (g(mSt , St , Yt )|Ft ) = Gt (mSt , St ),
where
Gt (m, s) = E∗ g(m, s, Yt ) = E∗ min(m, sYt ),
From (14.21) we see that
Vt = vt (mSt , St ),
Vt
0 < m ≤ s.
may now be expressed as
where
vt (m, s) = s − e−r(T −t) Gt (m, s).
The remainder of the section is devoted to evaluating
To this end, x
t, m
and
(14.22)
s
A = {mŴ
τ ≤ a},
with
0<m≤s
τ := T − t,
(14.23)
Gt .
and set
a :=
1 m
ln
.
σ
s
∗
+ β(u − t), we see
Ŵu − Ŵt = Wu∗ − Wt∗ + β(u − t) and Ŵu−t = Wu−t
∗
that Ŵu − Ŵt and Ŵu−t have the same distribution under P . Therefore Yt
Ŵ
σmτ
∗
and e
have the same distribution under P . It follows that
h
i
h
i
Ŵ
Ŵ
Gt (m, s) − m = E∗ min m, seσmτ − m = E∗ min 0, seσmτ − m .
Since
Noting that
Ŵ
Ŵ
min 0, seσmτ − m = seσmτ − m
i
mŴ
τ ≤ a,
h
i
Ŵ
Gt (m, s) = E∗ (seσmτ − m)IA + m
Ŵ
= sE∗ eσmτ IA + m (1 − E∗ IA ) .
It remains then to evaluate
the following lemmas.
we see that
(14.24)
Ŵ
E∗ eσmτ IA and E∗ (IA ). For this, we shall need
Other Options
Lemma 14.5.4. The joint density
by
fm (x, y)
187
∗
of (Ŵτ , mŴ
τ ) under P is given
fm (x, y) = eβx− 2 β τ gm (x, y)IE (x, y), where
(x − 2y)2
2(x − 2y)
exp −
gm (x, y) = √
and
2τ
τ 2πτ
1
2
E = {(x, y) | y ≤ 0, y ≤ x}.
Proof.
By Girsanov's Theorem,
probability measure
P̂
(Ŵu )0≤u≤τ
is a Brownian motion under the
given by
1
2
dP̂ = Zτ dP∗ , Zτ = e−β Ŵτ + 2 β τ .
τ , the joint density of (Ŵτ , mŴ
τ ) under P̂
Ŵ
is gm (x, y)IE (x, y), where gm and E are dened as above. The cdf of (Ŵτ , mτ )
∗
under P is therefore
∗
P∗ Ŵτ ≤ x, mŴ
≤
y
=
E
I
Ŵ
τ
{Ŵτ ≤x,mτ ≤y}
= Ê I{Ŵτ ≤x,mŴ ≤y} Zτ−1
τ
1 2
= Ê I{Ŵτ ≤x,mŴ ≤y} eβ Ŵτ − 2 β τ
τ
Z x Z y
1 2
=
eβu− 2 β τ gm (u, v)IE (u, v) dv du,
By Lemma 14.5.1 with
T
replaced by
−∞
−∞
verifying the lemma.
∗
Lemma 14.5.5. The density fm of mŴ
τ under P is given by
fm (z) = gm (z)I(−∞,0] (z), where
2
z − βτ
z + βτ
2βz
√
√
gm (z) = √ ϕ
+ 2βe Φ
.
τ
τ
τ
Proof.
∗
mŴ
τ under P is
Z ∞Z z
≤z =
fm (x, y) dy dx
−∞ −∞
ZZ
2
= e−β τ /2
eβx gm (x, y) dA,
By Lemma 14.5.4, the cdf of
P∗ mŴ
τ
D
D = {(x, y) | y ≤ min(0, x, z)}. Suppose rst that z ≤ 0. The integral
D may then be expressed as I 0 + I 00 , where
Z ∞Z z
Z z Z x
I0 =
eβx gm (x, y) dy dx and I 00 =
eβx gm (x, y) dy dx
where
over
z
−∞
−∞
−∞
Option Valuation: A First Course in Financial Mathematics
188
y
x
z
z
D
FIGURE 14.3:
D = {(x, y) | y ≤ min{x, z}}, z ≤ 0.
τ ),
2
z + βτ
√
I 0 = Iβ (z; z, ∞) − Iβ (−∞; z, ∞) = e2zβ+β τ /2 Φ
τ
2
z
−
βτ
√
I 00 = Iβ (0; −∞, z) − Iβ (−∞; −∞, z) = eβ τ /2 Φ
.
τ
(Figure 14.3). By Lemma 14.5.2 (with
Thus, if
T
replaced by
and
z ≤ 0,
z − βτ
z + βτ
√
√
+Φ
.
P∗ mŴ ≤ z = e2βz Φ
τ
τ
Dierentiating the expression on the right with respect to
z
and using the
identity
z + βτ
z − βτ
√
√
=ϕ
τ
τ
P∗ mŴ ≤ z = P∗ mŴ ≤ 0 for z > 0,
e2zβ ϕ
gm (z).
produces
Since
the conclu-
sion of the lemma follows.
We are now in a position to evaluate
(noting that
∗
E
a ≤ 0),
e
λmŴ
τ
Ŵ
E∗ eλmτ IA .
By Lemma 14.5.5
we have
IA =
a
2
eλz gm (z) dz = √ Jλ0 + 2βJλ00 ,
τ
−∞
Z
(14.25)
where
Z
2
z − βτ
e−β τ /2 a −z2 /(2τ )+(β+λ)z
√
dz = √
e
dz and
τ
2π
−∞
−∞
Z a
Z a
Z z+βτ
√
τ
2
z + βτ
1
00
(λ+2β)z
(λ+2β)z
√
Jλ =
e
Φ
dz = √
e
e−x /2 dx dz.
τ
2π −∞
−∞
−∞
Jλ0 =
Z
a
eλz ϕ
Other Options
189
By Exercise 11.12,
Jλ0
To evaluate
Jλ00 ,
1
Jλ00 = √
2π
=
√
τe
λβτ +λ2 τ /2
Φ
a − (λ + β)τ
√
τ
we reverse the order of integration: For
b
Z
e−x
2
/2
Z
(14.26)
λ 6= −2β ,
a
a + βτ
√
τ
τ x−βτ
√
2
e−x /2 e(λ+2β)a − e(λ+2β)( τ x−βτ ) dx
e(λ+2β)z dz dx,
√
−∞
.
b :=
Z b
1
√
=
(λ + 2β) 2π −∞
h
√ i
2
1
=
e(λ+2β)a Φ(b) − eλβτ +λ τ /2 Φ b − (λ + 2β) τ ,
(λ + 2β)
(14.27)
where to obtain the last equality we used Exercise 11.12 again.
From (14.26) and (14.27),
Jσ0 =
√
Jσ00 =
h
√ i
2
1
e(σ+2β)a Φ(b) − eσβτ +σ τ /2 Φ b − (σ + 2β) τ .
(σ + 2β)
τ eσβτ +σ
2
τ /2
Φ
a − (σ + β)τ
√
τ
and
Recalling that
a=
1 m
σ
r
ln
, β= − ,
σ
s
σ
2
and
b=
m
√
a + βτ
1
√
= √ ln
+ β τ,
s
τ
σ τ
we see that
σ2 τ
= τ r, (σ + 2β)a =
2
√
a − (σ + β)τ
√
= b − (σ + 2β) τ =
τ
σβτ +
2r m ln
and
σ2 s
1 1
m r
σ
√
ln
+
−
τ .
s
σ
2
τ σ
Therefore, setting
ln(s/m) + (r ± σ 2 /2)τ
√
σ τ
ln(m/s) + (r − σ 2 /2)τ
√
d = d(τ, m, s) =
,
σ τ
δ1,2 = δ1,2 (τ, m, s) =
and
we have
Jσ0
=
√
rτ
τ e Φ(−δ1 )
and
Jσ00
2
σ m 2r/σ
rτ
=
Φ(d) − e Φ(−δ1 ) .
2r
s
Option Valuation: A First Course in Financial Mathematics
190
From (14.25) then
Ŵ
2
E∗ eσmτ IA = √ Jσ0 + 2βJσ00
τ
2
σ2
m 2r/σ
= 2erτ Φ(−δ1 ) + 1 −
Φ(d) − erτ Φ(−δ1 )
2r
s
2
2
σ
σ 2 m 2r/σ
rτ
=e
1+
Φ(−δ1 ) + 1 −
Φ(d). (14.28)
2r
2r
s
Similarly, if
β 6= 0,
J00 =
√
a − βτ
√
τ
J000 =
√ 1 2βa
1 m σ2r2 −1
Φ(d) − Φ(−δ2 ) ,
e Φ(b) − Φ b − 2β τ =
2β
2β
s
τΦ
=
√
τ Φ(−δ2 )
and
hence
2
E∗ (IA ) = √ J00 + 2βJ000
τ
2r
m σ2 −1
= 2Φ(−δ2 ) +
Φ(d) − Φ(−δ2 )
s
m 2r2 −1
σ
= Φ(−δ2 ) +
Φ(d).
s
The reader may verify that (14.29) also holds if
β = 0.
(14.29)
From (14.24), (14.28),
and (14.29),
σ 2 m σ2r2
Gt (m, s) = s e
Φ − δ1 + 1 −
Φ d
2r
s
m σ2r2 −1
+ m 1 − Φ − δ2 −
Φ d
s
σ2
= serτ 1 +
Φ − δ1 (τ, m, s) + mΦ δ2 (τ, m, s)
2r
sσ 2 m σ2r2
−
Φ d(τ, m, s) .
(14.30)
2r
s
rτ
σ2
1+
2r
Finally, recalling (14.23), we have
Vt = v(mSt , St ),
where
vt (m, s) = s − e−rτ Gt (m, s)
sσ 2
= sΦ δ1 (m, s, τ ) − me−rτ Φ δ2 (m, s, τ ) −
Φ − δ1 (m, s, τ )
2r
sσ 2 m σ2r2
Φ d(m, s, τ ) .
+ e−rτ
2r
s
Other Options
14.5.3
An
191
Asian Options
Asian
or
average option
of the price process
S
has payo that depends on an average
A(S)
of the underlying asset. The most common types of
Asian options are
ˆ
the
xed strike average call
ˆ
the
oating strike average call
ˆ
the
xed strike average put
ˆ
the
oating strike average put
with payo
tj
satisfy
ˆ
with payo
with payo
(ST − A(S))+ ,
(K − A(S))+ ,
with payo
A(S) is typically one of
0 ≤ t1 < t2 < · · · < tn ≤ T :
The average
(A(S) − K)+ ,
and
(A(S) − ST )+ .
the following, where the discrete times
discrete arithmetic average : A(S) =
n
1X
St ,
n j=1 j
Z
1 T
ˆ continuous arithmetic average : A(S) =
St dt,
T 0
1/n

n
Y
ˆ discrete geometric average : A(S) = 
Stj  ,
j=1
ˆ
continuous geometric average : A(S) = exp
1
T
Z
!
T
ln St dt
.
0
In the continuous case, averaging intervals can be of the more general form
[T0 , T ].
Asian options are usually less expensive than standard options and have
the advantage of being less sensitive to manipulation of the underlying asset price on a particular day, that eect mitigated by the averaging process.
They are also useful as hedges for an investment plan consisting of a series of
purchases over time of a commodity with changing price.
The xed strike geometric average option readily lends itself to BlackScholes-Merton risk-neutral pricing, as the following theorems illustrate.
Theorem 14.5.6. The cost V0 of a xed strike continuous geometric average
call option is
V0 = S0 e−rT /2−σ
where
2
T /12
Φ (d1 ) − Ke−rT Φ (d2 ) ,
ln ( SK0 ) + (r − 12 σ 2 ) T2
q
d2 :=
,
σ T3
r
d1 := d2 + σ
(14.31)
T
.
3
Option Valuation: A First Course in Financial Mathematics
192
Proof.
By risk-neutral pricing,
−rT
V0 = e
+
∗
E [A(S) − K] ,
where
A(S) = exp
1
T
Z
T
!
ln St dt .
0
From Lemma 13.2.2,
σ2
ln St = ln S0 + σWt∗ + r −
t
2
hence
Z
σ T ∗
σ2 T
ln St dt = ln S0 +
Wt dt + r −
.
T 0
2
2
0
RT ∗
By Example 10.6.1,
Wt dt is normal under P∗ with mean zero and variance
0
3
T /3. Therefore
( r
)
σ2 T
T
Z+ r−
,
A(S) = S0 exp σ
3
2
2
1
T
where
T
Z
Z ∼ N (0, 1).
rT
e
Z
It follows that
( r
∞
V0 =
S0 exp σ
−∞
!+
)
T
σ2 T
z+ r−
−K
ϕ(z) dz.
3
2
2
z < −d2 ,
)
Z ∞
∞
T
σ2 T
rT
e V0 = S0
z+ r−
ϕ(z) dz − K
exp σ
ϕ(z) dz
3
2
2
−d2
−d2
)
( r
Z ∞
1 2
T
(r−σ 2 /2)T /2 1
√
z − z dz − KΦ (d2 )
= S0 e
exp σ
3
2
2π −d2
Since the integrand is zero when
( r
Z
= S0 erT /2−σ
2
T /12
Φ (d1 ) − KΦ (d2 ) ,
the last equality by Exercise 11.12.
To price the discrete geometric average call option, we rst establish the
following lemma.
Lemma 14.5.7. Let t0 := 0 < t1 < t2 < · · · < tn ≤ T . The joint density of
under P∗ is given by
(Wt∗1 , Wt∗2 , . . . , Wt∗n )
f (x1 , x2 , . . . , xn ) =
n
Y
j=1
fj (xj − xj−1 ),
where x0 = 0 and fj is the density of Wt∗j − Wt∗j−1 :
1
ϕ
fj (x) = p
2π(tj − tj−1 )
x
√
tj − tj−1
.
Other Options
Proof.
193
Set
A = (−∞, z1 ] × (−∞, z2 ] × · · · × (−∞, zn ]
and
B = {(y1 , y2 , . . . , yn ) | (y1 , y1 + y2 , . . . , y1 + y2 + · · · + yn ) ∈ A}.
By independent increments,
P∗ (Wt∗1 , Wt∗2 , . . . , Wt∗n ) ∈ A = P∗ (Wt∗1 , Wt∗2 − Wt∗1 , . . . , Wt∗n − Wt∗n−1 ) ∈ B
Z
= f1 (y1 )f2 (y2 ) · · · fn (yn ) dy.
B
xj = y1 + y2 + · · · + yj , we obtain
Z
∗
∗
∗
∗
P (Wt1 , Wt2 , . . . , Wtn ) ∈ A = f1 (x1 )f2 (x2 − x1 ) · · · fn (xn − xn−1 ) dx,
With the substitution
A
which establishes the lemma.
Corollary 14.5.8.
σn2 := 12 τn + 22 τn−1 + · · · + n2 τ1 ,
Proof.
∗
E
Wt∗1 + Wt∗2 + · · · + Wt∗n ∼ N (0, σn2 )
under P∗ , where
τj := tj − tj−1 .
By Lemma 14.5.7,
h
e
λ(Wt∗ +Wt∗ +···Wt∗n )
1
i
2
Z
∞
=
−∞
Z
∞
=
−∞
···
Z
···
Z
∞
eλ(x1 +···+xn ) f (x1 , . . . , xn ) dx
−∞
∞
eλ(x1 +x2 +···+xn−1 ) f1 (x1 ) · · · fn−1 (xn−1 − xn−2 )
−∞
Z ∞
−∞
eλxn fn (xn − xn−1 ) dxn dxn−1 · · · dx1 .
The innermost integral evaluates to
1
√
2πτn
Z
∞
e
λxn −(xn −xn−1 )2 /(2τn )
−∞
eλxn−1
dxn = √
2πτn
=e
Z
∞
2
eλxn −xn /(2τn ) dxn
−∞
λxn−1 +τn λ2 /2
.
Therefore,
h
i
∗
∗
∗
E∗ eλ(Wt1 +Wt2 +···+Wtn )
Z ∞
Z ∞
2
= eτn λ /2
···
eλ(x1 +···+xn−2 ) f1 (x1 ) · · · fn−2 (xn−2 − xn−3 )
−∞
−∞
Z ∞
e2λxn−1 fn−1 (xn−1 − xn−2 ) dxn−1 dxn−2 · · · dx1 .
−∞
Option Valuation: A First Course in Financial Mathematics
194
Repeating the argument, we arrive at
h
i
∗
∗
∗
2 2
E∗ eλ(Wt1 +Wt2 +···+Wtn ) = eσn λ /2 .
The corollary now follows from Theorem 12.4.3 and Example 12.4.2.
Theorem 14.5.9. The cost of a xed strike discrete geometric average call
option with payo (A(S) − K)+ , where
n
Y
A(S) :=
!1/(n+1)
,
Stj
tj :=
0
jT
, j = 0, 1, . . . , n,
n
is given by
(n)
V0
p
σ2 T
σ 2 T an
= S0 exp − r +
+
Φ σ an T + dn − KΦ (dn ) ,
2
2
2
where
dn :=
Proof.
ln (S0 /K) + 21 (r − σ 2 /2)T
√
,
σ an T
an :=
2n + 1
.
6(n + 1)
By Corollary 14.5.8,



