DEPARTMENT OF FINANCE
ARBITRAGE THEORY IN DISCRETE
AND CONTINUOUS TIME
Anna Battauz - Fulvio Ortu
Lecture notes for the course
Quantitative Methods for Finance
Cod. 20188
Stampa: Logo S.r.l., Borgoricco (PD)
Contents
Preface
I
vii
One-period Financial Markets
1 Basic Notation and De nitions
1.1 Time, Uncertainty and Prices . . . . . . . . . . . . . . . . . .
1.2 Investment Strategies . . . . . . . . . . . . . . . . . . . . . .
1.3 Law of One Price and No-arbitrage . . . . . . . . . . . . . . .
1
5
5
8
10
2 Characterization of No-arbitrage
17
2.1 State Price Vectors . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Risk-neutral Probabilities . . . . . . . . . . . . . . . . . . . . 21
2.3 The First Fundamental Theorem of Asset Pricing . . . . . . . 25
3 Complete One-Period Markets
29
3.1 De nition and Characterization . . . . . . . . . . . . . . . . . 30
3.2 The Second Fundamental Theorem of Asset Pricing . . . . . 33
3.3 Interpretation and Further Remarks . . . . . . . . . . . . . . 34
4 No-Arbitrage Valuation of Derivatives
39
4.1 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 The Case of Redundant Securities . . . . . . . . . . . . . . . 40
4.3 The Case of Non-redundant Securities . . . . . . . . . . . . . 43
5 The
5.1
5.2
5.3
One-period Binomial Model
47
Assumptions of the Model . . . . . . . . . . . . . . . . . . . . 48
Completeness and No-arbitrage . . . . . . . . . . . . . . . . . 49
Option Pricing in the Binomial Model . . . . . . . . . . . . . 52
iii
iv
CONTENTS
5.3.1
5.3.2
5.3.3
II
Call Option Pricing . . . . . . . . . . . . . . . . . . .
Put Option Pricing . . . . . . . . . . . . . . . . . . . .
Put-call Parity . . . . . . . . . . . . . . . . . . . . . .
Multi-period Financial Markets in Discrete Time
6 Stochastic Processes in Discrete Time
6.1 Information Structures . . . . . . . . . . . . . . . .
6.1.1 “Event-Tree”Representation . . . . . . . . .
6.1.2 Information Structure Generated by Market
6.2 Adapted Stochastic Processes . . . . . . . . . . . .
6.3 Conditional Expectations and Martingales . . . . .
52
55
57
59
. . . .
. . . .
Data
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
61
61
64
65
67
69
7 Multi-period Markets: Basic Notions
77
7.1 Price Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Dynamic Investment Strategies . . . . . . . . . . . . . . . . . 79
7.3 Multi-period Arbitrage Opportunities . . . . . . . . . . . . . 84
8 No-arbitrage in Multi-period Markets: the Characterization
8.1 State Price Vectors in the Multi-period Case . . . . . . . . .
8.2 Equivalent Martingale Measures . . . . . . . . . . . . . . . .
8.3 The First Fundamental Theorem of Asset Pricing in the Multiperiod Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Dynamically Complete Multi-period Markets
9.1 Dynamic Completeness: the Notion . . . . . . . . . . . .
9.2 The Characterization of Dynamic Completeness . . . . .
9.3 The Second Fundamental Theorem of Asset Pricing in
Multi-period Case . . . . . . . . . . . . . . . . . . . . .
87
87
88
92
101
. . . 101
. . . 103
the
. . . 111
10 No-arbitrage Valuation in the Multi-period Case
113
10.1 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.2 The Case of Redundant Securities . . . . . . . . . . . . . . . 114
10.3 The Case of Non-redundant Securities . . . . . . . . . . . . . 117
11 The
11.1
11.2
11.3
Multi-period Binomial Model
121
Description of the Model . . . . . . . . . . . . . . . . . . . . . 121
No-Arbitrage and Dynamic Completeness . . . . . . . . . . . 125
Option Pricing in the Multi-period Binomial Model . . . . . . 127
CONTENTS
v
11.3.1 Valuation of a Call Option via Replication . . . . . . . 127
11.3.2 Call Option Pricing via Risk-Neutral Valuation . . . . 130
11.3.3 Put-call Parity . . . . . . . . . . . . . . . . . . . . . . 132
III
Continuous-Time Financial Markets
135
12 Stochastic Processes in Continuous Time
137
12.1 Trajectories and Measurability of Stochastic Processes in Continuous Time 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 137
12.2 Wiener Processes . . . . . . . . . . . . . . . . . . . . . . . . . 142
12.3 Di¤usions and Stochastic Integration . . . . . . . . . . . . . . 147
12.4 Constructing the Stochastic Integral . . . . . . . . . . . . . . 149
12.5 Ito Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
12.6 Ito’s Formula(*) . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.7 Stochastic Di¤erential Equations . . . . . . . . . . . . . . . . 157
13 The
13.1
13.2
13.3
Black-Scholes Model
161
The Basic Securities . . . . . . . . . . . . . . . . . . . . . . . 161
Information and Investment Strategies . . . . . . . . . . . . . 169
No-Arbitrage Analysis . . . . . . . . . . . . . . . . . . . . . . 173
13.3.1 The No-Arbitrage Property . . . . . . . . . . . . . . . 173
13.3.2 Equivalent Martingale Measures . . . . . . . . . . . . 173
13.3.3 The Equivalence between No-arbitrage and the Existence of an Equivalent Martingale Measure . . . . . . . 179
13.4 Completeness of the Black-Scholes Model . . . . . . . . . . . 184
14 No-Arbitrage Pricing in the Black-Scholes Market
187
14.1 No-arbitrage Valuation of European Derivatives in the BlackScholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
14.2 Pricing of European Derivatives via the Black-Scholes PDE . 196
14.2.1 The Black-Scholes PDE . . . . . . . . . . . . . . . . . 196
14.2.2 The Classical Derivation of the Black-Scholes Equation 198
14.3 Market Price of Risk . . . . . . . . . . . . . . . . . . . . . . . 202
1
The symbol 3 will indicate the sections of the chapter that are not part of the syllabus.
Preface
Part I
One-period Financial
Markets
3
We devote the rst part of this work to the description of a one-period nancial market under uncertainty. In the rst chapter, we introduce some basic
concepts, such as states of the world, price systems, investment strategies,
value and gain processes associated with investment strategies, etc.. Based
on such de nitions, we move on to describing violations of the law of one
price and arbitrage opportunities of the rst and second type. We analyze the relations linking these concepts and provide a formal de nition of
absence of arbitrage opportunities (or simply no-arbitrage).
The next step is the discussion of the First Fundamental Theorem of
Asset Pricing, yielding a characterization of arbitrage-free nancial markets.
We do that in the second chapter, where, for the sake of pedagogical clarity,
we split the result into two parts. The rst one characterizes no-arbitrage
in terms of state-price vectors, thus leading immediately to the equivalence
between no-arbitrage and the existence of linear operators for the pricing of
nancial securities. The second part characterizes the absence of arbitrage in
probabilistic terms, in the sense that no-arbitrage is shown to be equivalent
to the existence of a speci c probability usually called risk-neutral. The key
feature of such probability is that it makes current security prices equal to
the expected discounted value of their future prices.
A concept that is key to most applications in nancial market analysis
is the one of market completeness. We introduce it in the third chapter,
providing both a formal de nition and an economic/ nancial interpretation.
We then prove the so called Second Fundamental Theorem of Asset Pricing,
which states that arbitrage-free complete markets are characterized by the
existence of a unique state-price vector or, equivalently, of a unique riskneutral probability measure.
In the fourth chapter, we show how the results obtained in the previous
chapters can be regarded as the foundations upon which the theory of noarbitrage pricing can be built. This is particularly relevant for that class
of securities called derivative instruments. We discuss the pricing by noarbitrage of securities that can be perfectly replicated by investing on the
securities available in the market, and of securities for which no perfect
replication is always possible, a situation arising in the case of incomplete
markets. As an example, we develop pricing formulas the most commonly
traded derivative securities, the options.
In the fth and last chapter, we consider the simplest model possible of
one-period nancial market, i.e. that with only two states of the world at
the end of the period. Such model, usually called Binomial Model, is simple
yet rich enough to analyze all the concepts, de nitions and results described
in greater generality in the preceding chapters.. We deem the Binomial
4
Model not just a useful example allowing us to verify the understanding of
the topics introduced, but also the building block of multi-period markets,
which will be treated in the second part of this work.
Chapter 1
Basic Notation and
De nitions
In the rst section of this chapter we introduce the basic de nitions of a
one-period nancial market model. In particular, we introduce the basic
timing of the model and we explain how to describe uncertainty, the key
issue in nancial decisions making. Once time and uncertainty are described
formally, we discuss how to formalize the prices of nancial securities.
In the second section we introduce the concept of investment strategy,
which describes formally the way investors carry on their decisions. With
each investment strategy we associate two quantities that will play an important role in the theory of no-arbitrage pricing: the value process and the
discounted gain process of an investment strategy.
The quantities introduced in the second section underpin the de nition
of investment strategies yielding nonnegative pro ts with nonpositive costs.
Such strategies are generated when the law of one price is violated, or when
there exist arbitrage opportunities of the rst or of the second type. These
concepts are formally de ned in the third section, where we explore the
general concept of no-arbitrage in a one-period nancial market.
1.1
Time, Uncertainty and Prices
By one-period nancial market we mean a model in which investors are
faced with two trading dates only, namely t = 0 and t = 1. At time t = 0,
investors make their investment choices, while at time t = 1 they just receive
the liquidation value of the asset allocation decided at time t = 0. At
time 0, investors can choose among N + 1 securities, for which we use the
5
6
CHAPTER 1. BASIC NOTATION AND DEFINITIONS
index j, with j = 0; : : : ; N . From now on, the security corresponding to
j = 0 will represent a riskless asset: for example, a secured loan deposit,
or a pure discount bond whose issuer carries no default risk (e.g. BOT,
T-Bills, etc.). We will denote by B(0) , 1 the time-0 price of the riskless
asset, by B(1) , 1 + r its time-1 price.1 As a consequence, the quantity
r must be interpreted as the risk-free rate available on the market, and we
will require r > 01 so that 1 + r > 0. With regard to the remaining N
securities, we use the symbol Sj (0) for their price at time 0 and Sj (1) for
their price at time 1, with j = 1; : : : ; N . Such securities can be thought of as
assets yielding random returns, such as stocks, defaultable bonds, derivative
contracts (options, forwards, futures, etc.).
In order to describe the uncertainty a¤ecting the securities indexed by
j = 1; : : : ; N , we assume that by time 1 the market uncertainty will resolve
in one, and only one, of K possible states of the world. We use the symbol
! k to indicate the generic k-th state of the world at time 1, and the symbol
to indicate the set of all states of the world, i.e.
= f! 1 ; : : : ; ! K g. The
! k ’s are to be interpreted as possible economic/ nancial scenarios that are
relevant for the determination of the price of our risky securities. We require
that these scenarios be mutually exclusive and that only one of them will
occur at time 1. We assume that all investors agree on the set of scenarios
possible at time 1. We also assume that all market participants agree on the
probability of each of the K scenarios, and we denote this probability by
P(! k ). Furthermore, we assume that investors are unanimous in assigning
a positive probability to each scenario, i.e. P(! k ) > 0 for all k = 1; : : : ; K.
Finally, we use the symbol Sj (1)(! k ) to indicate the time-1 price of the j-th
security in scenario ! k .
Summing up, so far we have introduced the following quantities:
Dates: t = 0; 1.
States:
= f! 1;111 ; ! K g.
Probabilities: P (! k ) > 0; k = 1; : : : ; K.
Riskless security:
– B(0) , 1: price at time t = 0;
– B(1) , 1 + r: price at time t = 1 of the risk-free asset;
– r: risk-free rate.
1
Henceforth, the symbol , will mean ‘equal by de nition’.
1.1. TIME, UNCERTAINTY AND PRICES
7
N risky securities:
– S1 (0) ; : : : ; SN (0): price at time t = 0;
– S1 (1); : : : ; SN (1): price at time t = 1;
– Sj (1) (! k ), j = 1; : : : ; N; k = 1; : : : ; K: time-1 price of the j-th
risky security in scenario ! k .
A one-period nancial market can be synthetically described by means
of a so-called payo¤ matrix, denoted by M. The payo¤ matrix M has K + 1
rows and N + 1 columns, and is constructed as follows:
2
6
6
6
M,6
6
4
01
1+r
1+r
..
.
0S1 (0)
S1 (1)(! 1 )
S1 (1)(! 2 )
..
.
0S2 (0)
S2 (1)(! 1 )
S2 (1)(! 2 )
..
.
111
111
111
1 + r S1 (1)(! K ) S2 (1)(! K ) 1 1 1
0SN (0)
SN (1)(! 1 )
SN (1)(! 2 )
..
.
3
7
7
7
7
7
5
(1.1)
SN (1)(! K )
Reading M by rows, we nd in the rst row the opposite of the time-0 prices
of the N + 1 securities, in the second row the time-1 prices in scenario ! 1 , in
the third row the time-1 prices in scenario ! 2 , and so on up to the last row
where we nd the time-1 prices in last possible scenario ! K . Reading M by
columns, the generic column of M has the opposite of the time-0 price of a
given security in the rst entry, while the other entries are the time-1 prices
of the that security, ordered by scenario.