n

X
A(S) = S0 exp σ(n + 1)−1
Wt∗j + (r − 21 σ 2 )(n + 1)−1
tj


j=1
j=1
= S0 exp σσn (n + 1)−1 Z + r − 12 σ 2 t ,
where
n
X
Z ∼ N (0, 1),
σn2
n
T X 2
(n + 1)(2n + 1)T
=
j =
= (n + 1)2 an T,
n j=1
6
and
n
t=
Since
1 X
T
tj = .
n + 1 j=1
2
σn (n + 1)−1 =
(n)
erT V0
√
an T ,
risk-neutral pricing implies that
= E∗ (A(S) − K)+
p
+
Z ∞
σ2 T
=
S0 exp σ an T z + r −
−K
ϕ(z) dz
2
2
−∞
Z
Z
∞
∞
2
√
1 2
1
r− σ2 T2
√
= S0 e
eσ an T z− 2 z dz − K
ϕ(z) dz
2π −dn
−dn
p
2
r− σ2 T2 + 12 σ 2 an T
= S0 e
Φ σ an T + dn − KΦ(dn ),
the last equality by Exercise 11.12.
Other Options
195
Arguments similar to those of Theorems 14.5.6 and 14.5.9 may be used to
establish formulas for the value of the options at arbitrary time
[10]).
In the arithmetic case,
t ∈ [0, T ]
(see
A(S) does not have a lognormal distribution, hence
arithmetic average options are more dicult to price. Indeed, there is no
known closed form pricing formula as in the geometric average case. Techniques used to price arithmetic average options typically involve approximation, Monte Carlo simulation, or partial dierential equations. Accounts of
these approaches along with references may be found in [1, 10, 12, 17].
14.6 Quantos
A
quanto is a derivative with underlying asset denominated in one currency
and payo denominated in another. Consider the case of a foreign stock with
price process
Se
denominated in euros. Let
Q
denote the exchange rate in
dollars per euro, so that the stock has dollar price process
S := S e Q.
A
standard call option on the foreign stock has payo
(STe − K)+
euros
= (ST − KQT )+
dollars,
but this value might be adversely aected by the exchange rate. Instead,
(STe − K)+
−1
e
dollars. Here, the strike price K is in dollars and ST = ST QT is interpreted
e
as a dollar value (e.g., ST = 5 euros is interpreted as 5 dollars).
e
More generally, consider a claim with dollar value H = f (ST ), where
2
f (x) is continuous and E H < ∞. The methods of Chapter 13 may be easily
a domestic investor could buy a quanto call option with payo
adapted to nd the fair price of such a claim. Our point of view is that of a do-
mestic investor, that is, one who constantly computes investment transactions
in dollar units.
To begin, assume that
S
and
Q are geometric Brownian motion processes,
say
2
St = S0 eσ1 W1 (t)+(µ1 −σ1 /2)t
σ1 , σ2 , µ1 , and µ2
(Ω, F, P). For ease of
and
2
Qt = Q0 eσ2 W2 (t)+(µ2 −σ2 /2)t ,
W1
W2 are Brownian motions
W1 and W2 to be indepen3
r t
r t
dent processes. As in Section 14.1, Dt = e d and Et = e e denote the price
processes of a dollar bond and a euro bond, respectively, where rd and re are
where
are constants and
on
exposition, we shall take
and
the dollar and euro interest rates. Set
α1 :=
µ1 − rd
σ1
and
α2 :=
µ2 + re − rd
.
σ2
3
p A more realistic model assumes that the processes are correlated, that is, W2 = %W1 +
1 − %2 W3 , where W1 and W3 are independent Brownian motions and 0 < |%| < 1. See,
for example, [17].
Option Valuation: A First Course in Financial Mathematics
196
By the general Girsanov's Theorem (Remark 12.5.3), there exists a single
probability measure
P∗
on
Ω
(the so-called
measure ) relative to which the processes
W1∗ (t) := W1 (t) + α1 t
domestic risk-neutral probability
W2∗ (t) := W2 (t) + α2 t, 0 ≤ t ≤ T,
and
(Gt ) generated
S e = SQ−1 may be
are independent Brownian motions with respect to the ltration
by
W1
W2 .
and
In terms of
W1∗
and
W2∗ ,
the process
written
Ste
σ22
σ12
t − σ2 W2 (t) − µ2 −
t
=
exp σ1 W1 (t) + µ1 −
2
2
σ2
σ2
= S0e exp σ1 W1∗ (t) − σ2 W2∗ (t) + re − 1 + 2 t
2
2
2
σ
= S0e exp σW ∗ (t) + re + σ22 −
t ,
(14.32)
2
S0e
where
σ := (σ12 + σ22 )1/2
W ∗ := σ −1 (σ1 W1∗ − σ2 W2∗ ) .
W2∗ are independent, W ∗ is easily seen to be a Brownian motion
with respect to (Gt ). By the continuous parameter version of Theorem 8.4.6,
∗
W ∗ is also a Brownian motion with respect to the Brownian ltration (FtW ).
e
Now let H be a claim of the form f (ST ) and let (φ, θ) be a self-nancing,
replicating trading strategy for H based on the dollar bond and a risky asset
with dollar price process X given by
σ2
Xt := eζt Ste = S0e exp σW ∗ (t) + rd −
t , ζ := rd − re − σ22 .
2
Since
W1∗
and
and
V = φD + θX denote the value process of the portfolio and set f1 (x) =
f e−ζT x , so that H = f1 (XT ). Note that the processes D−1 X and D−1 V
∗
are P -martingales. By Corollary 13.1.3, the value of the claim at time t is
Let
∗
Vt = e−rd (T −t) E∗ (H|FtW ) = e−rd (T −t) G1 (t, Xt ),
where
Z
∞
G1 (t, s) :=
−∞
Dene
eζt s
in
n √
o
f1 s exp σ T − t y + (rd − σ 2 /2)(T − t) ϕ(y) dy.
G(t, s) := G1 t, eζt s , so that G1 (t, Xt ) = G(t, Ste ). Replacing s
the denition of G1 (t, s), we arrive at the following result:
Theorem 14.6.1. Let H = f (STe ), where f is continuous and E H 2
Then the dollar value of H at time t is Vt = e−r (T −t) G(t, Ste ), where
d
Z
∞
G(t, s) :=
−∞
by
< ∞.
n √
o
f s exp σ T − t y + (re + σ22 − σ 2 /2)(T − t) ϕ(y) dy.
Other Options
Example 14.6.2.
197
f (x) = (x − K)+
re + σ22 , we obtain
(Quanto Call Option). Taking
rem 14.6.1 and using (11.13) with
r
replaced by
in Theo-
2
G(t, s) = se(re +σ2 )(T −t) Φ d1 (s, T − t) − KΦ d2 (s, T − t) ,
where
d1,2 (s, τ ) =
ln (s/K) + (re + σ22 ± σ 2 /2)τ
√
.
σ τ
The dollar value of a quanto call option at time
t
is therefore
2
Vt = e−(rd −re −σ2 )(T −t) Ste Φ d1 (Ste , T − t) − e−rd (T −t) KΦ d2 (Ste , T − t) .
14.7 Options on Dividend-Paying Stocks
In this section, we determine the price of a claim based on a dividendpaying stock. We consider two cases: continuous dividends and periodic dividends. We begin with the former, which is somewhat easier to model.
14.7.1
Continuous Dividend Stream
Assume our stock pays a dividend of
to
t + dt,
where
δ
δSt dt
in the time interval from
t
is a constant between 0 and 1. Since dividends reduce the
value of the stock, the price process must be adjusted to reect this reduction.
Therefore we have
dSt = σSt dWt + µSt dt − δSt dt = σSt dWt + (µ − δ)St dt.
(14.33)
(φ, θ) be a self-nancing trading strategy with value process V =
φB +θS . The denition of self-nancing must be modied to take into account
the dividend stream, as the change in V depends not only on changes in the
Now let
stock and bond values but also on the dividend increment. Thus, we require
that
dVt = φt dBt + θt dSt + δθt St dt.
From (14.33),
dV = φ dB + θ σS dW + (µ − δ)S dt + δθS dt
= φ dB + θS σ dW + µ dt .
Now set
θ̂t = e−δt θt
and
Ŝt = eδt St = S0 exp σWt + (µ − 12 σ 2 )t .
(14.34)
Option Valuation: A First Course in Financial Mathematics
198
Note that
Ŝt
is the price process of the stock without dividends (or with
V = φB + θ̂Ŝ is the value process of a trading
(φ, θ̂) based on the bond and the non-dividend-paying version of our
stock. Since dŜ = Ŝ(σ dW + µ dt), we see from (14.34) that dV = φ dB + θ̂ dŜ ,
that is, the trading strategy (φ, θ̂) is self-nancing. The results of Chapter 13
therefore apply to Ŝ and we see that the value of a claim H is the same as
−r(T −t) ∗
that for a stock without dividends, namely, e
E (H|Ft ), where P∗ is
the risk-neutral measure. The dierence here is that the discounted process S̃
∗
is no longer a P -martingale, as seen from the representation of S as
dividends reinvested) and
strategy
∗
St = e−δt Ŝt = S0 eσWt +(r−δ−σ
2
/2)t
.
(14.35)
∗ 2
H = f (ST ), where f (x) is continuous and E H < ∞.
f1 (x) = f e
x , so that H = f1 (ŜT ). By Corollary 13.1.3, the value of
−r(T −t)
claim at time t is Vt = e
G1 (t, Ŝt ), where
Z ∞
o
n √
G1 (t, s) :=
f1 s exp σ T − t y + (r − σ 2 /2)(T − t) ϕ(y) dy.
Now suppose that
Set
the
−δT
−∞
G(t, s) = G1 t, eδt s , so that G1 (t, Ŝt ) = G(t, St ).
δt
denition of G1 by e s, we obtain the following result:
Dene
Replacing
Theorem 14.7.1. Let H = f (ST ), where f is continuous and
Then the value of H at time t is Vt = e−r(T −t) G(t, St ), where
Z
∞
G(t, s) :=
−∞
in the
E H 2 < ∞.
n √
o
f s exp σ T − t y + (r − δ − σ 2 /2)(T − t) ϕ(y) dy.
Example 14.7.2.
(x − K)+
s
f (x) =
r − δ , we
(Call option on a dividend-paying stock). Taking
r
in Theorem 14.7.1 and using (11.13) with
replaced by
have
G(t, s) = se(r−δ)(T −t) Φ (d1 (s, T − t)) − KΦ (d2 (s, T − t)) ,
where
d1,2 (s, τ ) =
ln (s/K) + (r − δ ± σ 2 /2)τ
√
.
σ τ
The time-t value of a call option on a dividend-paying stock is therefore
Ct = e−δ(T −t) St Φ d1 (St , T − t) − e−r(T −t) KΦ d2 (St , T − t) .
14.7.2
Discrete Dividend Stream
Now suppose that our stock pays a dividend only at the discrete times
tj ,
where
0 < t1 < t2 < · · · < tn < T .
Set
t0 = 0
tn+1 = T . Between
dSt = σSt dWt +µSt dt.
and
dividends the stock price is assumed to follow the SDE
At each dividend-payment time, the stock value is reduced by the amount of
Other Options
199
the dividend, which we again assume is a fraction
δ ∈ (0, 1)
of the value of
the stock. The price process is no longer continuous but has jumps at the
dividend-payment times
tj .
This may be modeled by the equations
St = Stj eσ(Wt −Wtj )+(µ−σ
Stj+1 = Stj e
Setting
Ŝt = S0 eσWt +(µ−σ
Stj+1 =
j = n
/2)(t−tj )
tj ≤ t < tj+1 , j = 0, 1, . . . , n,
,
σ (W (tj+1 )−Wtj )+(µ−σ 2 /2)(tj+1 −tj )
St =
If
2
2
/2)t
Stj Ŝt
Ŝtj
(1 − δ), j = 0, 1, . . . , n − 1.
we can rewrite these as
tj ≤ t < tj+1 ,
,
Stj Ŝtj+1
Ŝtj
(1 − δ),
j = 0, 1, . . . , n,
j = 0, 1, . . . , n − 1.
the rst formula also holds for
t = tn+1 (= T ).
(14.36)
Let
(φ, θ)
be a
trading strategy based on the stock and the bond. Between dividends, the
value process is given by
Vt = φt Bt + θt St = φt Bt + θt
Stj Ŝt
Ŝtj
,
tj ≤ t < tj+1 .
At time tj+1 the stock portion of the portfolio decreases in value by the amount
δθtj
Stj Ŝtj+1
Ŝtj
,
but the cash portion increases by the same amount because of the dividend.
Since the net change is zero,
V
is a continuous process. Moreover, assuming
tj ≤ t < tj+1
dVt = φt dBt + θt σSt dWt + µSt dt = rφt Bt + µθt St dt + σθt St dWt .
that
(φ, θ)
is self-nancing between dividends, we have for
From (14.36), for
have
1≤m≤n
and
tm ≤ t < tm+1 ,
or
m=n
and
t = T,
we
m
m
St
St Y Stj
Ŝt Y Ŝtj
Ŝt
=
= (1 − δ)m
= (1 − δ)m ,
S0
Stm j=1 Stj−1
S0
Ŝtm j=1 Ŝtj−1
that is,
St = (1 − δ)m Ŝt , tm ≤ t < tm+1 ,
In particular,
ST
is the value at time
T
and
of a non-dividend-paying stock with
the geometric Brownian motion price process
can replicate a claim
ST = (1 − δ)n ŜT .
X = (1 − δ)n Ŝ .
Therefore, we
H = f (ST ) = f (XT ) with a self-nancing strategy based
on a stock with this price process and the original bond. By Corollary 13.1.3,
the value of the claim at time
Vt = e
where
−r(T −t)
G(t, s
G (t, Xt ) = e
t
is
−r(T −t)
is given by (13.3).
G t, (1 − δ)n−m St , tm ≤ t < tm+1 ,
Option Valuation: A First Course in Financial Mathematics
200
14.8 American Claims in the BSM Model
Recall that the holder of an American claim has the right to exercise the
claim at any time
t ≤ T . Many of the results concerning valuation of American
claims in the binomial model carry over to the BSM setting. The verications
are more dicult, however, and require advanced techniques from martingale
theory. In this section, we give a brief overview of the main ideas, without
proofs. For details the reader is referred to [8, 12, 13].
t is of
g(t, St ). The holder will try to choose an exercise time that optimizes
Assume the payo of a (path-independent) American claim at time
the form
his payo. The exercise time, being a function of the price of the underlying,
is a random variable, the determination of which cannot rely on future information. As in the discrete case, such a random variable is called a
time, formally dened as a function τ
on
Ω
with values in
[0, T ]
stopping
such that
{τ ≤ t} ∈ Ft , 0 ≤ t ≤ T.
Since
that
{τ > t − 1/n} = {τ ≤ t − 1/n}0 ∈ Ft
∞
\
{τ = t} =
for any positive integer
n it follows
{t − 1/n < τ ≤ t} ∈ Ft , 0 ≤ t ≤ T.
n=1
Now let Tt,T denote the set of all stopping times with values in the interval
[t, T ], 0 ≤ t ≤ T . If at time t the holder of the claim chooses the stopping time
τ ∈ Tt,T , her payo will be g(τ, Sτ ). The writer will therefore need a portfolio
that covers this payo. By risk-neutral pricing, the time-t value of the payo
∗
is E
e−r(τ −t) g(τ, Sτ )|Ft . Thus, if the writer is to cover the claim for any
choice of stopping time, then the value of the portfolio at time t should be
Vt = max E∗ e−r(τ −t) g(τ, Sτ )|Ft .4
(14.37)
τ ∈Tt,T
One can show that a trading strategy
(φ, θ)
exists with value process given
by (14.37). With this trading strategy, the writer may hedge the claim, so it
is natural to take the value of the claim at time
t
to be
Vt .
In particular, the
fair price of the claim is
V0 = max E∗ e−rτ g(τ, Sτ ) .
τ ∈T0,T
It may be shown that price
V0
(14.38)
given by (14.38) guarantees that neither
the holder nor the writer of the claim has an arbitrage opportunity. More
precisely,
4 Since
the conditional expectations in this equation are dened only up to a set of
probability one, the maximum must be interpreted as the
standard texts on real analysis.
essential supremum,
dened in
Other Options
ˆ
201
it is not possible for the holder to initiate a self-nancing trading strategy
(φ0 , θ0 )
with initial value
V00 := φ00 + θ00 S0 + V0 = 0
such that the terminal value of the portfolio obtained by exercising the
claim at some time
τ
and investing the proceeds in risk-free bonds,
namely,
VT0 := er(T −τ ) [φ0τ erτ + θτ0 Sτ + g(τ, Sτ )] ,
is nonnegative and has a positive probability of being positive;
ˆ
for any writer-initiated self-nancing trading
(φ0 , θ0 )
with initial value
V00 := φ00 + θ00 S0 − V0 = 0,
there exists a stopping time
τ
for which it is
not
the case that the
terminal value of the portfolio resulting from the holder exercising the
claim at time
τ,
namely,
VT0 := er(T −τ ) (φ0τ erτ + θτ0 Sτ − g(τ, Sτ )) ,
is nonnegative and has a positive probability of being positive.
Finally, it may be shown that after time
t
the optimal time for the holder
to exercise the claim is
τt = inf{u ∈ [t, T ] | g(u, Su ) = Vu }.
Explicit formulas are available in the special case of an American put, for
which
g(t, s) = (K − s)+ .
Further Directions
Because of the limited scope of the text we have described only a few of
the many intricate options available in the market. Omissions include
ˆ
call or put options based on stocks with jumps,
thus incorporating into
the model the realistic possibility of market shock;
ˆ
stock index option, where the underlying is a weighted average of stocks
with interrelated price processes;
ˆ
exchange option, giving the holder the right to exchange one risky asset
for another;
Option Valuation: A First Course in Financial Mathematics
202
ˆ
basket option,
the payo a weighted average of a group of underlying
assets;
ˆ
Bermuda option, similar to an American option but with a nite set of
prescribed exercise dates.
ˆ
Russian option,
with payo the discounted maximum value of the un-
derlying up to exercise time;
ˆ
rainbow option, the payo usually based on the maximum or minimum
value of a group of correlated assets.
The interested reader may nd descriptions of these and other options, as well
as expositions of related topics, in [1, 10, 12, 17].
Other Options
203
14.9 Exercises
Q be the exchange rate process in dollars per euro, as given by (14.1).
1. Let
Show that
where
W∗
Qt = Q0 exp σWt∗ + (rd − re − σ 2 /2)t ,
is dened in (14.3). Use this to derive the SDEs
dQ
= σ dW ∗ +(rd −re ) dt
Q
(a)
Remark.
σ
the term
Q−1
Since
2
and (b)
dQ−1
= σ dW ∗ +(re −rd +σ 2 ) dt.
Q−1
is the exchange rate process in euros per dollar,
in (b) is at rst surprising, as it suggests an asymmetric
relationship between the currencies. This phenomenon, known as
paradox,
P∗
Siegel's
may be explained by observing that the probability measure
is risk neutral when the dollar bond is the numeraire. This is the
appropriate measure when pricing in the domestic currency and for that
P∗
reason
is called the
domestic risk-neutral probability measure.
Both
(a) and (b) are derived in this context. When calculating prices in a
foreign currency, the
foreign risk-neutral probability measure
used. This is the probability measure under which
must be
Wt∗ −σt is a Brownian
motion, and is risk-neutral when the euro bond is taken as the numeraire.
With respect to this measure, the Ito-Doeblin formula gives the expected
form of
2. Let
V0cp
dQ−1 .
denote the price of a call-on-put option with strike price
T0 , where the underlying put has strike price K
T > T0 . Show that
Z y1
cp
−rT0
V0 = e
[PT0 (g(y)) − K0 ] ϕ(y) dy,
maturity
at time
K0
and
and matures
−∞
where
y1 := sup{y | PT0 (g(y)) > K0 },
Φ − d2 (T − T0 , s) − sΦ − d1 (T − T0 , s) ,
−r(T −T0 )
PT0 (s) = Ke
and
g
and
d1,2
are dened as in Section 14.4.
pc
cc
and V0
denote, respectively, the prices of a put-on-call and
3. Let V0
a call-on-call option with strike price K0 and maturity T0 , where the
underlying call has strike price
K
and matures at time
T > T0 .
the put-on-call, call-on-call parity relation
V0pc
where
g
and
−
C
V0cc
+e
−rT0
Z
∞
C (g(y)) ϕ(y) dy = K0 e−rT0 ,
−∞
are dened as in Section 14.4.
Prove
Option Valuation: A First Course in Financial Mathematics
204
4. Let
Ctdi
and
Ctdo
denote, respectively, the time-t values of a down-and-
in call option and a down-and-out call option, each with underlying
strike price
K,
and barrier level
c.
Show that
C0do + C0di = C0 ,
where
S,
C0
is the price of the corresponding standard call option.
Down-and-out forward ).
5. (
(ST − K)IA ,
M = c.
payo
with
6. Let
Ctdo
and
Ptdo
where
Show that the price
A = {mS ≥ c},
V0
of a derivative with
is given by (14.9) and (14.10)
denote, respectively, the time-t values of a down-and-
out call option and a down-and-out put option, each with underlying
S , strike price K , and barrier level c. Let A = {mS ≥ c}.
C0do − P0do = V0 , where V0 is as in Exercise 5.
Currency barrier option ).
7. (
Referring
to
Section
14.1
Show that
and
Subsec-
tion 14.5.1, show that the cost of a down-and-out option to buy one
K
euro for
T is
h
i
/2)T
S0 Ê eγ ŴT IB − K Ê eβ ŴT IB ,
dollars at time
C0do = e−(rd +β
2
where
S = QE,
β=
σ
r
− , r = rd − re ,
σ
2
and
γ = β + σ,
Conclude that


C0do = S0 e−re T Φ(d1 ) −

2r +1
2
c σ

Φ(δ1 )
S0


− Ke−rd T Φ(d2 ) −
where
8. A
d1,2
δ1,2
and

2r −1
c σ2

Φ(δ2 ) ,
S0
are dened as in (14.10) with
xed strike lookback put option
value of the option at time
t
has payo
is
r = rd − re .
(K − mS )+ .
Show that the
Vt = e−r(T −t) E∗ (K − mS )+ |FtS
= e−r(T −t) K − Gt (min mSt , K , St ) ,
where
Gt (m, s)
is given by (14.30).
Û := −Ŵ is a P̂-Brownian
ˆ
fM (x, y) of (ŴT , M Ŵ ) under P̂ is
9. Referring to Lemma 14.5.1, use the fact that
motion to show that the joint density
given by
fˆM (x, y) = −ĝm (x, y)I{(x,y)|y≥0,y≥x} .
Other Options
205
10. Referring to Subsection 14.5.1, carry out the following steps to nd the
price
C0ui
of an up-and-in call option for the case
ui
(a) Show that CT
= (ST − K)IB ,
where
a := σ −1 ln (K/S0 ),
B = {ŴT ≥ a, M Ŵ ≥ b},
and
S0 < K < c:
b := σ −1 ln (c/S0 ),
b > a > 0.
(b) Show that
C0ui = e−(r+β
where
β :=
2
/2)T
i
h
S0 Ê e(β+σ)ŴT IB − K Ê eβ ŴT IB ,
r
σ
− .
σ
2
(c) Use Exercise 9 and Lemma 14.5.2 to show that
Ê e
λŴT
Z
b
Z
∞
Z
λx
∞
Z
∞
e ĝm (x, y) dy dx −
eλx ĝm (x, y) dy dx
a
b
b
x
b + λT
2b − a + λT
2bλ+λ2 T /2
√
√
−Φ
=e
Φ
T
T
2
−b
+
λT
√
.
+ eλ T /2 Φ
T
IB = −
(d) Conclude from (b) and (c) that
C0ui
= S0
c
S0
2r2 +1
σ
−K
−rT
Φ(e1 ) − Φ(e3 ) + S0 Φ(e5 ) − K −rT Φ(e6 )
c
S0
2r2 −1
σ
Φ(e2 ) − Φ(e4 ) ,
where
e1,2
e3,4
e5,6
ln c2 /(KS0 ) + (r ± σ 2 /2)T
√
=
σ T
ln(c/S0 ) + (r ± σ 2 /2)T
√
=
σ T
ln(S0 /c) + (r ± σ 2 /2)T
√
=
.
σ T
11. Find the price of an up-and-in call option for the case
S0 < c < K .
12. In the notation of Section 14.6 and Subsection 14.5.1, a
quanto call option
has payo
VT = (STe − K)IB ,
where
down-and-out
STe
and
K
are
denominated in dollars. Carry out the following steps to nd the cost
V0
of the option for the case
K > c:
Option Valuation: A First Course in Financial Mathematics
206
(a) Replace
St
in (14.12) by
Ste = S0e exp [σW ∗ (t) + βt] ,
β := re + σ22 − 21 σ 2
(see (14.32)).
(b) Find a formula analogous to (14.18).
(c) Conclude from (b) that
"
V0 =
S0e e(s−rd )T
Φ(d1 ) −
c
S0
Φ(δ1 )
"
−rd T
− Ke
where
#
%+1
Φ(d2 ) −
c
S0
#
%−1
Φ(δ2 )
s = re + σ22 , % = 2s/σ 2 ,
ln (S0 /K) + (s ± 12 σ 2 )T
√
, and
σ T
ln c2 /(S0 K) + (s ± 12 σ 2 )T
√
=
.
σ T
d1,2 =
δ1,2
13. Let
C0do
be the price of a down-and-out barrier call option, as given by
(14.9). Show that
lim C0do = 0
c→S0−
where
C0
and
lim C0do = C0 ,
c→0+
is the cost of a standard call option. Interpret.
14. In the notation of Theorems 14.5.6 and 14.5.9, show that
(n)
V0 = lim V0 .
n→∞
15. Referring to Subsection 14.7.2, show that if dividend payments are made
at the equally spaced times
tj = jT /(n + 1), j = 1, 2, . . . , n,
then
St = (1 − δ)b(n+1)t/T c Ŝt and
Vt = e−r(T −t) G (1 − δ)n−b(n+1)t/T c S(t) ,
where
bxc
denotes the greatest integer in
16. Referring to Subsection 14.7.1, show that
0 ≤ t < T,
x.
e(δ−r)t St
is a
P∗ -martingale.
Other Options
207
Barrier option on a stock with dividends ). Find a formula for the price
17. (
of a down-and-out call option based on a stock that pays a continuous
stream of dividends.
18. Referring to Section 14.3, nd the probability under the risk-neutral
measure
P∗
that the call is chosen at time
T0 .
19. Referring to Subsection 14.5.1, nd the probability under
barrier
20. A
c
P
that the
is breached.
shout option
is a European option that allows the holder to shout
to the writer at some time
current price
Sτ
τ
before maturity her wish to lock in the
of the security. For a call option, the holder's payo at
maturity, assuming that a shout is made, is
VT := max(Sτ − K, ST − K),
where
K
is the strike price. (If no shout is made, then the payo is the
usual amount
(ST − K)+ .)
Show that
VT = Sτ − K + (ST − Sτ )+
and use this to nd the value of the shout option at time
+
(or ratchet ) option is a derivative with payo (ST0 − K)
T0 and payo (STj − STj−1 )+ at time Tj , j = 1, . . . , n, where
0 < T0 < T1 < · · · < Tn . Thus, the strike price of the option is initially
set at K , but at times Tj , 0 ≤ j ≤ n − 1, it is reset to STj . Find the cost
21. A
cliquet
t ≥ τ.
at time
of the option.
V0 of a derivative
A = {mS ≥ c}, c < S0 .
22. Use the methods of Subsection 14.5.1 to nd the price
with payo
VT = (ST − mS )IA ,
where
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Appendix A
Sets and Counting
Basic Set Theory
A
set
is a collection of objects called the
members
a member of the set
or
A, B ,
elements
of the set.
x is
A, we write x ∈ A; otherwise, we write x 6∈ A. The empty
Abstract sets are usually denoted by capital letters
and so forth. If
set, denoted by ∅, is the set with no members.
A set can be described either by words, by listing the elements, or by
{x | P (x)}, which
P (x) is a well-dened
set-builder notation. Set builder notation is of the form
is read the set of all
x
property that
x
such that
P (x),
where
must satisfy to belong to the set. For example, the set of
all even positive integers can be described as
2m
for some positive integer
A set
n,
A
is
nite
if either
m}.
A is
{2, 4, 6, . . .}
In eect, this means that the members of
1, 2, . . . , n,
so that
A
{n | n =
the empty set or, for some positive integer
there is a one-to-one correspondence between
bers
or as
A
A
and the set
{1, 2, . . . , n}.
may be labeled with the num-
may be described as, say,
{a1 , a2 , . . . , an }.
A set
countably innite if its members may be labeled with the positive integers
1, 2, 3, . . .; countable if it is either nite or countably innite; and uncountable
is
N of all positive integers is obviously countably innite, as
Q of all rational
countably innite. The set R of all real numbers is uncountable,
otherwise. The set
is the set
Z
numbers is
of
all
integers. Less obvious is the fact that set
as is any (nontrivial) interval of real numbers.
B are said to be equal, written A = B , if every member of A
B and vice versa. A is a subset of B , written A ⊆ B , if every
1 A ⊆ B and B ⊆ A.
member of A is a member of B . It follows that A = B i
Sets
A
and
is a member of
Note that the empty set is a subset of every set. Hereafter, we shall assume
that all sets under consideration in any discussion are subsets of a larger set
S,
sometimes called the
1 Read
universe (of discourse ) for that discussion.
if and only if.
209
Option Valuation: A First Course in Financial Mathematics
210
The basic set operations are
A∪B
A∩B
A0
A−B
A×B
=
=
=
=
=
{x | x ∈ A or x ∈ B},
{x | x ∈ A and x ∈ B},
{x | x ∈ S and x 6∈ A},
{x | x ∈ A and x 6∈ B},
{(x, y) | x ∈ A and y ∈ B},
union of A and B;
intersection of A and B;
the complement of A;
the dierence of A and B;
the product of A and B.
the
the
Similar denitions may be given for the union, intersection, and product of
three or more sets, or even for innitely many sets. For example,
∞
[
n=1
∞
\
n=1
∞
Y
n=1
An = A1 ∪ A2 ∪ · · · = {x | x ∈ An
for some
An = A1 ∩ A2 ∩ · · · = {x | x ∈ An
for all
n},
n},
An = A1 × A2 × · · · = {(a1 , a2 , . . . ) | an ∈ An , n = 1, 2, . . . }.
We usually omit the cap symbol in the notation for intersection, writing, for
AB 0 for A − B , etc.
A collection of sets is said to be pairwise disjoint if AB = ∅ for each pair
of distinct members A and B of the collection. A partition of a set S is a
collection of pairwise disjoint nonempty sets whose union is S .
example,
ABC
instead of
A ∩ B ∩ C.
Similarly, we write
Counting Techniques
The number of elements in a nite set
|∅| = 0.
A
is denoted by
|A|.
In particular,
The following result is easily established by mathematical induction.
Theorem A.1. If A1 , A2 , . . . , Ar are pairwise disjoint nite sets, then
|A1 ∪ A2 ∪ · · · ∪ Ar | = |A1 | + |A2 | + . . . + |Ar |.
Corollary A.2. If A and B are nite sets, then
Proof.
|A ∪ B| = |A| + |B| − |AB|.
Note that
A ∪ B = AB 0 ∪ A0 B ∪ AB, A = AB 0 ∪ AB,
and
B = A0 B ∪ AB,
where the sets comprising each of these unions are pairwise disjoint. By Theorem A.1,
|A∪B| = |AB 0 |+|A0 B|+|AB|, |A| = |AB 0 |+|AB|,
and
|B| = |A0 B|+|AB|.
Sets and Counting
211
Subtracting the second and third equations from the rst and rearranging
yields the desired formula.
Theorem A.3. If A1 , A2 , . . . , An are nite sets then
|A1 × A2 × · · · × An | = |A1 ||A2 | · · · |An |.
Proof.
n.) Consider the case n = 2. For each xed x1 ∈ A1
|A2 | elements of the form (x1 , x2 ), where x2 runs through the set A2 .
Since all the members of A1 × A2 may be listed in this manner it follows that
|A1 × A2 | = |A1 ||A2 |. Now suppose that the assertion of the theorem holds for
n = k−1. Since A1 ×A2 ×· · ·×Ak = B×Ak , where B = A1 ×A2 ×· · ·×Ak−1 , the
case n = 2 implies that |B × Ak | = |B||Ak |. But by the induction hypothesis
|B| = |A1 ||A2 | · · · |Ak−1 |. Therefore, the assertion holds for n = k .
(By induction on
there are
Theorem A.3 is the basis of the so-called
Multiplication Principle, described
as follows:
When performing a task requiring
r
steps, if there are
n2
are nr
to complete step 1, and if for each of these there are
complete step 2,
. . .,
and if for each of these there
complete step r, then there are
task.
Example A.4.
n1 n2 · · · nr
n1
ways
ways to
ways to
ways to perform the
How many three-letter sequences can be made from the letters
of the word formula if no letter is used more than once?
Solution: We have the following three-step process:
Step 1: Select the rst letter: 7 choices.
Step 2: Select the second letter: 6 choices, since the letter chosen in Step
1 is no longer available.
Step 3: Select the third letter: 5 choices.
By the multiplication principle, there are a total of
sequences.
The sequences in Example A.4 are called
7 · 6 · 5 = 210
permutations,
possible
formally dened
as follows.
Denition A.5. Let n and r be positive integers with 1 ≤ r ≤ n. A permutation of n items taken r at a time is an ordered list of r items chosen from