To x ideas, we conclude this section with a simple numerical example
describing the concepts examined so far.
Example 1 Consider a one-period market with N = 2 and K = 3. That
means we have three securities (one riskless asset and two risky securities)
and three possible scenarios at time 1. Assume that B(1) = 1:1 so that since
B(0) = 1 and B(1) = 1 + r, we have r = 0:1, i.e. the risk-free rate is 10%.
Suppose also that the time-0 prices of the risky securities are S1 (0) = 1:563
and S2 (0) = 1:636. Moreover, let the time-1 prices of the rst risky security
in the three possible scenarios be the following ones:
S1 (1)(! 1 ) = 1:5
S1 (1)(! 2 ) = 2
S1 (1)(! 3 ) = 1:4;
8
CHAPTER 1. BASIC NOTATION AND DEFINITIONS
while for the second risky security assume the following prices:
S2 (1)(! 1 ) = 2
S2 (1)(! 2 ) = 1
S2 (1)(! 3 ) = 3:
The payo¤ matrix of this one-period market is then the following 423 matrix:
3
2
01 01:563 01:636
7
6 1:1
1:5
2
7
M=6
5
4 1:1
2
1
1:1
1:4
3
1.2
Investment Strategies
In a one-period nancial market investors take their positions on the N + 1
securities at time 0, while at time 1 they see which one among the K possible
scenarios has actually occurred, and close the positions taken in 0 at the
prices of the securities in that scenario. As a result, in the one-period case
an investment strategy is quite simply a list of N + 1 variables, one for each
security available in the market, where the variables represent the units of
each security bought, or sold short, at time 0. From now on, we will denote
by #0 the position in the risk-free asset and by #1 ; : : : ; #N the positions in
the N risky securities.
We assume that the nancial market is competitive and frictionless, i.e.
there are no indivisibilities, short-selling constraints, margin requirements,
bid-ask spreads, taxation on dividends and/or capital gains, and so on. This
assumption implies that the set of all investment strategies that can be set
up in our one-period market coincide with RN +1 , i.e the space of N + 1dimensional vectors of real numbers, since N + 1 is the overall number of
securities available. Formally, therefore, the generic investment strategy is
represented by a (column) vector # 2 RN +1 , that is
1
0
#0
B # C
B 1 C
C
B
C
B
(1.2)
# = B #2 C
B 1 C
C
B
@ 1 A
#N
With each investment strategy #, we associate two fundamental quantities, the value process and the discounted gain process of that strategy,
de ned formally as follows.
1.2. INVESTMENT STRATEGIES
9
De nition 2 Value Process. Given an investment strategy #, we call
Value Process of # the quantities V# (0), V# (1) (! 1 ), :::,V# (1) (! K ) de ned
as follows:
N
X
V# (0) = #0 +
#j Sj (0)
(1.3)
j=1
V# (1) (! k ) = #0 (1 + r) +
N
X
#j Sj (1) (! k );
k = 1; :::; K
(1.4)
j=1
We note that in the de nition of value process we have exploited the fact
that B(0) = 1 and B(1) = 1 + r regardless of the scenario.
From the economic/ nancial viewpoint, the value process has the following interpretation. At time 0, the quantity V# (0) represents the cost an
investor must face to set up the investment strategy #. It must be stressed
that such cost may take negative values: this is a consequence of our assumption that short-seling is allowed. Indeed, the short-sale of the generic j-th
security translates into #j < 0 and hence #j Sj (0) < 0. Therefore V# (0) < 0
may occur when the out‡ows incurred to buy the stocks are lower than the
in‡ows generated by the short-sales, while we would get V# (0) 0 otherwise.
At time 1, and conditional on the occurrence of the generic scenario ! k ,
the realization V# (1) (! k ) of the value process represents the net cash‡ows
given by the liquidation of the investment strategy decided at time 0. From
the nancial point of view, we would expect strategies with negative cost at
time 0 to produce a negative liquidation value in at least one state of the
world, and strategies with no cost at time 0 to give rise to liquidation values
equal to zero in every state. Indeed, this is exactly what happens in markets
where arbitrage opportunities are ruled out, as will be explained thoroughly
in the next section.
We now complete this section with the de nition of discounted gain process associated with an investment strategy.
De nition 3 Discounted Gain Process. Given an investment strategy
#, we call Discounted Gain Process of # the quantities G# (0), G# (1) (! 1 ),:::,G# (1) (! K )
de ned as follows:
G# (0) = 0
(1.5)
G# (1) (! k ) =
V# (1) (! k )
0 V# (0) ;
1+r
k = 1; :::; K
(1.6)
10
CHAPTER 1. BASIC NOTATION AND DEFINITIONS
The nancial meaning of the discounted gain process is quite intuitive:
it represents the gain generated by an investment strategy during its “lifetime” (which in our simple model consists in only one period). The need
for discounting is apparent: the mere di¤erence V# (1) (! k ) 0 V# (0) has no
nancial content, since it involves amounts referred to di¤erent points in
V# (1) (! k )
time. The di¤erence
0 V# (0) represents instead the correct gain
1+r
generated by the strategy between time 0 and time 1 if the state ! k occurs,
gain evaluated at time 0 since V# (1) (! k ) is discounted back to time 0 at
the risk-free rate r.
1.3
Law of One Price and No-arbitrage
In order to be useful, a nancial market model must satisfy some requirements enabling to approximate, despite its theoretical abstractness, the dynamics of real-world nancial markets. This section is devoted to such requirements. In particular, we start by describing a number of situations
that, if not eliminated from the nancial market model, would make it not
only quite unrealistic, but also useless from the practical point of view. We
group such situations in three categories: violations of the law of one price,
arbitrage opportunities of the rst type and arbitrage opportunities of the
second type.
De nition 4 Violations of the Law of One Price. A one-period nancial market permits violations of the law of one price if there exist two
investment strategies # and #0 such that
V# (0) 6= V#0 (0)
(1.7)
and
V# (1) (! k ) = V# (1) (! k ) ;
for all k = 1; : : : ; K
(1.8)
In words, a one-period nancial market gives rise to violations of the
law of one price if two investment strategies can be setup, having di¤erent
costs at time 0 despite providing the same liquidation value in any state of
the world at time 1. In a model in which such violations are allowed it is
impossible to de ne univocally the prices of nancial securities, thus making
the model useless for application purposes. We will therefore assume in the
sequel that the law of one price holds.
1.3. LAW OF ONE PRICE AND NO-ARBITRAGE
11
De nition 5 Arbitrage Opportunities of the First Type. A oneperiod nancial market permits arbitrage opportunities of the rst type if
there exists an investment strategy # such that
V# (0) 0
and
V# (1) (! k ) 0,
for all k = 1; : : : ; K;
V# (1) (! k ) > 0,
for some k:
(1.9)
(1.10)
Hence, a one-period nancial market admits arbitrage opportunities of
the rst type when one can set up an investment strategy satisfying three
requirements. First, the initial cost is non positive, i.e. the strategy does not
require any out‡ow at time 0. Second, the liquidation value is nonnegative
in every state at time 1. Third, the liquidation value is strictly positive in
at least one of the states at time 1.
Why is it fundamental for the model to rule out such arbitrage opportunities? We rst note that any investor, regardless of his wealth, could
exploit such arbitrage opportunities: since their cost is zero or even negative, no money is required to engage in an arbitrage strategy. Moreover,
every non-satiated investor (in the sense that more wealth is always preferred
to less) would exploit the arbitrage opportunities available in the market.
With what consequences then? The demand for securities allowing arbitrage would go up to in nity (if the strategy # leads to an arbitrage of the
rst type, the same is true for the strategy #, for any > 0::::), and hence
there would be no equilibrium in the market: a clear example of a useless
market model!
De nition 6 Arbitrage Opportunities of the Second Type. A oneperiod nancial market permits arbitrage opportunities of the second type if
there exists an investment strategy # such that
V# (0) < 0
and
V# (1) (! k ) 0,
for all k = 1; : : : ; K:
(1.11)
(1.12)
The de nition says that a strategy leads to an arbitrage opportunity of
the second type if it has strictly negative initial cost (i.e. it yields a strictly
positive in‡ow at time 0) and nonnegative liquidation value at time 1 in
12
CHAPTER
1.
BASIC NOTATION AND DEFINITIONS
every state of the world. Why must the model rule out such opportunities?
Clearly, for the same reasons mentioned with regard to arbitrage opportunities of the first type. Why do we need to differentiate between the two types
of arbitrage opportunities? The reason is that the set of arbitrage opportunities of the first or second type are neither one subset of the other, despite
having non empty intersection. In particular, every arbitrage of the first
type such that Vs (0) = 0 is not an arbitrage of the second type and, vice
versa, every arbitrage of the second type such that Vy (1) (w,) = 0 for all
k =1,...,K is not an arbitrage of the first type. This is why, in general, we
must impose that the model rules out both types of arbitrage opportunities.
It is quite useful and instructive to analyze the relation between violations of the law of one price and existence of arbitrage opportunities.
The
following result shows that the absence of arbitrage opportunities of the
second type is a sufficient (but not necessary) condition for the law of one
price to hold.
Proposition 7 In a one-period financial market, the absence of arbitrage
opportunities of the second type implies that the law of one price holds.
Proof. To prove the result we show that the violation of the law of one price
implies the existence of secon-type arbitrage.
To see this, let the strategies
0, 9 violate the law of one price, in that Vs (0) < Vy (0), Vo (1) (wx) =
Vg (1) (we), for all k = 1,...,K (the case. in which Vy (0) > Vy (0) is fully
symmetric). Consider then the investment strategy 3” = 3 — 0’ and note
that
Von (1) (we)
= 9G(L +7) + DML, 84S; (1) (wr)
= (80 ~ 9)(L +1) + Dyan (8; — 99); (1) (we)
= Bo(1 +r) + DMs VS; (1) we) - [81 +r) + DL 5; (1) we)]
= Vg (1) (wR) — Vg (1) (We) = 0,
for all k=1,...,K.
On the other hand, we have that
Von
(0)
=
0%
+
Ae
04S; ; (0 )
= (80 — 9) + DjLa(8; — 84); (0)
= Bo + Dj1 855; (0) - | 0+ Dja 8555
0)
= Vo (0) — Vg (0) < 0,
1.3. LAW OF ONE PRICE AND NO-ARBITRAGE
13
As a consequence, V#00 (0) < 0 and V#00 (1) (! k ) = 0 for all k = 1; : : : ; K, and
hence #00 is an arbitrage of the second type.4
We are now ready to introduce the key concept of this rst chapter, i.e.
the de nition of no-arbitrage.
De nition 8 No-Arbitrage. A one-period nancial market satis es the
no-arbitrage condition if it neither permits arbitrage opportunities of the rst
type nor of the second type. Furthermore, by Proposition 7 every one-period
nancial market satisfying the no-arbitrage condition satis es the law of one
price.
We conclude this chapter by providing a result that allows us to verify
numerically the absence of arbitrage in a one-period market. Such result
relies on an argument from matrix algebra, and involves the payo¤ matrix
M of formula (1.1) and the vector representation of an investment strategy
# given by formula (1.2). Before stating the result, we need to clarify the
meaning of the symbols , >, when employed in the context of the vector
space Rm : In particular, for any vector x 2 Rm , where
0
1
x1
B 1 C
C
x=B
@ 1 A
xm
we set:
x 0 if and only if xi 0 for all i = 1; :::; m (i.e. all coordinates of x
are nonnegative)
x > 0 if and only if x 0 and for at least one i we have xi > 0 (i.e. all
coordinates of x are nonnegative and at least one is strictly positive)
x 0 if and only if xi > 0 for all i = 1; :::; m (i.e. all coordinates of x
are strictly positive).
Now, since the payo¤s matrix M is of order (K + 1) 2 (N + 1) and the
investment strategy # is an (N + 1)0dimensional column vector, their matrix product M1# is well de ned and gives as a result a (K + 1)0dimensional
column vector. We are now ready to state and prove the following characterization of no-arbitrage in a one-period nancial market.
Proposition 9 In a one-period nancial market the no-arbitrage condition
is satis ed if and only if there exists no strategy # such that M1# > 0.
14
CHAPTER 1. BASIC NOTATION AND DEFINITIONS
Proof. We begin by analyzing the vector resulting from the matrix product
M1#:
2
01
6 1+r
6
M1# =6
6 1+r
4 111
1+r
0
B
B
B
=B
B
@
0S1 (0)
S1 (1) (! 1 )
S1 (1) (! 2 )
111
S1 (1) (! K )
0S2 (0)
S2 (1) (! 1 )
S2 (1) (! 2 )
111
S2 (1) (! K )
P
0#0 0 N
#j Sj (0)
j=1P
# S (1) (! 1 )
#0 (1 + r) + N
Pj=1 j j
#0 (1 + r) + N
j=1 #j Sj (1) (! 2 )
111
P
#0 (1 + r) + N
j=1 #j Sj (1) (! K )
111
111
111
111
111
1
0SN (0)
SN (1) (! 1 )
SN (1) (! 2 )
111
SN (1) (! K )
3
0
B
7B
7B
7B
7B
5B
B
@
#0
#1
#2
1
1
#N
1
C
C
C
C
C
C
C
A
C
C
C
C
C
A
0
1
0V# (0)
B V# (1) (! 1 ) C
C
=B
@ 111
A
V# (1) (! K )
(1.13)
where the rst equality follows from the usual "rows by columns" rule and
the second equality follows from the de nition of value process of # (see
(1.3) and (1.4) in the previous section). Therefore:
1
0V# (0)
B V# (1) (! 1 ) C
C
M1#=B
A
@ 111
V# (1) (! K )
0
Suppose then that there exists # such that M1# > 0. Based on the de nition given above to the symbol >, this means that there exists a strategy
# such that V# (0) 0, V# (1) (! k ) 0 for all k = 1; : : : ; K; and that at
least one of the following facts holds:0 either
V# (0) < 0, or there exists at
1
least one scenario k such that V# (1) ! k > 0. If V# (0) < 0, then # is an
arbitrage opportunity of the second type, while in the other case # is an
arbitrage opportunity of the rst type.