the n items. The number of such lists is denoted by (n)r .
An argument similar to that in Example A.4 shows that
(n)r = n(n − 1)(n − 2) · · · (n − r + 1).
Option Valuation: A First Course in Financial Mathematics
212
This may be written compactly as
(n)r =
where the symbol
m!,
n!
,
(n − r)!
read m factorial, is dened as
m! = m(m − 1)(m − 2) · · · 2 · 1.
By convention,
0! = 1,
a choice that ensures consistency in combinatorial
n items taken n at a
n!
0! = n!, which is what
one obtains by directly applying the multiplication principle.
formulas. For example, the number of permutations of
time is, according to the above formula and convention,
Example A.6.
In how many ways can a group of 4 women and 4 men line
up for a photograph if no two adjacent people are of the same gender?
Solution: There are two alternatives, corresponding to the gender of, say,
the rst person on the left. For each alternative there are
of the women and
2(4!)2 = 1152
4!
items.
n
arrangements
arrangements.
Denition A.7. Let
bination of
4!
arrangements of the men. Thus, there are a total of
n
and r be positive integers with 1 ≤ r ≤ n. A comr at a time is a set of r items chosen from the n
items taken
In contrast to a permutation, which is an
be viewed as an
unordered
tations, the number of combinations of
be
ordered
list. Since each set of size
n
list, a combination may
r
things taken
gives rise to
r
r!
permu-
at a time is seen to
(n)r
n!
=
.
r!
(n − r)!r!
The quotient on the right is called a
n
r , read n choose
asserts that
r.
binomial coecient and is denoted by
binomial theorem, which
Its name derives from the
n
(a + b) =
n X
n
j=0
j
aj bn−j .
(This may be proved combinatorially as follows: The expression
(a + b)n = (a + b)(a + b) · · · (a + b)
|
{z
}
n factors
is the sum of all products of the form
ith
0 ≤ j ≤ n.
in the
x1 x2 · · · xn ,
where
xi
factor. Each of these products may be written
For each
j
j
of the
n
or
b
term
for some
n
j such terms, corresponding to the
factors in x1 x2 · · · xn may be chosen as a.)
there are exactly
number of ways exactly
a
aj bn−j
is the
Sets and Counting
Example A.8.
213
A restauranteur needs to hire a sauté chef, a sh chef, a
vegetable chef, and three grill chefs. If there are 10 applicants equally qualied
for the positions, in how many ways can the positions be lled?
Solution: We apply the multiplication principle: First, select a sauté chef:
10 choices; second, select a sh chef: 9 choices; third, select a vegetable chef: 8
choices; nally, select three grill chefs from the remaining 7 applicants:
35
choices. Thus, there are a total of
Example A.9.
10 · 9 · 8 · 35 = 25, 200
7
3
=
choices.
A bag contains 5 red, 4 yellow, and 3 green marbles. In how
many ways is it possible to draw 5 marbles at random from the bag with
exactly 2 reds and no more than 1 green?
Solution: We have the following decision scheme:
Case 1: No green marbles.
5
2
Step 1: Choose the 2 reds:
Step 2: Choose 3 yellows:
4
3
= 10
=4
possibilities.
possibilities.
Case 2: Exactly one green marble.
Step 1: Choose the green:
3
possibilities.
5
2
Step 2: Choose the 2 reds:
Step 3: Choose 2 yellows:
Thus, there are a total of
Example A.10.
4
2
= 10
=6
40 + 180 = 220
possibilities.
possibilities.
possibilities.
How many dierent 12-letter arrangements of the letters of
the word arrangements are there?
Solution:
Notice that there are duplicate letters, so the answer
12!
is in-
correct. We proceed as follows:
Step 1: Select positions for the a's:
Step 2: Select positions for the r's:
Step 3: Select positions for the e's:
Step 4: Select positions for the n's:
12
2
= 66
choices.
10
= 45
choices.
2
8
2
= 28
choices.
6
= 15
choices.
2
Step 5: Fill the remaining spots with the letters g, m, t, s:
Thus there are
66 · 45 · 28 · 15 · 24 = 29, 937, 600
4!
choices.
dierent arrangements.
We conclude this section with the following application of the binomial
theorem.
Theorem A.11. A set
∅).
S
with n members has 2n subsets (including S and
Option Valuation: A First Course in Financial Mathematics
214
Proof.
n
r
Since
subsets of
S
is
gives the number of subsets of size
By the binomial theorem, this quantity is
Remark.
r,
the total number of
n
n
n
+
+ ··· +
.
0
1
n
(1 + 1)n = 2n .
n: The conclun = 0 or 1. Assume the theorem holds for all sets with n ≥ 1
members, and let S be a set with n + 1 members. Choose a xed element s
from S . This produces two collections of subsets of S : those that contain s and
those that don't. The latter are precisely the subsets of S − {s}, and, by the
n
induction hypothesis, there are 2 of these. But the two collections have the
One can also prove Theorem A.11 by induction on
sion is obvious if
same number of subsets, since one collection may be obtained from the other
by either removing or adjoining
subsets of
S,
s.
Thus, there are a total of
completing the induction step.
2n + 2n = 2n+1
Appendix B
Solution of the BSM PDE
In this appendix, we solve the Black-Scholes-Merton PDE
vt + rsvs + 12 σ 2 s2 vss − rv = 0, s > 0, 0 ≤ t ≤ T,
(B.1)
with boundary conditions
v(T, s) = f (s), s ≥ 0,
where
f
and
v(t, 0) = 0, 0 ≤ t ≤ T,
(B.2)
is continuous and satises suitable growth conditions (see footnote
on page 217).
Reduction to a Diusion Equation
As a rst step, we simplify Equation (B.1) by making the substitutions
s = ex , t = T −
2τ
,
σ2
and
v(t, s) = u(τ, x).
(B.3)
By the chain rule,
vt (t, s) = uτ (τ, x)τt = − 21 σ 2 uτ (τ, x),
vs (t, s) = ux (τ, x)xs = s−1 ux (τ, x),
vss (t, s) = s−2 [uxx (τ, x) − ux (τ, x)] .
Substituting these expressions into (B.1) produces the equation
− 21 σ 2 uτ (τ, x) + rux (τ, x) + 12 σ 2 [uxx (τ, x) − ux (τ, x)] − ru(τ, x) = 0.
Dividing by
σ 2 /2
and setting
k = 2r/σ 2 ,
we obtain
uτ (τ, x) = (k − 1)ux (τ, x) + uxx (x, τ ) − ku(x, τ ).
In terms of
u,
(B.4)
the conditions in (B.2) become
u(0, x) = f (ex ),
and
lim u(τ, x) = 0, 0 ≤ τ ≤ T σ 2 /2.
x→−∞
Equation (B.4) is an example of a
diusion equation.
215
Option Valuation: A First Course in Financial Mathematics
216
Reduction to the Heat Equation
The diusion equation (B.4) may be reduced to a simpler form by the
substitution
u(τ, x) = eax+bτ w(τ, x)
for suitable constants
a
and
b.
To determine
a
and
(B.5)
b,
we calculate the partial
derivatives
uτ (τ, x) = eax+bτ [bw(τ, x) + wτ (τ, x)]
ux (τ, x) = eax+bτ [aw(τ, x) + wx (τ, x)]
uxx (τ, x) = eax+bτ [awx (τ, x) + wxx (τ, x)] + aeax+bτ [aw(τ, x) + wx (τ, x)]
= eax+bτ [a2 w(τ, x) + 2awx (τ, x) + wxx (τ, x)].
Substituting these expressions into (B.4) and dividing by
eax+bτ
yields
bw(τ, x) + wτ (τ, x) = (k − 1) aw(τ, x) + wx (τ, x) + a2 w(τ, x)
+ 2awx (τ, x) + wxx (τ, x) − kw(τ, x),
which simplies to
wτ (τ, x) = a(k − 1) + a2 − k − b w(τ, x) + [2a + k − 1]wx (τ, x) + wxx (τ, x).
The terms involving
w(τ, x)
a = 21 (1 − k)
With these values of
a
and
may be eliminated by choosing
b = a(k − 1) + a2 − k = − 14 (k + 1)2 .
and
and
wx (τ, x)
b,
we obtain the PDE
wτ (τ, x) = wxx (τ, x),
Since
w(0, x) = e−ax u(0, x),
τ > 0.
the boundary condition
(B.6)
u(0, x) = f (ex )
w0 (x) := w(0, x) = e−ax f (ex ).
Equation (B.6) is the well-known
heat equation
becomes
(B.7)
of mathematical physics.
Solution of the Heat Equation
To solve the heat equation, we begin with the observation that the function
2
1
1
κ(τ, x) = √ e−x /4τ = √ ϕ
2 πτ
2τ
x
√
2τ
Solution of the BSM PDE
217
is a solution of (B.6), as may be readily veried. The function
kernel
κ
is called the
of the heat equation. To construct a solution of (B.6) that satises
(B.7) we form the
convolution
of
κ
Z
∞
with
w0 :
w0 (y)κ(τ, x − y) dy.
w(τ, x) =
−∞
(B.8)
1 we obtain
Dierentiating inside the integral,
∞
Z
w0 (y)κτ (τ, x − y) dy,
wτ (τ, x) =
−∞
∞
Z
w0 (y)κx (τ, x − y) dy,
wx (τ, x) =
−∞
∞
Z
w0 (y)κxx (τ, x − y) dy.
wxx (τ, x) =
−∞
Since
κt = κxx
we see that
1
w(τ, x) = √
2τ
Z
w
and
satises (B.6). Also,
∞
w0 (y)ϕ
−∞
y−x
√
2τ
Z
∞
dy =
√ w0 x + z 2τ ϕ(z) dz,
−∞
hence,
Z
lim w(τ, x) =
τ →0+
√ lim w0 x + z 2τ ϕ(z) dz
∞
−∞ τ →0+
∞
Z
=
w0 (x)ϕ(z) dz
−∞
= w0 (x).
Therefore,
w
has a continuous extension to
condition (B.7).
From (B.7) we see that the solution
1
w(τ, x) = √
2 τπ
w
y
f (e )e
making the substitution
z=e
2
we obtain
e−ax+a τ
√
w(τ, x) =
2 τπ
1
−ay− 12
y−x
√
2τ
2
dy.
−∞
Rewriting the exponent in the integrand as
y
that satises the initial
in (B.8) may now be written
∞
Z
R+ × R
Z
0
∞
1
−ax + a2 τ − 4τ
(y − x + 2aτ )2
1
f (z)e− 4τ {ln z−x+2aτ }
2
dz
.
z
and
(B.9)
That limit operations such as dierentiation may be moved inside the integral is justi-
ed by a theorem of real analysis, which is applicable in the current setting provided that
w0 does not grow too rapidly. In the case of a call option, for example, one can
|w0 (x)| ≤ M eN |x| for suitable positive constants M and N and for all x. This
show that
inequality
is sucient to ensure that the appropriate integrals converge, allowing the interchange of
limit operation and integral.
Option Valuation: A First Course in Financial Mathematics
218
Back to the BSM PDE
The nal step is to unravel the substitutions that led to the heat equation.
From (B.3), (B.5), and (B.9),
v(t, s) = u(τ, x) = eax+bτ w(τ, x)
Z
2
2 dz
1
e(a +b)τ ∞
√
f (z)e− 4τ {ln z−x+2aτ }
=
.
z
2 τπ 0
Recalling that
k=
1−k
σ 2 − 2r
(k + 1)2
2r
, a=
=
, b=−
,
2
2
σ
2
2σ
4
and
τ=
σ 2 (T − t)
,
2
we have
(k − 1)2 − (k + 1)2
τ = −kτ = −r(T − t) and
4
σ 2 − 2r σ 2 (T − t)
(σ 2 − 2r)(T − t)
2aτ =
·
=
= −(r − 21 σ 2 )(T − t).
2
σ
2
2
(a2 + b)τ =
Since
x = ln s,
we obtain the following solution for the general Black-Scholes-
Merton PDE
e−rτ
v(t, s) = p
σ 2πτ )
where
τ := T − t.
∞
Z
0
1
f (z) exp −
2
ln (z/s) − (r − σ 2 /2)τ
√
σ τ
2 !
dz
,
z
Making the substitution
y=
ln (z/s) − (r − σ 2 /2)τ
√
σ τ
and noting that
√
z = s exp yσ τ + (r − σ 2 /2)τ