Conversely, suppose the no-arbitrage condition is violated, that is there
exist arbitrage opportunities of the rst or of the second type. In any case,
1.3. LAW OF ONE PRICE AND NO-ARBITRAGE
15
0
1
0V# (0)
B V# (1) (! 1 ) C
C > 0,
this implies the existence of a strategy # such that B
@ 111
A
V# (1) (! K )
and hence by (1.13) there is a strategy # such that M 1 # > 0:4
The result we have just proved not only provides a practical rule to
ascertain the absence of arbitrage opportunities in a one-period market, but
it lies also at the heart of the arguments used to show that no-arbitrage
is equivalent to the existence of state-prices and risk-neutral probabilities.
The next chapter is devoted to the discussion of this equivalences.
Chapter 2
Characterization of
No-arbitrage
In this chapter we provide two alternative characterizations of no-arbitrage,
one based on state-price vectors, the other on risk-neutral probabilities. The
central result, called the First Fundamental Theorem of Asset Pricing, is
that in a one-period market no-arbitrage is equivalent to the existence of
both state-price vectors and risk-neutral probabilities.
In the rst two sections we introduce the de nitions of state-price vectors and risk-neutral probabilities, we examin their economic and nancial
meaning and we give an intuitive grasp of their practical use. In particular,
in the rst section we clarify the concept of linear valuation, which is common practice in the valuation of nancial securities. In the second section,
we provide a rst exam of a key issue in asset pricing, i.e. the fact that
under no-arbitrage the discounted security prices are martingales.
In the third section, we rst state (without proof) a mathematical result,
known as Stiemke’s Lemma, that constitutes the mathematical tool at the
basis of the First Fundamental Theorem of Asset Pricing, and then we state
and prove the theorem.
2.1
State Price Vectors
The concept of state-price vectors is a cornerstone of modern asset pricing
theory and of its applications. We begin by providing a formal de nition in
the context of our one-period nancial market.
17
18
CHAPTER 2. CHARACTERIZATION OF NO-ARBITRAGE
De nition 10 State-Price Vectors. A vector
2 RK is a state-price
vector if and only if its coordinates (! 1 ) ; : : : ; (! K ) are all strictly positive, i.e. (! k ) > 0 for k = 1; :::; K, and they satisfy the following equations
K
X
(! k ) =
k=1
Sj (0) =
K
X
1
;
1+r
(! k ) Sj (1) (! k ) ;
for all j = 1; :::; N:
(2.1)
(2.2)
k=1
Hence, in a one-period nancial market, state-price vectors are characterized by two properties. First, by (2.1) the sum of their coordinates must
be equal to the risk-free discount factor. Second, by (2.2) the current price
of every risky security must be equal to a linear combination of the prices
Sj (1) (! k ) in the K possible states, with weights the coordinates (! k ).
In order to have a better idea of the economic/ nancial meaning of stateprice vectors, and to motivate the terminology, consider a one-period market
in which the time01 price of one security, say the j-th security, is such that
Sj (1) (! k ) = 1, while Sj (1) (! l ) = 0 for l 6= k. In other words, if we hold
one unit of security j to be liquidated at time 1, we will receive one unit
of money if state ! k occurs, zero if any other state will. Applying equation
(2.2) to such security a security we obtain
X
(! l ) 1 0 + (! k ) 1 1
Sj (0) =
l6=k
=
(! k ) :
In this case, therefore, (! k ) represents the time00 market of one unit of
money to be received at time 1 if and only if scenario ! k occurs. This
justi es the naming of (! k ) as price of the state (scenario) ! k . In particular, (! k ) is the market price of state ! k in case the security described is
traded, while in case such security is not traded (nor can be replicated by a
suitable investment strategy. . . ) then (! k ) must be correctly interpreted
as the shadow price of state ! k .
In the economic/ nancial literature risky securities that pay 1 unit if
and only if a given scenario is revealed as true, and 0 in all other scenarios,
are also called Arrow-Debreu securities, from the names of two economists
(Kenneth Arrow and Gerard Debreu, both winners of the Nobel prize for
Economics) that rst introduced them. Consistently, the coordinates of
state-price vectors are usually called Arrow-Debreu state-prices.
2.1. STATE PRICE VECTORS
19
It is useful to provide an alternative characterization of the conditions
allowing to identify the state-price vectors. Such characterization yields
additional insights into their economic/ nancial meaning and constitutes
the basis for their practical applications.
Remark 11 A vector
2 RK with all strictly positive coordinates is a
state-price vector if and only if the value process V# (0), V# (1) (! 1 ), :::,V# (1) (! K )
of any investment strategy # satis es the condition
V# (0) =
K
X
(! k ) V# (1) (! k )
(2.3)
k=1
Proof. Given any state-price vector
have that
V# (0) = #0 +
N
X
and any investment strategy #, we
#j Sj (0)
j=1
= #0 +
N
X
#j
j=1
= #0 (1 + r)
K
X
!
(! k ) Sj (1) (! k )
k=1
K
X
(! k ) +
k=1
=
K
X
0
=
(! k )
k=1
(! k ) @#0 (1 + r) +
N
X
N
X
j=1
#j Sj (1) (! k )
1
#j (1) Sj (1) (! k )A
j=1
k=1
K
X
K
X
(! k ) V# (1) (! k )
k=1
The rst equality simply repeats the de nition of value process
PK at time 0.
The second one employs equation (2.2) to replace Sj (0) with k=1 (! k ) Sj (1) (! k ).
P
The third one exploits (2.1), from which we have (1 + r) K
k=1 (! k ) = 1;
and inverts the order of summation. The fourth equality regroups the terms
in the sum. The nal one exploits the de nition of value of a strategy at
time 1 in scenario ! k . This chain of equalities shows that if is a state-price
vector, then (2.3) is satis ed for every investment strategy #.
Conversely, suppose that 2 RK , with all strictly positive coordinates,
is such that equation (2.3) holds for every strategy #. We show that in
this case is a state-price vector. To this aim, consider rst the strategy
20
CHAPTER 2.
CHARACTERIZATION OF NO-ARBITRAGE
that involves buying one unit of risk-free asset, and zero untis of every risky
security, i.e. the strategy Jo = 1 and ¥; = 0 for all j = 1,...,N. By applying
to such strategy equation (1.3) in the Definition 2 of value process, we have
Vy (0) = 1 and Vy (1) (wx) = (1+7) for all &. Substituting our strategy into
(2.3) we obtain (2.1).
Consider then the strategy that consists in buying
1 unit of the j-th risky security, and zero units of any other security, i.e.
9o = 0, 0; = 1, Un = 0 for all n = 1,...,N, n #7.
strategy equation
By applying to such
(1.4) in the Definition 2 of value process, we now have
Vo (0) = S;(0), Vo (1) (we) = Sj (1)(we), for all k, from which, substituting
into (2.3), we obtain (2.2), thus completing the proof.
From standpoint of interpretation, the result just proved shows that
state-price vectors can be equivalently characterized by two properties. The
first one is that the cost for setting up any strategy with time-1 nonnegative
liquidation value in all states of the world, and positive in at least one of the
states, must be positive. Indeed, since ¢ (wx) > 0 for all k, if Vs (1) (wz) > 0
for all k and Vy (1) (wz) > 0 for some k, then Vg (0) > 0.
The second one is a linearity property. In fact, given the time-1 values
Vo (1), Vg (1) of any two strategies J and 0’, consider their linear com-
bination with weights a and 8, so to obtain the quantity aVg (1) (wz) +
BV (1) (we) in each scenario k.
Substituting into the right-hand side of
(2.3), we get
K
k=1
K
b (we) [aVo (1) (we)
+ BV
(1) (we)
=
aS)
b we) Vo (1) (we)+
k=1
K
+850 v (we) Vo (1) we)
k=1
From (2.3) we have
K
Vo (0) = $> (we) Vo (1) (wr)
k=l
and
K
Vor (0) = Sov (wx) Vor (1) (we) k=1
Putting these three equations together we end up with:
K
Sv (we) [Ve (1) (we) + BV (1) (we)] = aVe (0) + Vy" (0).
k=1
2.2. RISK-NEUTRAL PROBABILITIES
21
The interpretation is now straightforward: the existence of state-price vectors implies that any linear combination of time-1 value processes must have
a time-0 cost equal to the linear combination of the strategies underlying
the value processes combined at time 1.
Summing up, the existence of state-price vectors implies the value-additivity
(hence linearity) of the time00 costs of investment strategies, since the market cost of combining two strategies must be equal to the sum of the costs of
the two strategies considered separately. Moreover, the attentive reader will
have certainly noticed that this linearity implies that the law of one price
must hold, while the strict positivity of the coordinates of the state-price
vectors implies the absence of arbitrage, in particular of the rst type. We
will discuss these issues in detail in the third section. Before doing that,
however, we introduce the additional fundamental concept of risk-neutral
probabilities.
2.2
Risk-neutral Probabilities
The concept of risk-neutral probability is the second building block of modern
asset pricing theory. We begin with a formal de nition.
De nition 12 Risk-neutral probability. In a one-period nancial market a risk-neutral probability is a strictly positive probability Q on the K
states, i.e. Q(! k ) > 0 for all k = 1; :::; K, such that
1
Sj (0) =
(2.4)
EQ [Sj (1)] , j = 1; :::; N
1+r
where EQ denotes expectation with respect to Q, i.e.
Q
E [Sj (1)] ,
K
X
Q(! k )Sj (1) (! k ):1
k=1
In words, a strictly positive probability on the K states at time 1 is a riskneutral probability if, when employed to compute the expected discounted
value of any risky security at time 1; we obtain the price of the security at
time 0.
Why risk-neutral? In order to understand the motivation and interpretation, we observe that simple algebraic manipulations on (2.4) yield
K
X
Sj (1) (! k )
r=
Q(! k )
01
Sj (0)
k=1
1
Recall that the symbol , means "equal by de nition".
22
CHAPTER 2. CHARACTERIZATION OF NO-ARBITRAGE
or, more synthetically
Q Sj (1) 0 Sj (0)
= r,
E
Sj (0)
j = 1; :::; N
(2.5)
The right-hand side represents the expected (mean) return from buying one
unit of the j-th risky security at time 0 and reselling it at time 1, while the
left-hand side is the net return on the risk-free asset. From the interpretative
viewpoint, a risk-neutral probability has the property of making all expected
(mean) returns on risky securities equal to the risk-free rate. Hence, the
use of a risk-neutral probability makes the valuation of a risky security
only depend on its expected (mean) return, meaning that risk (in terms of
variance and higher moments) can be disregarded.
At rst sight, the reader familiar with models of portfolio choice and
investment valuation, such as the mean-variance model and the CAPM
(Capital Asset Pricing Model), may nd the interpretation just given in
contradiction with such models. To understand why this is not the case,
we take a step back and recall the basic notation used in the rst section
of the rst chapter for our one-period market. In particular, we recall that
P (! k ) > 0; k = 1; : : : ; K indicates the (positive) probability employed by
the investors to make their nancial decisions (to compute e¢cient frontiers, to write CAPM equations, and so on). Risk-neutral probabilities, when
they exist, are in general di¤erent from the probabiliti P, i.e. in general
Q(! k ) 6= P (! k ). The existence of risk-neutral probabilities, therefore, is
perfectly consistent with the fact that agents incorporate risk in their nancial asset valuations. In fact, risk-neutral probabilities enable us to value
risky securities as if investors were neutral to risk, although their risk aversion clearly arises when the probabilities P (! k ), those actually used to make
nancial decisions, are employed.
How are risk-neutral probabilities actually employed? The answer is
twofold. On one hand, they allow us to characterize arbitrage-free markets,
and this will be the object of the First Theorem of Asset Pricing to be proved
in the next section. On the other hand, they enable us to simplify the pricing
of several derivative securities, exactly because by employing risk-neutral
probabilities we can act as if investors were neutral to risk. We will examine
these concepts thoroughly in the next chapters. Before doing so, however,
we discuss some equivalent characterizations of risk-neutral probabilities,
thus providing a better understanding of their economic/ nancial meaning.
Remark 13 The following conditions are all equivalent:
2.2. RISK-NEUTRAL PROBABILITIES
23
1. Q is a risk-neutral probability (as de ned in De nition 12);
2. the Value Process of any investment strategy # satis es the condition
V# (0) =
where EQ [V# (1)] ,
1
EQ [V# (1)]
1+r
(2.6)
PK
k=1 Q(! k )V# (! k );
3. the Discounted Gain Process of any investment strategy # satis es the
condition
(2.7)
EQ [G# (1)] = 0
P
K
where EQ [G# (1)] , k=1 Q(! k )G# (1) (! K ).
Proof. To prove that (2.4) implies (2.6), for any given strategy #0 , #1 ,:::, #N
P
1
Sj (1) (! k ), with j = 1; :::; N .
we rewrite (2.4) as Sj (0) = K
k=1 Q(! k )
1+r
Multiplying both sides by #j and summing over j, we obtain
N
X
#j Sj (0) =
j=1
N
X
j=1
#j
K
X
k=1
Q(! k )
1
Sj (1) (! k )
1+r
(2.8)
PK
P
1
#0 (1 + r). Summing
= 1, then #0 = K
k=1 Q(! k )
1+r
P
1
#0 (1 + r) to the right-hand
#0 to the left-hand side and K
k=1 Q(! k )
1+r
side of (2.8) and grouping terms we obtain
0
1
N
K
N
X
X
X
1 @
#0 (1 + r) +
#j Sj (0) =
Q(! k )
#j Sj (1) (! k )A :
#0 +
1+r
Since
k=1 Q(! k )
j=1
j=1
k=1
Since from equations
(1.3) and (1.4) in the De nition P
2 of value process we
P
N
#
S
(0)
=
V
(0)
and
#
(1
+
r)
+
have that #0 + N
0
#
n=1 n n
j=1 #j Sj (1) (! k ) =
V# (1) (! k ) ; our last equations becomes therefore
V# (0) =
K
X
k=1
Q(! k )
1
V# (1) (! k )
1+r
=
1
EQ [V# (1)]
1+r
To show that (2.6) implies (2.4), consider the strategy that consists in buying
one unit of the generic j-th security, and zero units of any other security, i.e.
#0 = 0, #j = 1, #n = 0 for all n = 1; :::; N , n 6= j. Applying the de nition
24
CHAPTER 2. CHARACTERIZATION OF NO-ARBITRAGE
of value process to such strategy we have V# (0) = Sj (0), V# (1) (! k ) =
Sj (1)(! k ) for all k. Substituting such strategy into (2.6) yields (2.4), thus
proving the implication.
To show
(2.6) is equivalent to condition (2.7), recall that EQ [G# (1)] =
Pthat
K
0 means
k=1 Q(! k )G# (1) (! K ) = 0; and that, from equation (1.6) in
1
V# (1) (! k ) 0 V# (0). Taken toDe nition 3, we have G# (1) (! k ) =
1+r
gether, these two equations imply
K
X
k=1
and hence
K
X
k=1
1
V# (1) (! k ) 0 V# (0) = 0
Q(! k )
1+r
K
X
1
V# (1) (! k ) =
Q(! k )
Q(! k )V# (0)
1+r
k=1
The equivalence is then proved upon recalling that
P
1
EQ [V# (1)] and K
k=1 Q(! k ) = 1:4
1+r
PK
k=1 Q(! k )
1
V# (1) (! k ) =
1+r
Condition (2.6) is easily interpreted at the light of the comments following the de nition of risk-neutral probability. Indeed, equation (2.6) is
equivalent to saying that for any strategy # the following must hold
Q V# (1) 0 V# (0)
E
= r:
V# (0)
Hence, a probability is risk-neutral if, when computing the expected return
on an investment strategy, it leads to an expected return equal to the riskfree rate.
Condition (2.7) expresses the same nancial intuition from the viewpoint
of the expected discounted gain. Indeed, we interpret condition (2.7) as
saying that risk-neutral probabilities make the discounted gain from any
strategy equal to zero. The reader is asked once more to interpret such
property correctly. In particular, we are not saying that investors expect a
null gain from every investment strategy: if that were the case, they would
not have any reason to invest! In general, when the expected discounted gain
is computed employing the probability P there clearly exist strategies with
positive expected discounted gains. In fact, in general it is only when a riskneutral probabilities replaces P to compute expectations that all expected
discounted gains become null.
2.3. THE FIRST FUNDAMENTAL THEOREM OF ASSET PRICING25
We conclude this section by noting that condition (2.7) can be equivalently expressed as follows: for any investment strategy #, the discounted
gain process satis es the condition
G# (0) = EQ [G# (1)]
(2.9)
The equivalence between (2.7) and (2.9) is an immediate consequence of (1.5)
in the De nition 3 of discounted gain, according to which G# (0) = 0. Why
is it interesting to rewrite (2.7) in this way? The reason is that risk-neutral
probabilities are also commonly de ned equivalent martingale probability
measures. What is a martingale? We will introduce this concept formally
in the second part of the book, the one dealing with multi-period markets.
Here, we just mention that a martingale is, roughly speaking, a sequence
of random variables satisfying the following property: the value taken by
a variable at time t is equal to the expected value of the random variable
at time t + 1, conditional on all information available at time t. Consider
then the simple sequence fG# (0) ; G# (1)g: thanks to (2.9), we can say that
this simple sequence is actually a martingale. The martingale property will
be extended to multi-period markets, where risk-neutral probabilities will
be characterized in general as probabilities transforming discounted gain
processes into martingales.
2.3
The First Fundamental Theorem of Asset Pricing
In this section we state and prove the central result of this chapter, result
whose importance is made clear by its very same name: First Fundamental
Theorem of Asset Pricing. The proof we provide is based on a mathematical
result known as Stiemke’s Lemma, which we state below (without proof).
Lemma 14 Stiemke’s Lemma. Let A be a matrix with m rows and l
columns, let y denote the generic (column) vector in Rm and x the generic
(column) vector in Rl . Furthermore, let T denote transposition, e.g. y T =
(y1 ; :::; ym ): Then, one and only one of the following statements is true:
1. there exists y 0 (i.e. a vector with all strictly positive coordinates)
such that y T 1 A = 0
2. there exists x such that A1x > 0 (i.e. A1x is a vector with all nonnegative coordinates, and at least one strictly positive coordinate).
26
CHAPTER 2. CHARACTERIZATION OF NO-ARBITRAGE
Proof. See for example D. Gale (1960).4
The fundamental importance of state-price vectors and risk-neutral probabilities, de ned and discussed extensively in the two previous sections,
resides in that they both provide equivalent characterizations of one-period
arbitrage-free markets, as we now show.
Theorem 15 First Fundamental Theorem of Asset Pricing. In a
one-period nancial market the following statements are equivalent:
1. no-arbitrage holds;
2. there exist state-price vectors;
3. there exist risk-neutral probabilities.
Proof. We rst show the equivalence between 1. and 2. From Proposition 9
in the previous chapter we know that there no arbitrage holds if and only if
there exists no investment strategy # such that M 1 # > 0, where M is the
(K + 1) 2 (N + 1) payo¤s matrix de ned by (1.1) in the previous chapter.
We now apply Stiemke’s Lemma, with the payo¤s matrix M in place of the
generic matrix A, and hence with m = K + 1 and l = N + 1. According
to the lemma, there is no # such that M 1 # > 0 if and only if there exists
y 0 such that y T 1 M = 0. Denoting the K + 1 coordinates of y by y0;
y1; :::; yK , we can write explicitly the conditions y 0, y T 1 M = 0, thus
obtaining
2
8
>
>
>
6
>
>
6
>
>
< [y0; y1 ; : : : ; yK ] 6
6
4
>
>
>
>
>
>
>
:
y0; y1; :::; yK > 0;
01
1+r
1+r
111
1+r
0S1 (0)
S1 (1) (! 1 )
S1 (1) (! 2 )
111
S1 (1) (! K )
0S2 (0)
S2 (1) (! 1 )
S2 (1) (! 2 )
111
S2 (1) (! K )
111
111
111
111
111
0SN (0)
SN (1) (! 1 )
SN (1) (! 2 )
111
SN (1) (! K )
and hence, computing the product row by column we get
8
P
0y0 + (1 + r) K
>
k=1 yk = 0
>
>
<
P
j = 1; :::; N
0y0 Sj (0) + K
k=1 yk Sj (1) (! k ) = 0;
>
>
>
:
y0; y1; :::; yK > 0
3
7
7
7=0
7
5
2.3. THE FIRST FUNDAMENTAL THEOREM OF ASSET PRICING27
and, setting
(! k ) =
yk
for k = 1; : : : ; K, we have
y0
8
PK
1
>
>
>
k=1 (! k ) =
>
1
+
r
>
<
PK
>
k=1 (! k ) Sj (1) (! k ) = Sj (0);
>
>
>
>
: (! ) > 0; k = 1; :::; K
k
j = 1; :::; N
(2.10)
Summing up: thanks to Stiemke’s Lemma we have shown that no-arbitrage,
i.e. the absence of any # such that M 1 # > 0, is equivalent to the existence of (! 1 ) ; :::; (! K ) satisfying (2.10). According to De nition 10,
(! 1 ) ; :::; (! K ) are actually the coordinates of a state-price vector, thus
showing the equivalence of 1. and 2.
To prove the equivalence of 2. and 3., we rst suppose that there exists
a state-price vector with coordinates (! 1 ) ; :::; (! K ), and starting from it
we contruct the quantities Q (! k )’s as follows:
Q (! k ) =
(! k ) (1 + r);
k = 1; :::; K:
Since (! k ) > 0; k = 1; :::; K, we clearly have Q (! k ) > 0; k = 1; :::; K.
Furthermore, exploiting the rst of the two conditions in De nition 10 we
obtain
K
K
X
X
1
(1 + r) = 1;
Q (! k ) =
(! k ) (1 + r) =
(1 + r)
k=1
k=1
thus showing that the Q (! k )’s are probabilities on the states ! k . By using
such probabilities, for any risky security j we have
P
Sj (1)
Sj (1) (! k )
Q
= K
E
k=1 Q(! k )
1+r
1+r
=
=
PK
k=1
PK
k=1
(! k ) (1 + r)
Sj (1) (! k )
1+r
(! k ) Sj (1) (! k )
= Sj (0);
where the last equality uses (2.2) of De nition 10. The chain of equalities shows that, if there exists a state-price vector, then one can construct a probability that is strictly positive on all states and such that
28
CHAPTER 2. CHARACTERIZATION OF NO-ARBITRAGE
Sj (1)
. According to De nition 12, such probability is inSj (0) =
1+r
deed a risk-neutral probability.
Finally, if there exists a risk-neutral probability Q, we can set (! k ) =
P
1
Q(! k )
, k = 1; :::; K. Clearly (! k ) > 0 for all k, and K
.
k=1 (! k ) =
1+r
1+r
PK
Furthermore, for every risky security j we have k=1 (! k ) Sj (1) (! k ) =
PK
Sj (1) (! k )
= Sj (0); showing that if Q is indeed a risk-neutral
k=1 Q(! k )
1+r
Q(! k )
probability then the vector with coordinates (! k ) =
is a state-price
1+r
vector, concluding the proof.4
EQ
Summing up: no-arbitrage, existence of state-price vectors and existence of risk-neutral probabilities are all faces of the same coin, in that they
are all equivalent. From the operational viewpoint, therefore, verifying if
a nancial market admits arbitrage opportunities or not can be done directly, i.e. by checking whether there exist investment strategies whose value
process is nonnegative in any case, but positive at time 1 in some possible scenario, or negative at time 0. But it can also be done in a shorter
way, i.e. by verifying the existence of state-price vectors, hence by checking
whether the linear system y T 1 M = 0 admits strictly positive solutions.
Or, equivalently, it can be done by verifying whether there exist risk-neutral
probabilities. From the practical viewpoint, this means
PK checking if there
exist Q(! 1 ); :::; Q(! K ) all positive and such that
k=1 Q (! k ) = 1 and
1 PK
Sj (0) =
Q (! k ) Sj (1) (! k ) ; with j = 1; :::; N .
1 + r k=1
The following questions arise now naturally: provided state-price vectors
exist, can we have more than one? If the answer is yes, what are the conditions under which that happens? And can we draw the same conclusions
for risk-neutral probabilities? The next chapter is devoted to providing a
detailed answer to these and similar questions.
Chapter 3
Complete One-Period
Markets
The concept of complete nancial markets is a fundamental one in the theory
of nance. Complete markets, for instance, constitute the foundation of the
well known Binomial and Black-Scholes models used to value options and,
more generally, a wide range of derivative products.
In the rst section of this chapter, we present the formal de nition of
completeness of a one-period market and provide a practical criterion to
verify whether a nancial market is complete or not. In the second section,
we analyze completeness in light of the no-arbitrage property, i.e. we investigate the consequences of assuming that the market is both complete and it
admits state-price vectors (or, equivalently, it admits risk-neutral probabilities, see the First Fundamental Theorem of Asset Pricing). In this regard,
we state and prove a result known as the Second Fundamental Theorem of
Asset Pricing. This result states that a market that is both arbitrage-free
and complete is characterized by the existence of a unique state-price vector
or, equivalently, of a unique risk-neutral probability.
In the third and nal section we expand on the interpretation of the property of completeness. To this end, we formally introduce the general concept
of Arrow-Debreu securities and we discuss how completeness is equivalent
to such securities being available for trade. We also employ these securities to further interpret the notion of state-price vector (and risk-neutral
probability) in the case of complete and incomplete nancial markets.