and
√
dz
= σ τ dy,
z
we arrive at
v(t, s) = e
−rτ
Z
∞
−∞
√
f s exp σ τ y + (r − σ 2 /2)τ ϕ(y) dy.
It may be shown that the solution to the BSM PDE is unique within a
class of functions that do not grow too rapidly. (See, for example, [18].)
Appendix C
Analytical Properties of the BSM Call
Function
.
Recall that the Black-Scholes-Merton call function is dened as
C = C(τ, s, k, σ, r) = sΦ(d1 ) − ke−rτ Φ(d2 ), τ, s, k, σ, r > 0,
where
ln (s/k) + (r ± σ 2 /2)τ
√
.
σ τ
d1,2 = d1,2 (τ, s, k, σ, r) =
C(τ, s, k, σ, r)
is the price of a call option with strike price
s.
underlying stock price
k,
maturity
τ,
and
In this appendix, we prove Theorems 11.4.1 and
11.4.2, which summarize the main analytical properties of
C.
Preliminary Lemmas
Lemma C.1.
Proof.
Z
∞
eσ
√
τz
ϕ(z) dz = eσ
2
τ /2
−d2
Φ(d1 ).
The integral may be written
1
√
2π
Z
∞
eσ
√
τ z−z 2 /2
−d2
2
eσ τ /2
dz = √
2π
Z
∞
e−(z−σ
−d2
σ 2 τ /2 Z ∞
e
= √
2π
τ )2 /2
e−x
2
/2
dz
dx
−d2 −σ τ
√ Φ −d2 − σ τ ,
√
√
x = z − σ τ . Since d2 + σ τ = d1 ,
= eσ
where we have made the substitution
√
√
2
τ /2
the
conclusion of the lemma follows. (Alternately, one could use Exercise 11.12.)
Lemma C.2.
Z
∞
−d2
zeσ
√
τz
ϕ(z) dz = eσ
2
τ /2
√
σ τ Φ(d1 ) + ϕ(d1 ) .
219
Option Valuation: A First Course in Financial Mathematics
220
Proof.
Z
Arguing as in the proof Lemma C.1, we have
∞
zeσ
√
2
τz
−d2
eσ τ /2
ϕ(z) dz = √
2π
∞
Z
ze−(z−σ
−d2
σ 2 τ /2 Z ∞
e
= √
2π
σ 2 τ /2
e
= √
−d1
∞
xe
−d1
σ 2 τ /2 Z ∞
e
= √
= eσ
2
2π
τ /2
d21 /2
τ )2 /2
dz
√ 2
x + σ τ e−x /2 dx
Z
2π
√
√
Z
2
σ τ eσ τ /2 ∞ −x2 /2
√
dx +
e
dx
2π
−d1
−x2 /2
√
2
e−y dy + σ τ eσ τ /2 [1 − Φ(−d1 )]
√
2
ϕ(d1 ) + σ τ eσ τ /2 Φ(d1 ).
Lemma C.3. For positive τ, s, k, σ, r, dene
g(z) = g(τ, s, k, σ, r, z) := seσ
Then
C = e−rτ
√
τ z+(r−σ 2 /2)τ
∞
Z
g(z)ϕ(z) dz = e−rτ
Z
−d2
Proof.
− k.
∞
g + (z)ϕ(z) dz.
−∞
By Lemma C.1,
Z
∞
g(z)ϕ(z) dz = se(r−σ
2
/2)τ
−d2
Z
∞
eσ
√
τz
−d2
ϕ(z) dz − k
Z
∞
ϕ(z) dz
−d2
= serτ Φ(d1 ) − kΦ(d2 )
= erτ C.
Since
g+
is increasing in
z,
equals 0 if
assertion follows.
z ≤ −d2 ,
and equals
g
otherwise, the
Lemma C.4. With g as in Lemma C.3,
∂
∂x
Z
∞
Z
∞
g(τ, s, k, σ, r, z)ϕ(z) dz =
−d2 (τ,s,k,σ,r)
gx (z)ϕ(z) dz,
−d2
where x denotes any of the variables τ , s, k, σ, r.
Proof. Suppose that x = s. Fix the variables τ , k, σ, and r, and dene
Z
h(s) = d2 (τ, s, k, σ, r)
and
∞
F (s1 , s2 ) =
g(τ, s2 , k, σ, r, z)ϕ(z) dz.
−h(s1 )
By the chain rule for functions of several variables, the left side of the equation
in the assertion of the lemma for
x=s
is
d
F (s, s) = F1 (s, s) + F2 (s, s).
ds
Analytical Properties of the BSM Call Function
221
Since
and
F1 (s1 , s2 ) = g τ, s2 , k, σ, r, −h(s1 ) ϕ − h(s1 ) h0 (s1 )
g τ, s, k, σ, r, −h(s) = 0, we see that F1 (s, s) = 0. Noting that
Z
∞
gs (τ, s2 , k, σ, r, z)ϕ(z) dz
F2 (s1 , s2 ) =
−h(s1 )
we now have
d
F (s, s) = F2 (s, s) =
ds
Z
∞
gs (τ, s, k, σ, r, z)ϕ(z) dz,
−d2 (τ,s,k,σ,r)
which is the assertion of the lemma for the case
works for the variables
τ , k, σ,
and
x = s.
A similar argument
r.
Proof of Theorem 11.4.1
(i)
∂C
= Φ(d1 ):
∂s
By Lemmas C.1, C.3, and C.4,
∂C
= e−rτ
∂s
= e−τ σ
Z
∞
gs (z)ϕ(z) dz
−d2
2
/2
Z
∞
eσ
√
τz
ϕ(z) dz
−d2
= Φ(d1 ).
(ii)
(iii)
∂2C
ϕ(d1 )
= √ :
∂s2
sσ τ
This follows from the chain rule and part (i).
∂C
σs
= √ ϕ(d1 ) + kre−rτ Φ(d2 ): By Lemmas C.3 and C.4,
∂τ
2 τ
Z ∞
Z ∞
∂C
= e−rτ
gτ (z)ϕ(z) dz − re−rτ
g(z)ϕ(z) dz
∂τ
−d2
−d2
= A − B,
say.
By Lemma C.3
B = rC = rsΦ(d1 ) − rke−rτ Φ(d2 ),
Option Valuation: A First Course in Financial Mathematics
222
and by Lemmas C.1 and C.2
Z ∞
Z ∞ √
√
σ
σ τz
2
σ τz
√
A = se
ze
ϕ(z) dz + (r − σ /2)
e
ϕ(z) dz
2 τ −d2
−d2
σ σ2 τ /2 √
−σ 2 τ /2
2
σ 2 τ /2
√ e
= se
σ τ Φ(d1 ) + ϕ(d1 ) + (r − σ /2)e
Φ(d1 )
2 τ
sσ
= √ ϕ(d1 ) + rsΦ(d1 ).
2 τ
−σ 2 τ /2
(iv)
(v)
√
∂C
= s τ ϕ(d1 ): By Lemmas C.1C.4,
∂σ
Z ∞
∂C
gσ (z)ϕ(z) dz
= e−rτ
∂σ
−d2
Z ∞
√
√
2
= se−σ τ /2
z τ − στ eσ τ z ϕ(z) dz
−d
o
n√ 2 2
√
2
−σ 2 τ /2
τ eσ τ /2 σ τ Φ(d1 ) + ϕ(d1 ) − στ eσ τ /2 Φ(d1 )
= se
√
= s τ ϕ(d1 ).
∂C
= kτ e−rτ Φ(d2 ): By Lemmas C.1, C.3, and C.4,
∂r
Z ∞
Z ∞
∂C
−rτ
−rτ
=e
gr (z)ϕ(z) dz − τ e
g(z)ϕ(z) dz
∂r
−d
−d2
Z ∞2
= e−rτ
gr (z)ϕ(z) dz − τ C
−d2
Z ∞ √
2
= τ se−σ /2t
eσ τ z ϕ(z) dz − τ sΦ(d1 ) + kτ e−rτ Φ(d2 )
−d2
= kτ e−rτ Φ(d2 ).
(vi)
∂C
= −e−rτ Φ(d2 ): By Lemmas C.3 and C.4,
∂k
Z ∞
Z ∞
∂C
= e−rτ
gk (z)ϕ(z) dz = −e−rτ
ϕ(z) dz = −e−rτ Φ(d2 ).
∂k
−d2
−d2
Proof of Theorem 11.4.2
The proofs of the limit formulas make use of
lim Φ(z) = 1,
z→∞
and the limit properties of
d1,2 .
lim Φ(z) = 0,
z→−∞
Analytical Properties of the BSM Call Function
(i)
lim [C(τ, s, k, σ, r) − (s − ke−rτ )] = 0:
s→+∞
223
Note rst that
C(τ, s, k, σ, r) − s + ke−rτ = s (Φ(d1 ) − 1) − ke−rτ (Φ(d2 ) − 1) .
Since
lims→∞ d1 = lims→∞ d2 = ∞, lims→∞ Φ(d1,2 ) = 1. It
lims→∞ s (Φ(d1 ) − 1) = 0 or, by l'Hospital's rule,
remains to
show that
lim s2
s→+∞
∂
Φ(d1 ) = 0.
∂s
(†)
√
∂d1
= (sσ τ )−1 ,
∂s
Since
2
∂
∂d1
se−d1 /2
s
Φ(d1 ) = s2 ϕ(d1 )
= √
.
∂s
∂s
σ 2πτ
2
Now,
hence
d1 is of the form (ln (s/k) + b)/a
s = keln (s/k) = kead1 −b . It follows
for suitable constants
a
and
b
that
2
2
lim se−d1 /2 = ke−b lim ead1 −d1 /2 = 0,
s→+∞
s→+∞
verifying (†) and completing the proof of (i).
(ii)
(iii)
(iv)
lim C(τ, s, k, σ, r) = 0:
Immediate from
lim C(τ, s, k, σ, r) = s:
Follows from
s→0+
τ →∞
lim C(τ, s, k, σ, r) = (s − k)+ :
Since
τ →0+
implies that
Z
lim+ C(τ, s, k, σ, r) =
τ →0
(v)
lim C(τ, s, k, σ, r) = 0:
k→∞
lim d1,2 = −∞.
s→0+
lim d1,2 = ±∞.
τ →∞
lim g + = (s − k)+ ,
τ →0+
Lemma C.3
∞
−∞
This
(s − k)+ ϕ(z) dz = (s − k)+ .
follows
from
Lemma
C.3
and
from
+
lim g = 0.
k→∞
(vi)
(vii)
(viii)
lim C(τ, s, k, σ, r) = s:
Follows from
lim C(τ, s, k, σ, r) = s:
Immediate from
k→0+
σ→∞
lim C(τ, s, k, σ, r) = (s − e−rτ k)+ :
σ→0+
Lemma C.3 we have
lim+ C(τ, s, k, σ, r) = e−rτ
σ→0
(ix)
lim C(τ, s, k, σ, r) = s:
r→∞
Z
lim d1 = +∞.
k→0+
lim d1,2 = ±∞.
σ→∞
Since
lim g + = (serτ − k)+ ,
σ→0+
by
∞
−∞
(serτ − k)+ ϕ(z) dz = (s − e−rτ k)+ .
Follows immediately from
lim d1 = +∞.
r→∞
224
Option Valuation: A First Course in Financial Mathematics
Appendix D
Hints and Solutions to Odd-Numbered
Problems
Chapter 1
1. Rounding to two decimal places,
(a)
1500(1 + .06)3 = $1786.52;
(b)
1500(1 + .06/4)12 = $1793.43;
(c)
1500(1 + .06/12)36 = $1795.02;
(d)
1500(1 + .06/365)3·365 = $1795.80;
(e)
1500e3(.06) = $1795.83.
3. (a) 12.55%; (b) 12.68%; (c) 12.75%.
5.
A5 =
7.
n = 64
$29,391;
A10
= $73,178.
is the smallest value satisfying
400
9.
A5 = $130, 229.97
and
(1.005)n − 1
≥ 30, 000.
.005
A10 = $36, 120.65.
The account will be drawn
down to zero after 139 withdrawals. (The last withdrawal will be
$1,941.85.)
11.
n = 39.
13. The time-n value of the withdrawal made at time
where
j = 1, 2, . . . , N − n.
15. The rate
i = r/12
17.
is
P e−rj/12 ,
Add these to obtain the desired result.
must satisfy
1800 = 300, 000
This gives
n+j
i
.
1 − (1 + i)−360
r ≈ .06.
A0 = $42, 035.
225
Option Valuation: A First Course in Financial Mathematics
226
19. Paying $6000 now and investing $2000 for 10 years gives
$2000e10r
with
r0 that would allow you
10r0
to cover the $6000 exactly satises the equation 2000e
= 6000, which
ln 3
has solution r0 =
≈
0.11.
10
which to pay o the remaining $6000. The rate
P ≈ $600. AfA120 ≈ $83, 686. You must nance the amount
21. Your current monthly payments for the 6% mortgage are
ter 10 years you still owe
(1.03)A120 for 20 years. Payments for the new 4% mortgage are therefore Q ≈ $522. The monthly rate for which Q = $600 is approximately
.0047. Therefore, an annual mortgage rate above 12(.47) = 5.64% would
make renancing unwise.
23.
Bt =
PN
n=m+1
e−r(tn −t) Cn + F e−r(T −t) .
25. The rate of return for Plan A is 22.68% while that of Plan B is 22.82%.
Therefore, Plan B is slightly better.
Chapter 2
1. Use the inclusion-exclusion rule.
3. Let
Aj
be the event that Jill wins in
only one way,
A4
in 3 ways, and
A5
j
races,
j = 3, 4, 5. A3
occurs in
in 6 ways. Therefore, Jill wins with
probability
P(A3 ) + P(A4 ) + P(A5 ) = q 3 + 3pq 3 + 6p2 q 3 = q 3 (1 + 3p + 6p2 ).
5. The probability
pn
that at least two out of
is
1−
Since
7. Let
C
p7 ≈ .53
and
balls land in the same jar
30 · 29 · · · · · (30 − n + 1)
.
(30)n
p8 ≈ .64,
at least 8 throws are needed.
be the event that both tosses are heads,
one toss comes up heads, and
heads. Then
P(C|A) =
B
p
2−p
A
the event that at least
the event that the rst toss comes up
and
The probabilities are not the same since
9. Let
n
P(C|B) = p.
p
=p
2−p
implies
p=0
or
A be the event that the slip numbered 1 was drawn twice and B
1.
the
event that the sum of the numbers on the three slips drawn is 8. Then
P(AB) = 3/63
and
P(B) = 21/63
so
P(A|B) = 3/21.
Hints and Solutions
11.
P(A) = .5, P(B) = (.1)2 ,
227
P(AB) = (.5)(.1)2 . Therefore,
the inequality to x < .49 makes
and
are independent. Changing
the events
the events
dependent.
13. For (c),
P(A0 B 0 ) = 1 − P(A ∪ B) = 1 − [P(A) + P(B) − P(AB)]
= [1 − P(A)][1 − P(B)] = P(A0 )P(B 0 ).
15. (a) Let
x = r/s.
Then
P(E) = xP(E 0 ) = x(1 − P(E))
hence
x
r
P(E) =
=
.
x+1
r+s
(b) If
E
occurs, then the bettor receives
1+
r+s
1
s
=
=
.
r
r
P(E)
Chapter 3
1. The number of heads in
parameters
(n, .5);
n
Yn with
P(Yn ≥ 2) ≥ .99
tosses is a binomial random variable
hence the smallest
n
for which
satises
P(Yn = 0) + P(Yn = 1) = 2−n (1 + n) ≤ .01.
n = 11.
n
pX (k) =
AN BN , where
k
Np
Np − 1
Np − k + 1
AN =
···
→ pk and
N
N −1
N −k+1
Nq
Nq − 1
Nq − n + k + 1
BN =
···
→ q n−k
N −k
N −k−1
N −n+1
Therefore,
3.
as
N → ∞.
5. For
a > 0,
y−b
y−b
FY (y) = P X ≤
= FX
.
a
a
Dierentiating yields the desired result in this case.
7. Since
0
[Φ(x) + Φ(−x)] = 0, Φ(x) + Φ(−x) = 2Φ(0) = 1.
If
X ∼ N (0, 1),
P(−X ≤ x) = P(X ≥ −x) = 1 − P(X ≤ −x) = 1 − Φ(−x) = Φ(x),
so
−X ∼ N (0, 1).
Option Valuation: A First Course in Financial Mathematics
228
9.
r ≤ 0. For r > 0,


1, 2
FZ (r) = πr4 ,

 2
r arcsin 1r −
FZ (r) = 0
Therefore,
for
fZ = gZ I[0,√2] ,
(
gZ (r) =
11. Let
√
r ≥ 2,
0 ≤ r ≤ 1,
√
√
+ r2 − 1, 1 ≤ r ≤ 2.
πr 2
4
where
πr
2 ,
2r arcsin
1
r
−
α = Φ µσ −1 .
n
(a) There are
choices of times for the
k
0≤r≤1
√
1 ≤ r ≤ 2.
πr
2 ,
k
increases, and each of these
has probability
(b)
P(Z1 > 1, Z2 > 1, . . . , Zk > 1, Zk+1 < 1, . . . , Zn < 1).
n k
Therefore, the desired probability is
α (1 − α)n−k .
k
The k consecutive increases can start at times 1, 2, . . . , n − k + 1
k
n−k
hence the required probability is (n − k + 1)α (1 − α)
.
(c) Assuming that
k < n,
the event in question is the union of the
mutually exclusive events
{ Z1 > 1, Z2 > 1, . . . , Zk > 1, Zk+1 < 1},
{ Zn−k < 1, Zn−k+1 > 1, . . . , Zn > 1},
and
{ Zj < 1, Zj+1 > 1, . . . , Zj+k > 1, Zj+k+1 < 1},
where j = 1, 2, . . . , n − k − 1. Therefore,
2αk (1 − α) + (n − k − 1)αk (1 − α)2 .
the required probability is
13. By independence,
P(max(X, Y ) ≤ z) = P(X ≤ z, Y ≤ z) = P(X ≤ z)P(Y ≤ z)
and
P(min(X, Y ) > z) = P(X > z, Y > z) = P(X > z)P(Y > z).
Chapter 4
1. Suppose that
C0 > S 0 .
We then buy the security for
option, and place the prot
C0 − S0
S0 ,
write a call
into a risk-free account yielding
Hints and Solutions
229
erT (C0 − S0 ) at time T . If ST > K , we must sell the security for K . If
ST ≤ K , we sell the security for ST . In any case, the total proceeds from
rT
these transactions are e
(C0 − S0 ) + min{ ST , K}, giving an arbitrage.
Therefore, C0 ≤ S0 . The other inequalities follow from the put-call parity formula.
3. If
S0 +P0e −C0 > Ke−rT
we sell short one share of the security, sell a put
option, and buy a call option. We deposit the resulting cash
in a risk-free account. If
ST < K
S0 +P0e −C0
the call option we bought is worthless
and the put option we sold will be exercised, requiring us to buy the
security for the amount
K . If ST ≥ K
the put option we sold is worthless
K . Since each
K , the transactions give us a positive prot
but we can exercise our call option and buy the security for
case requires a cash outlay of
5.
of
(S0 + P0e − C0 )erT − K ,
P
can be exercised at any time in the interval
contradicting the no-arbitrage assumption.
exercised only at times in the subinterval
exibility and hence greater value.
Strip Payoff
[0, T ], while P 0
[0, T ]. This gives P
0
can be
greater
Strap Payoff
2K
2K
K
K
K
2K
ST
K
(a)
2K
ST
(b)
FIGURE D.1: Exercise 7
Strangle Payoff
K1
K1
K2
ST
FIGURE D.2: Exercise 9
11. If
P0 > P00 ,
buy the lower-priced option and sell the higher-priced one
P0 − P00 . The three possibilities at maturity, ST < K ,
K ≤ ST ≤ K 0 , and K 0 < ST , result in the respective payos K 0 − K ,
K 0 − ST , and 0. Therefore, the prot is at least P0 − P00 > 0, giving an
0
arbitrage. That P0 ≤ P0 is to be expected since a smaller strike price
for a cash prot of
gives a smaller payo.
Option Valuation: A First Course in Financial Mathematics
230
13. By Exercises 10, 11, and put-call parity,
0 ≤ C0 − C00 = P0 − P00 + (K 0 − K)e−rT
and
0 ≤ P00 − P0 = C00 − C0 + (K 0 − K)e−rT .
15. Consider a portfolio which is long in a put with strike price
in a put with strike price
F := K2 − K1 .
K2 > K1 ,
K1 ,
short
and long in a bond with face value
The payo is


0
+
+
(K1 − ST ) − (K2 − ST ) + F = ST − K1


K2 − K1
if
if
if
ST ≤ K1 ,
K1 ≤ ST ≤ K2 ,
ST > K2
+
= (ST − K1 ) − (ST − K2 )+ ,
which is the payo of a bull spread.
Chapter 5
1. The sample space consists of the permutations of 1, 2, and 3.
{(1, 2, 3), (1, 3, 2)}, {(2, 1, 3), (2, 3, 1)},
{(3, 1, 2), (3, 2, 1)}; F2 = F3 contains all subsets of Ω.
is generated by the sets
F1
and
3. By (v)
φn+1 = V0 −
Subtracting yields
n
X
S̃j ∆θj
and
j=0
hence
Gn =
n
X
j=1
Vn = V0 + Gn
∆Vn = ∆Gn =
n−1
X
S̃j ∆θj .
j=0
∆φn = −S̃n ∆θn .
5. If the portfolio is self-nancing, then
Conversely, if
φn = V0 −
n+1
X
j=1
∆Vj−1 = φj ∆Bj−1 + θj ∆Sj−1
∆Vj−1 = Vn − V0 .
for all
n,
then
(φj ∆Bj−1 + θj ∆Sj−1 ) −
= φn+1 ∆Bn + θn+1 ∆Sn
hence the portfolio is self-nancing.
n
X
j=1
(φj ∆Bj−1 + θj ∆Sj−1 )
Hints and Solutions
231
Chapter 6
X1 be the of number red marbles drawn before the rst white one and
Y = X1 + X2 ,
and X1 + 1 and X2 + 1 are geometric with parameter p = w/(r + w)
(Example 3.5.8). Therefore, E Y = E X1 + E X2 = (2/p) − 2 = 2r/w
1. Let
X2
the number of reds between the rst two whites. Then
(Example 6.1.5).
3. By Example 6.1.5,
E[X(X − 1)] =
E X = 1/p.
∞
X
n=2
Also,
n(n − 1)q n−1 p = pq
∞
X
d2 q 2
2q
d2 n
q
=
pq
= 2.
2
2 1−q
dq
dq
p
n=2
Therefore,
V X = E[X(X − 1)] + E X − E2 X = (2q + p − 1)/p2 = q/p2 .
N = r + w. The number X of marbles drawn is either 2 or
X = 2, then the marbles are either both red or both white, hence
 2
2

r + w

for (a)
2
N
P(X = 2) =
r(r − 1) + w(w − 1)


for (b).