29
30
3.1
CHAPTER 3. COMPLETE ONE-PERIOD MARKETS
De nition and Characterization
In general and intuitive terms, the property of completeness of a nancial
market translate formally the “spanning” achieved by a given market, i.e.
the “amplitude”of the set of future cash ‡ows that can be obtained by
suitably investing in the securities traded in that market.
To formalize this intuition, we call cash-‡ow, or contingent claim at time
1, any vector X in RK , whose coordinates are denoted by X (! 1 ) ; : : : ; X (! K ) :
We interpret X (! k ) as the amount of money that the holder of the contingent claim receives (or, possibly, is liable for) at time 1 and in the state
! k . Given a one-period nancial market and a contingent claim X, under what conditions is X actually traded (i.e. can be bought/sold) in the
market? The natural answer is that this is possible when there exists an
investment strategy yielding at time 1, scenario by scenario, a liquidation
value equal to the value of the contingent claim. Formally, therefore, we
say that X is traded when there exists an investment strategy # such that
V# (1) (! k ) = X (! k ), for all k = 1; : : : ; K. In this case, we also say that the
contingent claim X is attainable (or also redundant).
With this in mind the “spanning”, or “amplitude”, of a nancial market
is related to the dimension of the set of contingent claims attainable in that
market: the higher the dimension of the set of claims that can be attained by
suitably investing in the basic securities, the larger the ampler the market.
Clearly, the situation of “largest spanning” possible is when every contingent
claim is attainable. We call complete exactly those markets with this feature.
De nition 16 Complete One-Period Markets. We say that a oneperiod nancial market is complete if every contingent claim is attainable,
i.e. if for all X = (X (! 1 ) ; : : : ; X (! K ))T 2 RK there exists an investment
strategy # such that
V# (1) (! k ) = X (! k ) ,
for all k = 1; : : : ; K:
(3.1)
The fact that a market is complete or not depends on the securities
available for trade. In particular, we rst need a large enough number of
securities. For example, in the extreme case when the only security available
is the risk-free asset, and more than one scenario is possible at time 1, every
contingent claim not yielding the same amount of money in all states of
the world will not be attainable, and hence the market is incomplete. The
number of traded securities, however, is not the only factor a¤ecting the
completeness of a nancial market. Besides the number of securities, we
3.1.
DEFINITION AND CHARACTERIZATION
31
have to verify that such securities are “different” enough to be combined so
as to replicate any contingent claim.
To make things more precise we introduce the K x (N+1) matrix A, obtained from the payoffs matrix M (see (1.1) in the first chapter) by deleting
the first row of M. Formally, we have
Ltr
$(1)(1)
... Sy (1) (ws)
l+r
Sy (1) (wx)
...
SN (1) (wx)
In words, the columns of A collect the time-1 prices of the risk-free asset
and of the N risky securities in the K possible scenarios. Observing that
the product of the generic k-th row of A by the column vector # yields
Yo (1+r)+
ya 0;5; (1) (wz), and exploiting again the definition of value
process at time 1, we immediately see that
Ab
=
l+r
SS; (1) (w)
...
SN (1) (w1)
ltr
S| (1) (we)
...
SN (1) (we)
Vo
;
vy
Vg (1) (w1)
_
Ltr Sy(1)(wx) ... Sv(t)(wx) | \ oy
Vo (1) (we)
Vo (1) (wx)
As a consequence, the system of equations (3.1) that formalizes the attainability of a contingent claim can be written synthetically as A-v = X.
Therefore,
the Definition 16 of complete financial market is equivalent to
requiring the system A-v? = X to have solutions for any contingent claim
X. For this reason, the characterization of complete markets that we prove
herafter is based on a condition on the rank of matrix A (recall that the rank
of a matrix is the number of linearly independent columns, or equivalently
rows, of a matrix).
Proposition 17 A one-period financial market is complete if and only if
the rank of its matrix A is equal to the number K of scenarios possible at
time 1. Equivalently,
the market is complete
if and only if K
among
the
N +1 columns (rows) of A are linearly independent.
Proof. We first assume that the rank of A is K and we show that any
contingent claim X € R* is attainable, that is, thatthe system A-’ = X
admits solutions for all X € R*. To this end, we set up the complete
matrix of the system, ie.
the matrix of order K x (N + 2) obtained by
adding the column vector X to the columns of A, and we denote by p the
32
CHAPTER 3. COMPLETE ONE-PERIOD MARKETS
rank of the complete matrix. Clearly, it must be K, since adding a
column to a matrix cannot decrease the rank of that matrix. Furthermore,
since the rank of a matrix is anyway less than or equal to the minimum
of the number of rows and columns, then min fK; N + 2g. Putting all
this together, we immediately obtain = K, i.e. the complete matrix of the
system A1# = X has the same rank as the incomplete one. As a consequence
on the basic result on solutions of linear systems (often known as RouchéCapelli Theorem), the system A1# = X admits solutions and hence X is
attainable, no matter what X 2 RK we select.
Conversely, suppose that the system A1# = X has solutions for all X 2
RK . In other words, this means that every vector in RK can be expressed
as a suitable linear combination of the columns of the matrix A, i.e. the
columns of A span RK . A basic result from linear algebra tells us that every
set of vectors that span a given nite-dimensional vector space contains a
set of linearly independent vectors in number equal to the dimension of the
spanned space. In our case this amounts to saying that among the N + 1
columns of the matrix A there are K linearly independent vectors, i.e. the
rank of A is K, thus concluding the proof. 4
In the nancial literature, the result just proved is usually stated in
the following intuitive and informal way: a one-period nancial market is
complete if and only if the number of linearly independent traded securities
is equal to the number of possible scenarios at time 1. Proposition 17 and
its proof clarify the meaning of linearly independent securities: they are
securities whose time-1 prices in the K possible states can be arranged as
column vectors of RK , column vectors that are linearly independent. Finally,
we stress the fact that the number of securities being greater or at least equal
than the number of possible scenarios, i.e. N +1 K, is a necessary, but not
su¢cient, condition for completeness. In fact, a number of traded securities
(including the risk-free asset) greater or equal than the number of scenarios
does not guarantee market completeness unless K among the N +1 securities
are linearly independent.
What is the relation between no-arbitrage and completeness? The answer to this question is provided by the Second Fundamental Theorem of
Asset Pricing, which we state and prove in the next section.
3.2. THE SECOND FUNDAMENTAL THEOREM OF ASSET PRICING33
3.2
The Second Fundamental Theorem of Asset
Pricing
When coupled with no-arbitrage, the property of market completeness has
the e¤ect of dramatically restricting the set of state-price vectors or, equivalently, risk-neutral probabilities, as the following result shows.
Theorem 18 The Second Fundamental Theorem of Asset Pricing.
In a one-period nancial market the following statements are equivalent:
1. both no-arbitrage and market completeness hold;
2. there exists one, and only one, state-price vector;
3. there exists one, and only one, risk-neutral probability.
Proof. We rst prove the equivalence of 1. and 2. To this end, we reacll that
by (2.10) in the First Fundamental Theorem of Asset Pricing the state-price
vectors are the solutions to the following system:
8 PK
>
k=1 (1 + r) (! k ) = 1
>
>
<
PK
j = 1; :::; N
k=1 Sj (1) (! k ) (! k ) = Sj (0);
>
>
>
:
(! k ) > 0; k = 1; :::; K
This system can be rewritten equivalently as follows:
8 2
3
0
>
1+r
1+r
::: 1 + r
>
>
> 6 S (1) (! ) S (1) (! ) : : : S (1) (! ) 7
>
1
1
1
2
1
K
>
7@
< 6
5
4 :::
:::
::: :::
>
SN (1) (! 1 ) SN (1) (! 2 ) : : : SN (1) (! K )
>
>
>
>
>
: (! ) > 0; k = 1; :::; K
1
1
(! 1 )
B S1 (0) C
C
::: A = B
@ ::: A
(! K )
SN (0)
1
0
k
which, in base of the de nition of matrix A given by (3.2) in the previous
section, is also equivalent to
8
0
1
0
1
>
1
>
>
(! 1 )
>
B S1 (0) C
>
>
C
< AT @ : : : A = B
@ ::: A
(3.3)
(! K )
>
S
(0)
>
N
>
>
>
>
: (! ) > 0; k = 1; :::; K
k
34
CHAPTER 3. COMPLETE ONE-PERIOD MARKETS
Suppose now that both no-arbitrage and completeness hold. In this case system (3.3) admits solutions and the matrix A has rank K. Since transposing
a matrix leaves its rank unchanged, system (3.3) admits solutions and has
associated incomplete matrix with rank equal to the number of unknowns.
By Rouchè-Capelli Theorem, therefore, system (3.3) admits a unique solution, i.e. there exists one and only one state-price vector. This shows that
1. implies 2. Vice versa, if system (3.3) has a unique solution, again by
Rouchè-Capelli Theorem the matrix AT ; and hence A; has rank equal to
K; which means that both no-arbitrage (since (3.3) has solutions) and completeness (since the rank of A is K) hold, thus proving that 2. implies 1.
This concludes the proof of the equivalence between 1. and 2.
To show the equivalence of 2. and 3. we just need to recall from the proof
of the First Fundamental Theorem of Asset Pricing that Q (! 1 ) ; : : : ; Q (! K )
is a risk-neutral probability if and only if there exists a state-price vector
= ( (! 1 ) ; : : : ; (! k ))T such that Q (! k ) = (1 + r) (! k ) for k = 1; :::; K.
It is then apparent that there is only one state-price vector if and only if
there exists only one risk-neutral probability.4
In words, a one-period arbitrage free and complete nancial market is
characterized by the existence of a unique state-price vector or, equivalently,
by the existence of a unique risk-neutral probability. As a consequence, a
one-period nancial market that is arbitrage-free, but incomplete, admits
more than one state-price vector and more than one risk-neutral probability. How can we interpret the cases in which state-price vectors are not
uniquely determined? What is their meaning from the economic and nancial viewpoint in that case? We examine these issues in the nal section of
this chapter.
3.3
Interpretation and Further Remarks
In order to examine thoroughly the relation between state-price vectors and
completeness/incompleteness of an arbitrage free one-period market, we go
back to the concept of Arrow-Debreu securities, introduced brie‡y in Section
2.1 of the second chapter. We recall that the Arrow-Debreu security associated with scenario k is the contingent claim Ek that takes the following
values on the K scenarios:
(
1
if h = k
Ek (! h ) =
(3.4)
0
if h 6= k
3.3. INTERPRETATION AND FURTHER REMARKS
35
In words, the Arrow-Debreu security associated with the k-th scenario yields
one unit of money if, and only if, scenario k occurs at time 1, while it
yields nothing in all scenarios di¤erent from the k-th one. The existence
of an Arrow-Debreu security associated with a given scenario is usually
interpreted as the availability on the market of ‘perfect insurance’ against
the risks associated to a given scenario.
The following simple but e¤ective example is usually employed to describe the meaning given to the term ‘perfect insurance’. Suppose that only
two scenarios are possible at time 1, and assume that they represent risks
related to “weather conditions”. In particular, suppose that ! 1 is the scenario corresponding to the event “rain at time 1”, while ! 2 is the scenario
corresponding to event “sunshine at time 1” (for simplicity, a cloudy day
without rain is not taken into account...). At time 1 and in scenario ! 1
someone who has to go runs the risk of getting wet. We can of course insure
against this risk by buying today, i.e. at time 0, an umbrella. Insurance
may however be ‘imperfect’ or ‘perfect’. In what sense? Suppose rst that
there exists only the following “security”: we buy an umbrella today that
will be delivered to us tomorrow independently of the weather conditions.
Such insurance is imperfect since tomorrow we will own an umbrella even
in case of sunshine (scenario ! 2 ). This type of “security” is in fact a riskfree asset, o¤ering super-insurance (clearly expensive...). Suppose now that
there is also the following “security”: we can agree today that tomorrow
we will buy an umbrella if and only if it rains. This is what we mean by
‘perfect insurance’: we are able to get what we need if and only if we actually need it. Clearly, this second “security” is indeed an Arrow-Debreu
security associated with scenario ! 1 . Furthermore, it is fair to expect that,
in markets working reasonably well, the cost today of the second “security”
is going to be lower than that of the rst one (the risk-free asset). And
this is in fact the case when there exist state-price vectors, i.e. when the
no-arbitrage condition is satis ed.
In turns out that the property of completeness is equivalent to the
fact that the nancial market allows to buy (or sell...) perfect insurance
against the risks associated to each one of the K possible scenarios. One
implication is almost obvious: according to De nition 16, in a complete
market every contingent claim is attainable, hence so are all K ArrowDebreu securities E1 ; :::; EK associated with the K possible scenarios at
time 1. Vice versa, suppose that all Arrow-Debreu securities can be replicated by suitable investment strategies, and denote by #k , with coordinates
#k0 ; #k1 ::; #kN ; the strategy replicating the Arrow-Debreu security associated
with the generic k-th scenario. Given any contingent claim X with coordin-
36
CHAPTER 3. COMPLETE ONE-PERIOD MARKETS
ates X (! 1 ) ; : : : ; X (! K ) we argue that, if all K Arrow-Debreu securities can
be replicated, then also the generic contingent claim X can, i.e. the market
is in fact complete. To this end, consider the strategy #X with coordinates
X
X
#X
0 ; #1 ::; #N , de ned as follows:
1
K
#X
j = X (! 1 ) #j + : : : + X (! K ) #j ;
j = 0; 1; :::; N
(3.5)
In words, the units of the j-th security held in strategy #X are a linear
combination of those held according to the K strategies that replicate the
Arrow-Debreu securities, and the weights of this linear combination are the
values of the contingent claim X in the K scenarios. Now, by substituting
(3.5) into (1.4) in the De nition 2 of value process given in Chapter 1, after
some tedious but otherwise strightforward algebraic manipulations we get
V#X (1) (! h ) =
K
X
k=1
X (! k ) V#k (1) (! h ) ;
h = 1; :::; K
(3.6)
Recalling that the generic strategy #k replicates the Arrow-Debreu security
associated with the k-th state, and hence that
(
1
if h = k
(3.7)
V#k (1) (! h ) =
0
if h 6= k
by substituting into (3.6) we have
V#X (1) (! h ) = X (! h ) ;
h = 1; :::; K:
Therefore, if all Arrow-Debreu securities are attainable the market is indeed
complete.
Let’s go back now to the case in which completeness and no-arbitrage
hold together.The existence of a unique set of state-prices that holds in this
case can be clearly interpreted at the light of our previous discussion of
Arrow-Debreu securities. According to (2.3), in fact, for any given stateprice vector and any investment strategy #, we have that
V# (0) =
K
X
(! k ) V# (1) (! k )
k=1
If the market is complete and if in this equation we substitute a strategy #h
that replicates the generic h-th Arrow-Debreu security, we can exploit (3.7)
to get
V#h (0) = (! h )
3.3. INTERPRETATION AND FURTHER REMARKS
37
Therefore, all investment strategies that replicate the h-th Arrow-Debreu
security must have the same cost, given by the unique price of state h.
Indeed, if that were not the case, there would exist two strategies having
di¤erent costs, yet replicating the very same Arrow-Debreu security. But
this would violate the law of one price, leading to arbitrage opportunities
and preventing the existence of state-price vectors, against the fact that
no-arbitrage holds.
We conclude this section by interpreting market incompleteness in light
of the remarks made so far. We can say that a market is incomplete if
and only if there exists at least a scenario k for which no perfect insurance is available. In terms of prices, it follows that the price (! k ) of a
non-perfectly-insurable state cannot be univocally determined. The interpretation should now be clear: in the absence of arbitrage, the only prices
that are univocally determined are those of the tradeable contingent claims.
A natural question is now the following: is the set non-perfectly-insurable
state prices actually completely undetermined? Or can we give some bounds
to such prices, although they are not unique? The next chapter, which deals
with the pricing of attainable and non attainable claims, o¤ers an answer to
such questions.
Chapter 4
No-Arbitrage Valuation of
Derivatives
The most important application of the theoretical results introduced so far
is the pricing of nancial securities by no-arbitrage arguments. In the rst
section of this chapter we describe the context in which we tackle the pricing
problem. In particular, given a one-period nancial market, called the initial
market, we consider the situation in which a new security is added to the
market. The no-arbitrage pricing problem is then the following: under what
conditions on the initial price of the new security is the no-arbitrage property
preserved in the extended market? As we will see, the answer depends on
the new security being redundant or not.
We focus on the case of a redundant security in the second section. In
that section, we establish two (equivalent) conditions on the initial price of
the new security that are necessary and su¢cient for no-arbitrage to hold
in the extended market too. The rst condition requires the price of the
new security to be equal to the value process of any strategy that replicates
the new security. The second condition requires the price to be equal to
expected value of the discounted payo¤, the expectation being taken under
any of the risk-neutral probabilities of the initial market.
We consider the case of a non-redundant new security in the third section. As we will see there, the fundamental di¤erence between the case of
redundant and non redundant securities is that, while in the rst case there
is a unique initial price for the new security that preserves no-arbitrage in
the extended market, in the second case there is a whole (open) interval of
prices that guarantee such property.
39
40
4.1
CHAPTER 4. NO-ARBITRAGE VALUATION OF DERIVATIVES
The Framework
We take as initial input a one-period nancial market with a risk-free security
with index j = 0, and N risky securities indexed by j = 1; :::; N . From
now onwards we call this market the initial market and we assume it to be
arbitrage-free throughout the chapter. The situation we have in mind is
the following one: a nancial insitution (e.g. an investment bank) decides
to issue a new security in the initial nancial market. We call extended
market the one-period market obtained by adding the new security to the
already existing ones. The no-arbitrage pricing problem resides in providing
conditions under which the price of the new security can be set on the basis
of the initial market security prices, either univocally or in terms of a price
range.
To make things clearer, we assume that the time-1 payo¤ associated with
the new security is described by a contingent claim X, with coordinates
X (! 1 ) ; : : : ; X (! K ) : Furthermore, we denote by SX (0) the time-0 price of
the new security. Hence, by buying (respectively, shortselling) at time 0 the
new security at a price SX (0), we will receive (resp., we will have to pay) at
time 1 and in the generic state k a number X (! k ) of units of money. The
extended market will then include, besides the risk-free security, N + 1 risky
securities with initial prices S1 (0); :::; SN (0); SX (0) and with time-1 prices
S1 (1)(! k ); :::; SN (1)(! k ); X(! k ) in the states k = 1; :::; K.
We are now faced with the following problem: in what way, and under
what conditions, can we determine SX (0) starting from the initial market’s
security prices? As we will see, the answer to this question depends on the
fact that the contingent claim X is attainable or not. In the next section we
deal with the case of redundant (i.e. attainable) securities, while in Section
4.3 we deal with the case of non-redundant securities.
4.2
The Case of Redundant Securities
In this section we assume that the new security is attainable on the initial market, i.e. there exists an investment strategy #X with coordinX
ates #X
:::; #X
0 ; #1 ; P
N whose value process at time 1 satis es the condition
N
X
#0 (1 + r) + j=1 #X
j Sj (1) (! k ) = X(! k ); for k = 1; :::; K. In this case, we
can supply two equivalent necessary and su¢cient conditions on the initial
price SX (0) of the new security under which no-arbitrage is maintained in
the extended market.
4.2. THE CASE OF REDUNDANT SECURITIES
41
Proposition 19 If the new security is redundant (can be replicated), the
following three conditions are equivalent:
1. no-arbitrage holds in the extended market;
2. any strategy #X that replicates the new security satis es
SX (0) = V#X (0) ;
(4.1)
3. for every risk-neutral probability Q of the initial market we have
SX (0) =
1
EQ [X] :
1+r
(4.2)
Proof. Suppose that the extended market is arbitrage-free so that, in particular, the law of one price is satis ed. Since the strategy that consists only in
buying one unit of the new security and any strategy #X that replicates the
contingent claim X have the same value process at time 1, then condition
(4.1) is an immediate consequence of the law of one price.
Assuming now that (4.1) holds we show that (4.2) holds as well. To
this end, we note that for any strategy #X that replicates the contingent
claim X in the initial market, and for any risk-neutral measure Q of the
initial market, from (2.6) in Remark 13 of Chapter 2, we have that the value
process #X satis es the following condition:
V#X (0) =
2
3
1
EQ V#X (1) .
1+r
(4.3)
Since #X replicates X then V#X (1) = X. Substituting then V#X (1) = X
into (4.3) and exploiting the fact that by assumption SX (0) = V#X (0) ; we
obtain (4.2).
Finally, we prove that, if (4.2) holds, then the extended market is arbitrage free. Let then Q be any risk-neutral probability of the initial market,
i.e. Q(! k ) > 0 for all k = 1; :::; K and
Sj (0) =
1
EQ [Sj (1)] for all j = 1; :::; N
1+r
and suppose that Q satis es condition (4.2). It is then clear that Q is
a risk-neutral probability also for the extended market that consists of
frisk-free asset, S1 ; :::; SN , new security Xg. The application of the First
Fundamental Theorem of Asset Pricing to the extended market concludes
the proof.4
42
CHAPTER 4. NO-ARBITRAGE VALUATION OF DERIVATIVES
The result we have just proved states that there are two equivalent ways
to compute the unique price at which an attainable claim can be traded
in the extended market without introducing arbitrage opportunities. Both
ways have a strong practical relevance, since they point at two alternative
methods to price, for instance, all the so-called derivative securities, which
make up for a substantial portion of the assets commonly traded nowadays.
As we will see in the sequel, in fact, the above result is at the heart of the
pricing formulas for options and forward/futures contracts resulting from
the binomial and Black-Scholes model.
Condition (4.1) in Proposition 19 shows that the price of the new security can be obtained from the cost of an investment strategy that replicates
its future payo¤. According to this approach, therefore, to price the new security we have to determine a replicating strategy. The replicating startegy,
of fundamental importance in nancial market practice, is commonly called
(perfect) hedging strategy, and is the strategy that a nancial institution
issuing the new security should follow in order to optimally manage the liabilities that may arise in the future. Hence, the approach at the basis of
condition (4.1) has the strength of providing both the no-arbitrage price of
the new security and the hedging strategy to be followed to face the liabilities
involved by issuing the security.
In more complex models (such as the multi-period or the continuous time
ones) computing explicitly the replicating strategy of a contingent claim is
not always a simple exercise. In fact, in certain practical situation this costly
exercise is not even necessary and, if possible, should be avoided. Consider
for instance the case of an investor interested in taking a long position in the
new security. What really concerns the investor is whether the price of the
new security is fair, in the sense of not allowing arbitrage opportunities. If a
long position in the new security entails limited liability, i.e. no liabilities will
be incurred in the future, the knowledge of the hedging strategy is clearly
unnecessary.
The approach suggested by expression (4.2) in Proposition 19 matches
exactly the needs of an investor taking a long position in a limited liability
new security. According to this approach, indeed, to compute the ‘fair’ price
of the new security the investor needs only to know that the security can
be replicated, but he does not need to compute the replicating strategy.
Given a risk-neutral probability associated with the initial market, in fact,
the investor can compute the expected value at time 1 of the new security,
discount it at time 0 at the risk-free rate, and hence obtain the no-arbitrage
price of the new security. The trade-o¤ between the two approaches is clear:
in the rst case one obtains more information (price and hedging strategy)
4.3. THE CASE OF NON-REDUNDANT SECURITIES
43
but incurs a usually much greater computational burden. In the second
case, the computational burden is reduced, at the cost of obtaining only
the no-arbitrage price of the new security, but not its associated hedging
strategy.
4.3
The Case of Non-redundant Securities
Assume now that the new security cannot be replicated, i.e. that in the
initial market there exists no investment strategy whose value process in
every scenario k is equal to the contingent claim X(! k ). What can we
say in this case about the time-0 price of the new security? To answer
this question, we introduce the concept of super-replication of a contingent
claim: given a contingent claim X, we say that a trading strategy # in the
initial market super-replicates it if V# (1) (! k ) X (! k ) ; for k = 1; :::; K: In
words, a strategy involving securities from the initial market super-replicates
a contingent claim if the value process of such strategy is greater than or
at least equal to the value of the contingent claim in every possible time01
scenario.1
The following proposition states that, if the new security cannot be replicated, then there exists a whole set of initial prices of the new security
all of which guaranteeing that no-arbitrage in the extended market will be
maintained. Speci cally, this set is an open interval with endpoints de ned
in terms of optimal super-replication of the contingent claim X and of his
opposite 0X.
Proposition 20 If the new security is non-redundant (i.e. it cannot be
replicated), the following three conditions are equivalent:
1. no-arbitrage holds in the extended market;
2.
max 0 V# (0) < SX (0) < min V# (0)
(4.4)
where the maximum is taken over all the initial market strategies
that super-replicate 0X, and the minimum over all the initial market strategies that super-replicate X;
1
If a risk-free security exists, then every contingent claim can be super-replicated.
Indeed, if X(! k ) is the maximum value taken by the contingent claim over the time01
scenarios, buying X(! k ) units of the risk-free asset clearly allows us to super-replicate X.
44
CHAPTER 4. NO-ARBITRAGE VALUATION OF DERIVATIVES
3.
inf
Q
1
1
EQ [X] < SX (0) < sup
EQ [X]
1+r
Q 1+r
(4.5)
with the in mum and supremum are computed over the set of riskneutral probabilities associated with the initial market.
The endpoints of the interval de ned in (4.4) have an important nancial
meaning. In particular, the upper bound in (4.4) represents the minimum
cost to super-replicate in the initial market the time01 pay-o¤ of the new
security. As for the lower bound, recall that 0V# (0) represents the amount
of money that one obtains at time 0 by selling the strategy # in the initial
market, and hence by incurring in a time01 liability equal to V# (1) (! k ),
for k = 1; :::; K. As a consequence, the lower bound in (4.4) represents the
maximum amount that, in the initial market, can be obtained at time 0
against a time01 liability which will be anyway not greater than X.
The equivalence of 1. and 2. in Proposition 20 has then the following
interpretation: all and only the time-0 prices of the new security consistent
with no-arbitrage in the extended market are those satisfying the following
two properties. First, they must be greater than the maximum amount that
can be obtained by selling, in the initial market, a portfolio of securities that
entails a time01 liability at most equal to the payo¤ of the new security.
Second, they must be lower than the minimum cost incurred to superreplicate, again in the initial market, the time-1 payo¤ of the new security.