N (N − 1)
5. Let
The event
{X = 3}
3. If
consists of the outcomes RWR, RWW, WRR, and
WRW hence
 2
2r w + 2rw2



N3
P(X = 3) =
2r(r
−
1)w + 2w(w − 1)r



N (N − 1)(N − 2)
for (a)
for (b).
Therefore, for case (a)
EX = 2 ·
r2 + w2
r2 w + rw2
+
6
·
N2
N3
and for case (b)
EX = 2 ·
7. For any
r(r − 1) + w(w − 1)
r(r − 1)w + w(w − 1)r
+6·
.
N (N − 1)
N (N − 1)(N − 2)
A ∈ F,
V IA = E I2A − E2 IA = P(A) − P2 (A) = P(A)P(A0 )
hence the desired result follows from independence and Theorem 6.4.2.
Option Valuation: A First Course in Financial Mathematics
232
fX,Y (x, y) = I[0,1] (x)I[0,1] (y),
Z 1Z 1
4xy
4XY
E
=
dy dx
2 + y2 + 1
X2 + Y 2 + 1
x
0
0
Z 1
=
2x ln (x2 + 2) − ln (x2 + 1) dx
9. Since
0
Z
=
2
3
ln u du −
Z
2
ln u du
1
= ln (27/16).
11. By linearity and independence,
E (X + Y )2 = E X 2 + E Y 2 + 2(E X)(E Y ) = E X 2 + E Y 2
and
E (X +Y )3 = E X 3 +E Y 3 +3(E X 2 )(E Y )+3(E X)(E Y 2 ) = E X 3 +E Y 3 .
13. For (a), complete the square to obtain
Z
b
e
αx
ϕ(x) dx = e
α2 /2
a
b
Z
a
2
ϕ(x − α) dx = eα
/2
[Φ(b − α) − Φ(a − α)] .
For (b), integrate by parts and use (a) to obtain
Z
b
1 b αx
1 αx
e Φ(x) −
e ϕ(x) dx
α
α a
a
b
2
1 αx
.
=
e Φ(x) − eα /2 Φ(x − α)
α
a
b
Z
eαx Φ(x) dx =
a
15.
E X = 21 (α + β) (Example 6.2.2).
Z β
2
−1
E X = (β − α)
x2 dx = 31 (α2 + αβ + β 2 ),
V X = E X 2 − E2 X ,
where
Since
α
VX =
17.
2
1
3 (α
+ αβ + β 2 ) − 41 (α + β)2 =
E2 X = E X 2 − V X ≤ E X 2 ,
since
1
12 (α
− β)2 .
V X ≥ 0.
19. Referring to Example 3.2.5, the expectation is
X=
−1 X N
m
n
x,
z
x
z−x
x
where
max(z − n, 1) ≤ x ≤ min(z, m). Show that this may be written as
−1 X −1 N
m−1
n
N
N −1
m
=m
,
z
x−1
z−x
z
z−1
x
where
max(z − 1 − n, 0) ≤ x − 1 ≤ min(z − 1, m − 1).
Hints and Solutions
233
21. (a) By Equation (6.3),
The exact
P(Y = 50) ≈ Φ(.1) − Φ(−.1) ≈ 0.07966.
100 −100
probability is
2
= 0.07959.
50
(b) By Equation (6.2) with
p = .5,
P(40 < Y < 60) = P −2 <
2Y − 100
<2
10
≈ Φ(2) − Φ(−2) ≈ .95.
Chapter 7
1. If
A
denotes the Cartesian product, then
P(A) =
X
ω∈A
=
P1 (ω1 )P2 (ω2 ) · · · PN (ωN )
X
ω1 ∈A1
···
X
ωN ∈AN
P1 (ω1 ) · · · PN (ωN )
= P1 (A1 )P2 (A2 ) · · · PN (AN ).
Zn and Xn
Sn (ω) = ωn Sn−1 (ω).
3. Part (a) follows from the denitions of
ment of (7.2). For (c), use
and (b) is a restate-
Xn is FnS measurable hence, by denition of FnX ,
X
S
Fn ⊆ Fn . On the other hand, (7.3) implies that Sn is FnX measurable
X
S
hence Fn ⊆ Fn .
Part (c) implies that
5. If
S0 d < K < S0 u,
then
C0 = (1 + i)−1 (S0 u − K)p∗ =
hence
∂C0
(1 + i − d)(K − S0 d)
=
>0
∂u
(1 + i)(u − d)2
and
If
(1 + i − d)(S0 u − K)
(1 + i)(u − d)
∂C0
(1 + i − u)(S0 u − K)
=
< 0.
∂d
(1 + i)(u − d)2
d ≥ K/S0 ,
then
C0 =
(S0 u − K)p∗ + (S0 d − K)q ∗
K
= S0 −
.
1+i
1+i
Option Valuation: A First Course in Financial Mathematics
234
7. (a) call: $17.56; put: $19.98; (b) call: $19.70; put: $15.33.
k ≥ N/2,
k -run. Let A denote the event
k -run of u's, and Aj the event that the run started at
time j = 1, 2, . . . , N − k + 1, assuming that k < N − 1. The events Aj
k
are mutually exclusive with union A, P(A1 ) = P(AN −k+1 ) = p q , and
k 2
k
P(Aj ) = p q , j = 2, . . . , N −k . Therefore, P(A) = p q(2+(N −k −1)q).
The formula still holds if k = N − 1.
9. Since
there can be only one
that there was a
11. By Corollary 7.2.5 with
−N
V0 = (1 + i)
f (x) = xI(K,∞) (x),
N N X
X
N
N j N −j
j N −j ∗ j ∗ N −j
S0 u d
p q
= S0
p̂ q̂
.
j
j
j=m
j=m
13. Since
+
(SN − SM )+ = S0 uYN dN −YN − S0 uYM dM −YM
+
= S0 uYM dM −YM uYN −YM dL−(YN −YM ) − 1 ,
Corollary 7.2.2 and independence imply that
(1 + i)−N V0 = E∗ (SN − SM )+
+
= E∗ S0 uYM dM −YM E∗ uYN −YM dL−(YN −YM ) − 1
+
= E∗ (SM )E∗ uYN −YM dL−(YN −YM ) − 1 .
(α )
By Remark 7.2.3(b),
E∗ (SM ) = (1 + i)M S0 .
Since
(β )
YN − YM = XM +1 + . . . + XN ∼ B(p∗ , L),
+
+
E∗ uYN −YM dL−(YN −YM ) − 1
= E∗ uYL dL−YL − 1 .
The last expression is
L,
(1+i)L times the cost of a call option with maturity
strike price one unit, and initial stock value one unit. Therefore, by
the CRR formula,
+
E∗ uYN −YM dL−(YN −YM ) − 1
= (1 + i)L Ψ(k, L, p̂) − Ψ(k, L, p∗ ), (γ )
where
k
is the smallest nonnegative integer for which
desired expression for
V0
now follows from
(α), (β)
uk dL−k > 1.
(γ).
and
The
Hints and Solutions
235
f (x) = x(x − K)+ ,
N X
N
−N
V0 = (1 + i)
S0 uj dN −j (S0 uj dN −j − K)p∗ j q ∗ N −j
j
j=m
15. By Corollary 7.2.5 with
N X
S02
N
=
(u2 p∗ )j (d2 q ∗ )N −j
N
(1 + i) j=m j
−
= S02
where
v
1+i
q̃ = q ∗ d2 /v
N KS0 X N
(up∗ )j (dq ∗ )N −j
(1 + i)N j=m j
N X
N N
j=m
and
j
p̃j q̃ N −j − KS0
q̂ = 1 − p̂.
Since
N X
N
j=m
j
p̂j q̂ N −j ,
u2 p∗ + d2 q ∗ = v , (p̃, q̃)
is a
probability vector and the desired formula follows.
17. Use Exercise 3.12 and the law of the unconscious statistician.
19. By Exercise 17 with
f (x, y) =
1
2 (x
have
+ y) − K
(1 + i)N V0 =
+
,
m = 1,
and
n = N,
we
1
(A0 + A1 ),
2
where
N
−1 X
N − 1 ∗ k ∗ N −k
A0 :=
p q
(S0 d + S0 uk dN −k − 2K)+
k
k=0
N X
N − 1 ∗ k ∗ N −k
A1 :=
p q
(S0 u + S0 uk dN −k − 2K)+ .
k−1
and
k=1
S0 d + S0 uk dN −k > 2K for k = N − 1 hence
k N −k1
> 2K .
there exists a smallest integer k1 ≥ 0 such that S0 d+S0 u 1 d
k+1 N −k−1
N −1
Since S0 u + S0 u
d
> S0 d + S0 u
d for k = N − 1, there
k +1 N −k2 −1
exists a smallest integer k2 ≥ 0 such that S0 u + S0 u 2
d
> 2K .
The hypothesis implies that
Therefore
A0 =
N
−1
X
k=k1
N − 1 ∗ k ∗ N −k
p q
(S0 d + S0 uk dN −k − 2K)
k
= (S0 d − 2K)q
∗
N
−1
X
k=k1
N − 1 ∗ k ∗ N −1−k
p q
k
+ S0 dq
∗
N
−1
X
k=k1
∗
N −1
(up∗ )k (dq ∗ )N −1−k
k
= (S0 d − 2K)q Ψ(k1 , N − 1, p∗ ) + (1 + i)N S0 dq ∗ Ψ(k1 , N − 1, p̂),
Option Valuation: A First Course in Financial Mathematics
236
and
∗
A1 = p
N
−1
X
k=k2
N − 1 ∗ k ∗ N −1−k
p q
(S0 u + S0 uk+1 dN −1−k − 2K)
k
= (S0 u − 2K)p
∗
N
−1
X
k=k2
N − 1 ∗ k ∗ N −1−k
p q
k
+ S0 up
∗
N
−1
X
k=k2
N −1
(up∗ )k (dq ∗ )N −1−k
k
= (S0 u − 2K)p∗ Ψ(k2 , N − 1, p∗ ) + (1 + i)N S0 up∗ Ψ(k2 , N − 1, p̂).
Chapter 8
1. For any real number
a,
X
E g(X)I{X=a} =
g(x)I{a} (x)pX (x) = g(a)pX (a)
and
x
X
E Y I{X=a} =
pX,Y (a, y)y.
y
3. Since
XY = X 2 + X(Y − X),
conditioning on
G
yields
E(XY ) = E X 2 + XE(Y − X|G)
= E X 2 + XE(Y − X)
= E X 2 + E(X)E(Y − X)
= E X 2.
Therefore,
E(Y − X)2 = E Y 2 − 2E X 2 + E X 2 = E Y 2 − EX 2 .
5. By the iterated conditioning property, if
m>n
E(Mm |Fn ) = E[E(Mm |Fm−1 )|Fn ] = E(Mm−1 |Fn ).
2
+ 2Xn+1 Yn − σ 2 ,
Mn+1 − Mn = Xn+1
2
E Mn+1 − Mn |FnX = E Xn+1
+ 2Yn E Xn+1 |FnX − σ 2 = 0.
7. Since
9. Since
Mn+1 = Mn rXn+1 ,
q
p
= Mn .
E(Mn+1 |FnX ) = Mn E rXn+1 = Mn p + q
p
q
Hints and Solutions
237
(Am − An )2 , we have for n ≤ m
E (Am − An )2 |Fn = E A2m |Fn + E A2n |Fn − 2E (Am An |Fn )
= E A2m |Fn + A2n − 2An E (Am |Fn )
= E A2m |Fn + A2n − 2A2n
11. Expanding
= E (Bm |Fn ) + E (Cm |Fn ) − Bn − Cn
= E (Cm − Cn |Fn ) .
13. Condition on
Fk .
Chapter 9
1. For (a) let
Ak = {Sk ≤ (S0 + S1 + · · · + Sk−1 )/k},
Ak ∈ FkS
Then
k = 1, 2, . . . , N − 1.
and
{τa = n} = A1 A2 · · · An−1 A0n ∈ FnS ,
n = 1, 2, . . . , N − 1,
and
S
S
{τa = N } = A1 A2 · · · AN −1 ∈ FN
−1 ⊆ FN .
3. Price: $20.91. Optimal exercise time scenarios:
uudd
d
($27.00),
ud
($35.91);
($28.43).
5. We show by induction on
k
that
(„)
vk (Sk (ω)) = f (Sk (ω)) = 0
for all
k ≥ n (= τ0 (ω)). By denition
k ≥ n. Since
of
(†) holds for arbitrary
τ0 , (†)
holds for
k = n.
Suppose
vk (Sk (ω)) = max f (Sk (ω)), avk+1 (Sk (ω)u) + bvk+1 (Sk (ω)d)
and all terms comprising the expression on the right of this equation are
nonnegative,
vk+1 (Sk+1 (ω)) = 0.
Since
vk+1 (Sk+1 (ω))
= max f (Sk+1 (ω)), avk+2 (Sk+1 (ω)u) + bvk+2 (Sk+1 (ω)d) ,
f (Sk+1 (ω)) = 0.
Therefore, (†) holds for
k + 1.
Option Valuation: A First Course in Financial Mathematics
238
7. The proof of (a) is a straightforward modication of that of Corollary 9.1.3. To nd
C0
take
n = 0, m = N ,
and
(a). Then
−N
C0 = a
N X
N
j
j=0
p∗ j q ∗ N −j bN uj dN −j S0 − K
f (x) = (x − K)+
in
+
N
N N X
b
K X N ∗ j ∗ N −j
N ∗ j ∗ N −j j N −j
=
.
u d
− N
S0
p q
p q
a
a j=m j
j
j=m
Chapter 10
x−2 dx = sin t dt ⇒ x−1 = cos t + c ⇒ x = (cos t + c)−1 ; x(0) =
1/3 ⇒ c = 2. Therefore, x(t) = (cos t + 2)−1 , −∞ < t < ∞.
1. (a)
(b)
x(0) = 2 ⇒ c = −1/2 ⇒ x(t) = (cos t − 1/2)−1 , −π/3 < t < π/3.
2
2
2x dx
√ = (2t + cos t) dt ⇒ x = t + sin t + c. x(0) = 1 ⇒ c = 1 ⇒
x(t) = t2 + sin t + 1, valid for all t (positive root because x(0) > 0).
(c)
(x+1)−1 dx = cot t dt ⇒ ln |x + 1| = ln | sin t|+c ⇒ x+1 = ±ec sin t;
x(π/6) = 1/2 ⇒ x+1 = ±3 sin t. Positive sign is chosen because x(π/6)+
1 > 0. Therefore, x(t) = 3 sin t − 1.
(d)
3. Use the partitions
Pn
described in the example to construct Riemann-
Stieltjes sums that do not converge.
5. For the rst assertion use the identity
W (s) + W (t) = W (t) − W (s) + 2W (s),
independence, and Example 3.6.2.
7. By Theorem 10.6.3,
Xt =
Rt
0
F (s) dW (s)
has mean zero and variance
t
Z
E F 2 (s) ds.
V Xt =
0
(a)
E sWs2 = s2
hence
V Xt =
Rt
0
s2 ds = t3 /3.
W (s) ∼ N (0, s),
Z ∞
Z ∞
2
2
2
1
1
E exp (2Ws2 ) = √
e2x e−x /2s dx = √
e−αx /2 dx,
2πs −∞
2πs −∞
(b) Since
where
α = s−1 − 4.
If
s ≥ 1/4,
then
α≤0
and the integral diverges.
Hints and Solutions
239
V Yt =√ +∞ for t ≥ 1/4. If s ≤ t < 1/4, then, making
y = αx, we have
Z ∞
2
1
1
2
√
e−y /2 dy = √
E exp (2Ws ) =
= (1 − 4s)−1/2
sα
2πsα −∞
Therefore,
the
substitution
so that
t
Z
V Xt =
0
(c) For
(1 − 4s)−1/2 ds = 21 [1 −
√
1 − 4t].
s > 0,
2
E |Ws | = √
2πs
hence
r
V Xt =
9. (a) Use Version 1 with
(b) From Version 2,
2
π
Z
∞
−x2 /2s
xe
r
dx =
0
Z
t
0
√
2
s ds =
3
r
2s
π
2 3/2
t .
π
f (x) = ex .
d(tW 2 ) = 2tW dW + (W 2 + t) dt.
f (t, x, y) = x/y . Since ft = 0, fx = 1/y , fy =
−x/y 2 , fxx = 0, fxy = −1/y 2 , and fyy = 2x/y 3 , we have
X
dX
X
X
1
d
=
− 2 dY + 3 (dY )2 − 2 dX · dY.
Y
Y
Y
Y
Y
(c) Use Version 4 with
Factoring out
X
Y gives the desired result.
11. Taking expectations in (10.19) gives
α
E Xt = e−βt E X0 + (eβt − 1) .
β
Chapter 11
1. (a) call: $4.65; put: $0.50; (b) call: $1.27; put: $2.98.
3.
∂P
∂C
=
−1 = Φ(d1 )−1 < 0, lims→∞ P = 0, and lims→0+ P = Ke−rτ .
∂s
∂s
5. Taking
f (z) = AI(K,∞) (z)
in Theorem 11.3.2 yields
Vt = e−r(T −t) G(t, St ),
Option Valuation: A First Course in Financial Mathematics
240
where, as in the proof of Corollary 11.3.3,
∞
n √
o
AI(K,∞) s exp σ T − t y + (r − σ 2 /2)(T − t) ϕ(y) dy
−∞
= AΦ d1 (T − t, s, K, σ, r) .
Z
G(t, s) =
7. Since
VT = ST I(K1 ,∞) (ST ) − ST I[K2 ,∞) (ST ),
V0
9.
11.
is the dierence in the prices of two asset-or-nothing options.
Vt = C(T − t, St , F ) + (F − K)e−r(T −t) and, in particular, V0 = C0 +
(F − K)e−rT , where C0 is the cost of a call option on the stock with
rT
strike price F . Therefore, K = F + e
C0 .
ST > K
i
σWT +(µ−σ 2 /2)T > ln (K/S0 ) hence the desired probability
is
1−Φ
ln (K/S0 ) − (µ − σ 2 /2)T
√
σ T
13. The expression for
EC
=Φ
ln (S0 /K) + (µ − σ 2 /2)T
√
σ T
.
follows from Theorem 11.4.1(i) and the Black-
Scholes formula. To verify the limits, write
−1
EC
=1−α
Φ(d2 )
,
sΦ(d1 )
and note that (a) follows from
α := Ke−rT
lims→∞ Φ(d1,2 ) = 1.
For (b), apply
l'Hospital's Rule to obtain
sΦ(d1 )
sϕ(d1 )(βs)−1 + Φ(d1 )
= lim+
Φ(d2 )
ϕ(d2 )(βs)−1
s→0
s→0
√
sϕ(d1 )
Φ(d1 )
= lim
1+β
, β := σ T .
+
ϕ(d2 )
ϕ(d1 )
s→0
−1 −1
α(1 − EC
) = lim+
Since
d22 − d21 = (d2 − d1 )(d2 + d1 ) = −β(d2 + d1 ) = 2[ln(K/s) − rT ],
s
ϕ(d1 )
= s exp [ 12 (d22 − d21 )] = s exp [ln(K/s) − rT ] = α
ϕ(d2 )
hence
−1
1 − EC
−1
= α−1 lim+
s→0
sΦ(d1 )
Φ(d1 )
= 1 + β lim+
.
Φ(d2 )
s→0 ϕ(d1 )
Hints and Solutions
241
By l'Hospital's Rule,
Φ(d1 )
1
ϕ(d1 )(βs)−1
= − lim+
= lim+
= 0.
−1
ϕ(d
)
ϕ(d
)(−d
)(βs)
d
s→0
s→0
s→0
1
1
1
1
−1 −1
Therefore, lims→0+ 1 − EC
= 1, which implies (b).
lim+
15. Make the substitution
o
n √
z = s exp σ T − t y + (r − 12 σ 2 )(T − t) .
Chapter 12
E eλX = peλ + q
1. (a)
Ee
(b)
λX
= pe
λ
n
.
1 − qeλ
−1
.
3. For (a),
E(Ws Wt ) = E E Ws Wt |FsW = E Ws E Wt |FsW = E(Ws2 ) = s,
and for (b),
E(Wt −Ws |Ws ) = E E(Wt − Ws |FsW )|Ws ) = E [E(Wt − Ws )|Ws )] = 0.
5. Let
A = {(u, v) | v ≤ y, u + v ≤ x}
and
1
ϕ
f (x, y) = p
s(t − s)
√
By independent increments,
x
t−s
y
ϕ √ .
s
P(Wt ≤ x, Ws ≤ y) = P (Wt − Ws , Ws ) ∈ A
ZZ
=
f (u, v) du dv
A
Z
y
Z
x
=
−∞
7.
M
is a martingale i for all
−∞
f (u − v, v) du dv.
0 ≤ s ≤ t, E eα[W (t)−W (s)] |Fs = eh(s)−h(t) .
By independence and Exercise 6.14,
2
E eα[W (t)−W (s)] |Fs = E eα[W (t)−W (s)] = eα (t−s)/2 .
Therefore,
M
is a martingale i
h(t) − h(s) = α2 (s − t)/2.
242
Option Valuation: A First Course in Financial Mathematics
9. By Example 12.2.3 and iterated conditioning,
E(Wt2 |Ws ) = E[E(Wt2 −t|FsW )|Ws ]+t = E(Ws2 −s|Ws )+t = Ws2 +t−s.
Similarly, by Exercise 8,
E(Wt3 − 3tWt |Ws ) = E[E(Wt3 − 3tWt |FsW )|Ws ]
= E(Ws3 − 3sWs |Ws )
= Ws3 − 3sWs .
Therefore, by Exercise 3,
E(Wt3 |Ws ) = 3tE(Wt |Ws ) + Ws3 − 3sWs = Ws3 + 3(t − s)Ws .
11. For any
x,
P∗ (X ≤ x) = E∗ I(−∞,x] (X)
1
2
T
E I(−∞,x] (X)e−αWT
1
2
T
E(I(−∞,x] (X))E e−αWT
= e− 2 α
= e− 2 α
= P(X ≤ x),
the last equality from Exercise 6.14.
Chapter 13
1. By Lemma 13.2.2, the call nishes in the money i
WT∗ > σ −1 ln (K/S0 ) − (r − 21 σ 2 )T
Therefore, the
1−Φ
P∗ -probability
that the call nishes in the money is
ln (K/S0 ) − (r − 21 σ 2 )T
√
σ T
3. This follows from
e−(r+σ
2
)t
∗∗
St = S0 eσWt
= Φ d2 (T, S0 , K, σ, r) .
− 12 σ 2 t
and Example 12.2.6.
Chapter 14
1. Parts (a) and (b) follow from the Ito-Doeblin formula applied to
f (t, x) = exp σx + (rd − re − σ 2 )t
Hints and Solutions
243
and
f (t, x) = exp −σx − (rd − re − σ 2 )t ,
respectively.
3. Use Equation (14.8), its analog for a put-on-call option, and the identity
+
+
K0 − C(s) − C(s) − K0 + C(s) = K0 .
5. As a rst step,
V0 = e−rT Ê (ST − K)IA ZT−1
i
h
2
= e−(r+β /2)T S0 Ê eγ ŴT IA − K Ê eβ ŴT IA ,
where
Ê eλŴT IA
is given by (14.19). An obvious modication of
(14.16) shows that
Ê e
λŴT
ZZ
eλx ĝm (x, y) dA,
IA =
D := {(x, y) | b ≤ y ≤ 0, x ≥ y}.
D
Since
D
has the same form as in Figure 14.2, the integral evaluates to
(14.19) and hence, as in the text, leads to (14.9) and (14.10) with
M = c.
7. From (14.11), the cost of the option is
C0do = e−rd T E∗ [(ST − K)IB ],
where
S
is given by (14.12) with
r = rd − re .
The calculations leading
to Equation (14.9) yield, as in the text,
1
C0do = e− 2 (2rd +β
Since
9. Since
2
2rd + β 2 − γ 2 = 2re ,
−γ)T
S0 Φ(d1 ) − e2bγ Φ(δ1 )
− Ke−rd T Φ(d2 ) − e2bβ Φ(δ2 ) .
the desired conclusion follows.
M Ŵ = −mÛ ,
P̂(ŴT ≤ x, M Ŵ ≤ y) = P̂(ÛT ≥ −x, mÛ ≥ −y)
Z ∞Z ∞
=
fˆm (u, v) dv du.
−x
Therefore, if
−y < 0
fˆM (x, y) =
and
and
∂2
∂x∂y
−y
−y < −x,
Z ∞Z ∞
fˆm (u, v) dv du = ĝm (−x, −y),
−x
−y
fˆM (x, y) = 0 otherwise. Since ĝm (−x, −y) = −ĝm (x, y), the formula
follows.
Option Valuation: A First Course in Financial Mathematics
244
11. Since
call.
{ST ≥ K, M S ≥ c} = {ST ≥ K},
13. The rst assertion follows from
limc→0+ δ1,2 = −∞,
15. If
the price is that of a standard
limc→S − δ1,2 = d1,2 and the second from
0
limc→0+ Φ(δ1,2 ) = 0.
the latter implying that
mT /(n + 1) ≤ t < (m + 1)T /(n + 1),
m = bt(n + 1)/T c.
then
hence
m ≤ t(n + 1)/T < m + 1,
17. By (14.11) and (14.35),
C0do = e−rT E∗ (ST − K)IB ,
where
σ
r−δ
− .
σ
2
∗
St = S0 eσ(Wt +βt) , β :=
With this change in
β

C0do = e−δT S0 Φ(d1 ) −
c
S0

2(r−δ)
+1
2
σ
Φ(δ1 )

− Ke−rT Φ(d2 ) −
c
S0

2(r−δ)
−1
2
σ
Φ(δ2 ) ,
where
ln (S0 /M ) + (r − δ ± σ 2 )T /2
√
, and
σ T
ln c2 /(S0 M ) + (r − δ ± σ 2 )T /2
√
.
=
σ T
d1,2 =
δ1,2
19. The desired probability is
1 − P(C),
where
C := {mS ≥ c} = {mW ≥ b},
To nd
P(C),
recall that the measures
b := σ −1 ln (c/S0 ).
P∗ , P̂
and the processes
are dened by
1
dP∗ = e−αWT − 2 α
dP̂ = e
2
T
−βWT∗ − 12 β 2 T
WT∗ = WT + αT,
dP,
dP∗ ,
α=
µ−r
,
σ
and
ŴT = WT∗ + βT = WT + (α + β)T,
β=
σ
r
− .
σ
2
W ∗ , Ŵ
Hints and Solutions
It follows that
dP = U dP̂,
245
where
1
2
U := eλŴT − 2 λ
T
,
λ := α + β =
µ σ
− .
σ
2
Therefore,
2
P(C) = Ê(IC U ) = e−λ
T /2
ZZ
eλx ĝm (x, y) dA,
D
where
D = {(x, y) | b ≤ y ≤ 0, x ≥ y},
(see (14.16)). This is the region of integration described in Figure 14.2,
so by (14.19),
P(C) =
Since
b + λT
−b + λT
√
√
− e2bλ Φ
.
Φ
T
T
±b + λT
± ln (c/S0 ) + (µ − σ 2 /2)T
√
√
=
T
σ T
and
2bλ =
2µ
− 1 ln (c/S0 ),
σ2
the desired probability is
(
1−
Φ(d1 ) −
where
d1,2 =
c
S0
2µ2 −1
σ
)
Φ(d2 ) ,
± ln (S0 /c) + (µ − σ 2 /2)T
√
.
σ T
21. The total payo is that of a portfolio consisting of a call option maturing
at time
T0
and
n forward start options maturing at times T1 , T2 , . . . , Tn .
The cost of the cliquet is then the sum of the costs of these options,
which may be obtained by using the results of Section 14.2.
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Exotic Derivatives, World Scientic, London.
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Marcel
Finance/Mathematics
A First Course in Financial Mathematics
Option Valuation: A First Course in Financial Mathematics
provides a straightforward introduction to the mathematics and
models used in the valuation of financial derivatives. It examines
the principles of option pricing in detail via standard binomial and
stochastic calculus models. Developing the requisite mathematical
background as needed, the text introduces probability theory and
stochastic calculus at an undergraduate level.
Hugo D. Junghenn
Option Valuation
A First Course in
Financial Mathematics
Junghenn
Largely self-contained, this classroom-tested text offers a sound
introduction to applied probability through a mathematical finance
perspective. Numerous examples and exercises help readers
gain expertise with financial calculus methods and increase their
general mathematical sophistication. The exercises range from
routine applications to spreadsheet projects to the pricing of a
variety of complex financial instruments. Hints and solutions to
odd-numbered problems are given in an appendix.
A First Course in
Financial Mathematics
The first nine chapters of the book describe option valuation
techniques in discrete time, focusing on the binomial model. The
author shows how the binomial model offers a practical method
for pricing options using relatively elementary mathematical tools.
The binomial model also enables a clear, concrete exposition of
fundamental principles of finance, such as arbitrage and hedging,
without the distraction of complex mathematical constructs. The
remaining chapters illustrate the theory in continuous time, with
an emphasis on the more mathematically sophisticated Black–
Scholes–Merton model.
Option Valuation
Option Valuation
K14090
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