The fact that the only prices consistent with no-arbitrage in the extended
market are those in the interval de ned by (4.4) is easily justi ed. Indeed,
suppose SX (0) greater than or equal to the minimum super-replication cost
of X, and let #3 denote a minimum-cost super-replicating strategy. In this
case, the following arbitrage opportunity is readily available in the extended
market: buy #3 and sell the new security short. In this way, one has a
time-0 in‡ow equal to SX (0) 0 V#3 (0) 0, while at time 1 the value of
this strategy is V# (1) (! k ) 0 X (! k ) 0 in each scenario k, with a strict
inequality in at least one scenario since by assumption the contingent claim
is non attainable. In this way, therefore, one obtains an arbitrage of the
rst type. A symmetric argument shows that the same conclusion is reached
when the lower bound is violated, that is, when the price of the new security
does not exceeds the maximum amount that can be obtained by selling, in
the initial market, a portfolio of securities that entails a time01 liability at
most equal to X.
The fact that all prices in the interval de ned by (4.4) preserve noarbitrage in the extended market is a little bit harder to prove formally, and
4.3. THE CASE OF NON-REDUNDANT SECURITIES
45
will be not treated here. The intuitive idea, however, is not too di¢cult, and
is based on verifying that every extended market in which the price SX (0)
satis es condition (4.4) admits state-price vectors, and hence is arbitragefree by the First Fundamental Theorem of Asset Pricing.
We now conclude with a brief technical remark on restriction (4.5). The
proof of its equivalence to (4.4) (not supplied here) relies upon observing
that, from the de nition of value process, the super-replication problems
described above are in fact linear programming problems, in the sense that
both the objective function and the constraints are linear functions. Exploiting then the so-called Fundamental Theorem of Linear Programming, one
can show that the minimum super-replication price is actually equal to the
supremum of all possible risk-neutral valuations of X computed under the
risk-neutral probabilities associated with the initial market and, symmetrically, the maximum amount that can be obtained at time 0 against a time01
liability not exceeding X is equal to the in mum of these valuations. This
result, which might appear merely theoretical and abstract at rst glance,
is actually intensively employed in the theory of no-arbitrage valuation in
incomplete markets, as will be better explained in the sequel.
Chapter 5
The One-period Binomial
Model
The binomial model is the simplest example of one-period nancial market,
involving just two states at time 1 and only one risky security besides the
risk-free asset. We devote an entire chapter to this simple model for two
reasons. The rst reason is a pedagogical one: the binomila model is an
e¤ective “laboratory” in which the notions learned so far can be applied.
The second reason is more substantial: the binomial model constitutes in
fact the “staminal cell” of the modern theory of no-arbitrage valuation of
derivative securities. Indeed, we will see that such model is the building
block of the general binomial option pricing model and, under a suitable
notion of limit, of the celebrated Black-Scholes model.
In the rst section of this chapter we introduce the assumptions of the
model and then, in the second section, we analyze the conditions under
which no-arbitrage and market completeness hold. Under these conditions,
we derive the explicit expressions of the (unique) state-price vector and riskneutral probability associated with the binomial model.
In the third and nal section we introduce option contracts and discuss
their no-arbitrage valuation by employing the results seen in the previous
chapter. In particular, we derive pricing formulas for both call and put
options. Finally, we discuss the celebrated Put-Call Parity relation between
call and put option prices.
47
48
5.1
CHAPTER 5. THE ONE-PERIOD BINOMIAL MODEL
Assumptions of the Model
The one-period binomial model is based on two very simple assumptions:
only two scenarios, ! 1 and ! 2 , can occur at time 1, and besides the riskfree asset only one risky security is available for trade. For simplicity, in
this chapter we avoid both the time and the security index. Therefore, we
denote by S > 0 the time-0 price of the risky security, and by S (! 1 ) and
S (! 2 ) the time-1 prices of the risky asset in the two possible states of the
world.
In the one-period binomial model it is common to write the time01
prices in the two states in the following way:
S (! 1 ) = uS
and
S (! 2 ) = dS
The coe¢cient u is then interpreted as the gross return (and hence u 0 1 as
the net return) on the risky security in state ! 1 , while d as the gross return
(d 0 1 as the net return) on the risky security in state ! 2 . We assume u 6= d:
if it was u = d, the security would not be risky at all, and u = d = 1+r would
have to be satis ed in order to prevent arbitrage opportunities. Without loss
of generality, we assume u > d, with the mnemonic that u stands for ‘up’
and d for ‘down’.
The behaviour of the risk-free and risky security prices in the one-period
binomial model is usually summarized by means of the following “event-tree”
representation:
risk 0 f ree security
t=0
1
t=1
%
&
risky security
t=0
1+r
S
1+r
t=1
%
&
uS
dS
The questions that arise naturally now are: under what conditions is the
binomial an example of a complete market? Under what conditions is the
binomial market arbitrage-free? What are the relations between this two
sets of conditions? We answer to this questions in the following section.
5.2. COMPLETENESS AND NO-ARBITRAGE
5.2
49
Completeness and No-arbitrage
We rst analyze the conditions under which the binomial is an example of
complete market. To this end, we employ the characterization of market
completeness provided by Proposition 17 in the third chapter. We rst
observe that the pay-o¤ matrix, de ned by (1.1) in the rst chapter, in this
case has the following simple form:
2
3
01 0S
6
7
7
M=6
4 1 + r uS 5
1 + r dS
As a result, the matrix A obtained from M by eliminating the rst row, is
"
#
1 + r uS
A=
1 + r dS
Recalling that in the binomial model the number of scenarios, K, is just
2, according to to Proposition 17 we can say that the market is complete
if and only if the matrix A has rank equal to 2. Since in this case A is a
square matrix, this is equivalent to saying that A is non-singular, i.e. that
the determinant of A is non zero. It is now easy to see that
det (A) = (1 + r) S (d 0 u)
and hence det (A) 6= 0, since by assumption 1 + r > 0, S > 0 and d <
u. Summing up, the assumptions made on the one-period binomial model
ensure that it is complete.
It is now interesting to see if the assumptions ensuring completeness
imply that the law of one price is satis ed as well. To analyze this, we
consider the strategies # with coordinates #0 and #1 , and #0 with coordinates
#00 e #01 , and assume that they take the same value in 1 in both states. We
want to show that, under the condition d < u, then V# (0) = V#0 (0) must
hold, i.e. the law of one price is satis ed. In the one-period binomial model,
requiring the strategies # and #0 to take the same value in the two states at
time 1 means that the following two equations must hold:
(
#0 (1 + r) + #1 uS = #00 (1 + r) + #01 uS
#0 (1 + r) + #1 dS = #00 (1 + r) + #01 dS
50
CHAPTER 5. THE ONE-PERIOD BINOMIAL MODEL
that is
8
0
1
0
1
< (1 + r) #0 0 #00 0 uS #1 0 #01 = 0
: (1 + r) 0# 0 #0 1 0 dS 0# 0 #0 1 = 0
0
1
0
1
(5.1)
Subtracting the second equation from the rst we then obtain
1
0
S #1 0 #01 (d 0 u) = 0
0
#1 = #01 into
Since S > 0 and d < u, it
0 must 0then
1 be #1 = #1 . Substituting
(5.1), we obtain (1 + r) #0 0 #0 = 0; implying #0 = #00 since 1 + r > 0:
Finally, observing that V# (0) = #0 + #1 S and that V#0 (0) = #00 + #01 S, the
equalities #1 = #01 and #0 = #00 imply that V# (0) = V#0 (0) must hold, and
hence that under our assumptions the law of one price is satis ed.
Summing up, the assumptions 1 + r > 0, S > 0 and d < u ensure both
that the binomial market is complete, and that it satis es the law of one
price. Are these assumptions enough to ensure the absence of arbitrage
as well, or do we have to impose additional restrictions? The answer will
be a¢rmative, in the sense that an additional condition is needed for noarbitrage to hold. And even intuitively, in fact, one should not be surprised:
we know from the very rst chapter of this work that arbitrage opportunities
can well be present in a market that satis es the law of one price.
To determine the conditions under which no-arbitrage holds in the binomial market, we exploit the First Fundamental Theorem of Asset Pricing,
according to which no-arbitrage holds if and only if there exist state-price
vectors. Then, if we denote by (! 1 ) ; (! 2 ) the coordinates of the stateprice vectors, by de nition both such coordinates must be strictly positive
and solve the following system:
8
>
<
>
:
(! 1 ) +
(! 1 ) uS +
(! 2 ) =
1
1+r
(! 2 ) dS = S
It is immediate to verify that, under the condition d < u, the unique solution
to this system of equations is:
8
>
>
>
<
>
>
>
:
(! 1 ) =
1 (1 + r) 0 d
1+r u0d
1 u 0 (1 + r)
(! 2 ) =
1+r u0d
(5.2)
5.2.
COMPLETENESS AND NO-ARBITRAGE
51
In order to have w(w1) > 0 and w (we) > 0, we must thus have
(l+r)-d>0
fo ones
that is
d<lt+r<u
Therefore, no-arbitrage is ensured if and only if the gross return on the
risky security is strictly greater than the risk-free gross return in scenario
Ww, and strictly lower in the state w,. If that were not the case, it would be
a simple task to set up arbitrage strategies. For example, if 1+r> u, an
arbitrage (of the first type) can be set up by short selling the risky security
and reinvesting the money obtained in the risk-free asset. The time-0 value
of such strategy is zero, while its value at time 1 is S(1+r) —uS > 0 in state
wi, S(1+r)—dS > 0 in state we. Clearly, a symmetric arbitrage strategy
can be set up ifl+r<d.
Note that the condition d< 1+r < wis necessary and sufficient for the
one-period binomial market to admit a single state-price vector and hence,
by the Second Fundamental Theorem of Asset Pricing, to be both complete
and arbitrage-free. To put it in another way, the weaker condition d < u,
which we have seen to be necessary and sufficient for market completeness,
when paired with u—(1+ 7) > 0 and (1+ r)—d > 0 ensures also the absence
of arbitrage.
We conclude this section by deriving the expression for the risk-neutral
probability in the binomial model, probability that is unique by the Second
Fundamental Theorem of Asset Pricing.
Denoting by Q (wi), Q (we) the
risk-neutral probabilities of the states w; and we, from the First and Second
Fundamental Theorems of Asset Pricing we know that Q (wi) , Q (we) they
are linked to the state-price vector coordinates in the following way:
Q (#1) = (1+ 7)
(1)
Q (w2) = (1+)
v (w2)
Substituting the expressions for w= (w1) and w (we) obtained in (5.2), after
some simple algebra we obtain
_ (l+r)-d
Q (1) =
u-—d
QW)
=
(5.3)
52
CHAPTER 5. THE ONE-PERIOD BINOMIAL MODEL
Clearly, Q (! 1 ) + Q (! 2 ) = 1 and, to con rm our previous discussion, Q (! 1 )
and Q (! 2 ) are both positive if and only if d < 1 + r < u.
In the next section we analyze the problem of pricing and hedging option
contracts in the simple one-period binomial model.
5.3
Option Pricing in the Binomial Model
In this section we deal with the no-arbitrage valuation of European call and
put options on the risky security in the one-period binomial model. Recall
that the owner of a European call option on the risky security has the right
(but not the obligation) to buy at time 1 and at a xed price E one unit of
risky security, while the owner of a European put option has the right (but
not the obligation) to sell at time 1 and at a given price E one unit of risky
security. The counterpart of the owner (i.e., the writer) of a call option has
then the obligation to sell at time 1 and price E one unit of risky security, if
the owner of the call decides to exercise his right. Symmetrically, the writer
of a put option has the obligation to buy at time 1 and price E one unit
of risky security, if the owner of the put decides to exercise his right. The
price E xed in the contract is commonly called the strike price.
5.3.1
Call Option Pricing
We use the terminology introduced in Chapter 4 and name initial market
the one-period binomial market. We then consider an extended market
obtained by introducing a European call option on the risky security. The
time-1 contingent claim X associated with such option has the following
coordinates in the two possible states:
and
X (! 1 ) = max (uS 0 E; 0)
(5.4)
X (! 2 ) = max (dS 0 E; 0)
(5.5)
Since under our assumptions the binomial market model is complete, the
contingent claim just described is attainable. Furthermore, since the matrix
A is in a square matrix with non zero determinant, there exists one and
only one initial market strategy that replicates the maturity payo¤ of the
European call option. The strategy #c ; whose coordinates we denote by
#c0 ; #c1 , is obtained by solving the following linear system:
!
!
"
#
max (uS 0 E; 0)
#c0
1 + r uS
=
(5.6)
1 + r dS
#c1
max (dS 0 E; 0)
5.3. OPTION PRICING IN THE BINOMIAL MODEL
whose unique solution is
8
u max (dS 0 E; 0) 0 d max (uS 0 E; 0)
>
>
#c =
>
< 0
(1 + r) (u 0 d)
>
>
max (uS 0 E; 0) 0 max (dS 0 E; 0)
>
: #c1 =
S (u 0 d)
53
(5.7)
Now the inequality uS 0 E > dS 0 E holds since d < u; and the operator
max preserves weak inequalities, so that max (uS 0 E; 0) max (dS 0 E; 0),
which implies that #c1 0 under our assumptions. To verify the sign of #c0
we consider three possible cases. The rst case is dS 0 E < uS 0 E 0;
so that max (uS 0 E; 0) = max (dS 0 E; 0) = 0, and hence #c0 = 0 in this
case. The second case is when dS 0 E 0 < uS 0 E; so that the numerator
of #c0 is 0d(uS 0 E), so that #c0 < 0 in this second case. The third and
last case is when 0 < dS 0 E < uS 0 E, i.e. the numerator of #c0 becomes
E
u (dS 0 E) 0 d(uS 0 E) = 0(u 0 d)E; and hence #c0 = 0
< 0 holds.
1+r
Summing up, we can conclude that under our assumptions for any strike
price E we have #c1 0 and #c0 0:
The quantities #c0 and #c1 have an important the nancial interpretation.
Recall that #c , the hedging strategy for our European Call option, is the
strategy followed by the seller of the option (also called short position or
writer ) who wants to be covered against the liabilities that may have to be
faced at time 1. In fact, the short position will incur at time 1 in a liability
equal to 0 max (uS 0 E; 0) in state ! 1 and 0 max (dS 0 E; 0) in state ! 2 .
let’s go back then to the three cases examined above, i.e. dS 0E < uS 0E 0, dS 0 E 0 < uS 0 E and 0 < dS 0 E < uS 0 E. The rst case is when
the strike price E is so high that the option will never be exercised at time
1; and hence no liabilities will be incurred by the writer. In this case indeed,
from (5.7) it follows immediately that #c0 = #c1 = 0; i.e. that the short
position does not need any hedge. The third case 0 < dS 0 E < uS 0 E
is the other extreme case, in which the strike price is so low that the call
option will be exercised for sure at maturity. As we have seen above, in this
E
while #c1 = 1. This means that the short position sets up
case #c0 = 0
1+r
a hedge by buying one unit of risky security for every option sold and by
borrowing at the risk-free rate an amount equal to the discounted value of
the strike price. The value of such strategy at time 1 is indeed uS 0 E in
state ! 1 and dS 0 E in state ! 2 , thus leading to a perfect hedge. The most
interesting case is however the case dS 0E 0 < uS 0E; i.e. the option will
be exercised in state ! 1 and will not be exercised in state ! 2 . What are the
54
CHAPTER 5. THE ONE-PERIOD BINOMIAL MODEL
properties of the hedging strategy in this case? We have #c0 =
0d (uS 0 E)
(1 + r) (u 0 d)
uS 0 E
E
. It is easy to verify that 0
< #c0 0, while
S (u 0 d)
1+r
0 < #c1 < 1. This means that, for hedging purposes, the short position will
borrow at the risk-free rate a positive amount, although strictly lower than
the discounted value of the strike price, and will buy a number of units of
the underlying asset, but in a less than one-to-one proportion with respect
to the number of options sold.
We are now ready to employ no-arbitrage arguments to determine the
call option price in the one-period binomial model. In the sequel, we will
denote such price by c. Since the call option can be replicated, we apply
restriction (4.1) from the previous chapter to conclude that the introduction
of the call in the initial market does not introduce arbitrage opportunities
if and only if
and #c1 =
c = V#c (0) = #c0 + #c1 S
Substituting the expressions for #c0 and #c1 obtained in (5.7) into this equation, after some algebraic manipulations we get
(1 + r) 0 d
1
u 0 (1 + r)
max (uS 0 E; 0) +
max (dS 0 E; 0)
c=
(1 + r)
u0d
u0d
(5.8)
The no-arbitrage price of the European call option is then the discounted
value of a weighted average of the payo¤s of the option at maturity in the
two scenarios.
When commenting Proposition 19 in the previous chapter, we remarked
that the no-arbitrage valuation of an attainable claim can be carried out
in two equivalent ways. The rst relies on the computation of the perfect
hedging strategy, which has been described in details above in the case of
the call option in the binomial model. This approach provides a rich and
detailed answer to both the pricing and hedging problem, but involves a
certain computational burden. In case we are just interested in the noarbitrage price of the option, but not necessarily in the hedging strategy,
a second approach is preferable. According to this second approach, based
on equation (4.2) in Proposition 19, we can say that the extended binomial
model is arbitrage-free if and only if the call price c is equal to the discounted
value of the expected pay-o¤ at maturity, where the expectation is computed
under a risk-neutral probability measure of the initial market. In our case,
5.3. OPTION PRICING IN THE BINOMIAL MODEL
55
this means that we have to impose
c=
1
fQ (! 1 ) max (uS 0 E; 0) + Q (! 2 ) max (dS 0 E; 0)g
1+r
(5.9)
where Q (! 1 ) ; Q (! 2 ) are the (unique) risk-neutral probabilities of the two
scenarios in the binomial model. By substituting into (5.9) the expressions
of Q (! 1 ) ; Q (! 2 ) derived in (5.3) we immediately obtain expression (5.8),
thus showing how quick (although less informative...), this approach is.
5.3.2
Put Option Pricing
We now consider introducing in the binomial model a European put option
on the risky security. In other words, we consider an extended market where
three securities are available for trade: the risk-free asset, the risky security
and the put option. In the case of a European put option, the associated
time-1 contingent claim X has the following two coordinates in the two
possible states:
X (! 1 ) = max (E 0 uS; 0)
(5.10)
and
X (! 2 ) = max (E 0 dS; 0)
(5.11)
We denote by #p the (unique) strategy replicating the pay-o¤ at maturity
of the put. We let #p0 ; #p1 denote the coordinates of #p ; coordinates obtained
solving the following system:
"
!
!
#
#p0
max (E 0 uS; 0)
1 + r uS
=
1 + r dS
max (E 0 dS; 0)
#p1
The unique solution of this system is
8
u max (E 0 dS; 0) 0 d max (E 0 uS; 0)
>
>
#p =
>
< 0
(1 + r) (u 0 d)
>
>
max (E 0 uS; 0) 0 max (E 0 dS; 0)
>
: #p1 =
S (u 0 d)
(5.12)
By employing arguments similar to those used in the case of the call option,
it is easy to verify that in this case we have #p0 0 and #p1 0: In particular,
in the extreme case when 0 E 0 dS > E 0 uS, the put will expire without
being exercised, and hence the short position will need no hedge. Indeed, it
is straightforward to check that from (5.12) one gets in this case #p0 = #p1 = 0.
56
CHAPTER 5.
THE ONE-PERIOD BINOMIAL MODEL
The other extreme case is when EF — dS >
FE — uS > 0, and hence the put
will be exercised for sure. Expression (5.12) leads in this case to 0} = tor
and #{ = —1: the replicating strategy requires an investment in the riskfree asset equal to the discounted value of the exercise price, and the short
sale of one unit of risky security. Finally, in the more interesting case when
E-—dS >0>E-—-vuS, i.e. when the put is exercised in the state we and let
expire unexercised in state w1, expression (5.12) yields 0§ aa
=
and 3 = —{B ~ 45)
S(u—d) -
*
It is easy to verify that in this case 0 < 0} < ——
0
44+r
while 0 > 3 > —1. Hence, to set up a hedge, the short position will invest in
the risk-free asset an amount of money that is again positive, but lower than
the discounted value of the exercise price, and will short-sell a number of
units of the underlying this time proportional, but not equal, to the number
of options sold.
We now denote by p the no-arbitrage price of the European put option.
From restriction (4.1) in Proposition 19 of the previous chapter, we know
that the introduction of a put option in the binomial market will not generate
any arbitrage opportunity if and only if
p = Vg
(0) = 0h + 07S
Substituting the expressions for Jf and Vf obtained in (5.12) into this equation, after simple algebraic manipulations we get
__}!
Par
(+r)
[Gtr)-d
ud
max (E —_ uS,0)+
u-—(1+r)
Tod
max
(E
~ 48,0]
(5.13)
Similarly to what
seen for to call options,
we remark
that p can be
obtained directly by applying equation (4.2) in Proposition 19. Indeed, the
binomial model extended with a put option is arbitrage-free if and only if the
price p is equal to the discounted value of the expected payoff of the put at
maturity, where the expectation is taken under the risk-neutral probability
of the initial market. In our case, this amounts to imposing that
1
P=Ta7 {Q (w1) max (EZ — uS,0) + Q (we) max(£—dS,0)}
— (5.14)
where Q(w1),Q(w2) are the risk-neutral probabilities of the two scenarios of the binomial model. Substituting into (5.14) the expressions for
Q (w1) , Q (we) derived in (5.3) we immediately obtain (5.13). We note once
5.3. OPTION PRICING IN THE BINOMIAL MODEL
57
again that this approach provides the no-arbitrage price p of the option at
a very low computational cost, but gives no information about the hedging
strategy.
5.3.3
Put-call Parity
We conclude this chapter with an additional application of the no-arbitrage
principles studied in Chapter 4. The application is the famous put-call parity
relation. We take as initial market the binomial market and a call option,
and then extend it by introducing a put option available for trade besides
the risk-free asset, the risky asset and the call option. The initial market
is clearly complete and hence there exists a strategy, which we call #3 , that
replicates the put option by investing in the risk-free and risky assets and
in the call option. Denoting by #30 ; #31 ; #32 the coordinates of such strategy,
they must solve the following system
0 3 1
"
!
# #0
max (E 0 uS; 0)
1 + r uS max (uS 0 E; 0) B 3 C
B #1 C =
(5.15)
A
1 + r dS max (dS 0 E; 0) @
max (E 0 dS; 0)
#32
The matrix of the system has rank equal to two and, since there are three
unknowns, the system admits in nite solutions. We can then arbitrarily x
the value of one of the three variables and solve univocally the system for the
remaining two unknowns. Let’s set #32 = 1, so that system (5.15) reduces to
"
!
!
#
max (E 0 uS; 0) 0 max (uS 0 E; 0)
#30
1 + r uS
=
1 + r dS
#31
max (E 0 dS; 0) 0 max (dS 0 E; 0)
=
E 0 uS
!
E 0 dS
E
, #3 = 01. One
1+r 1
strategy replicating the pay-o¤ of the put at maturity, is then the following
one:
8
E
>
>
#30 =
>
>
1+r
<
(5.16)
#31 = 01
>
>
>
>
: 3
#2 = 1
Clearly, the unique solutiuon of this last system is #30 =
58
CHAPTER 5. THE ONE-PERIOD BINOMIAL MODEL
In other words, the pay-o¤ of the put can be replicated by a strategy requiring to invest in the risk-free asset an amount equal to the discounted
value of the strike price, to short sell one unit of risky security, and to buy
a European call option on the risky security.
If we assume that the initial market is arbitrage-free, we know from
restriction (4.1) in Proposition 19 that no arbitrage opportunities are introduced by adding the put option if and only if
p = #30 + #31 S + #32 c
Substituting into this equation the solutions for #30 ; #31 ; #32 obtained in (5.16),
we have
K
p = c 0 S1 (0) +
(5.17)
1+r
which is the famous Put-Call Parity relation.
Part II
Multi-period Financial
Markets in Discrete Time
Chapter 6
Stochastic Processes in
Discrete Time
6.1
Information Structures
In the de nition of discrete-time stochastic processes, we take as given three
fundamental objects. The rst one is the set of dates T = f0; 1; 2; :::T 0 1; T g,
whose generic element will be denoted by t. The second is the set
=
f! 1 ; :::; ! K g, the set of states of the world (scenarios) at the nal date T . The
third one is a strictly positive probability de ned on the possible scenarios
at the nal date T , probability that will be denoted by P, with P (! k ) > 0
for k = 1; :::; K.
The scenario that will be revealed true at time T is in general unknown
at every date t = 0; 1; :::; T 0 1: In particular, we want to describe the situation in which at time 0 an investor only knows that the scenario that will
be realized at time T is one among ! 1 ; :::; ! K , while at times t = 1; :::; T 0 1
the investor gathers more and more information enabling to consider as impossible some states of the world that were among the possible ones at time
0. We further assume that the investor has perfect memory, in the sense
that no information acquired by each time t is lost, so that the information owned at each time t includes all information available at all dates s
preceeding t.
The concept of information structure is what is needed to describe this
learning process undergone by investors. In order to provide a formal de nition, however, we need two more concepts:
1. de nition of a partition of
;
61
62
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
2. comparison between partitions:
De nition 21 We call partition of
sets Aj of such that
ner and coarser partitions.
a collection A = fA1 ; :::; Al g of sub-
Ai \ Aj = ;
[li=1 Ai =
= f! 1 ; ! 2 ; :::; ! 7 g. Then
Example 22 Set
8
9
>
>
<
=
A = f! 1 g; f! 2 ; ! 6 g; f! 4 ; ! 5 ; ! 7 g; f! 3 g
{z
} | {z }>
>
:| {z } | {z } |
;
A1
is a partition of
A2
A3
A4
: On the opposite,
8
>
<
9
>
=
0
A = f! 1 g; f! 1 ; ! 2 ; ! 6 g; f! 4 ; ! 5 ; ! 7 g; f! 3 g
{z
} |
{z
} | {z }>
>
:| {z0 } |
;
0
0
0
A1
A2
A3
A4
is not a partition of , since A01 \ A02 = f! 1 g 6= ;; and hence the (A0j )’s are
not all disjoints. Moreover,
9
>
=
00
A = f! 1 g; f! 2 ; ! 6 g; f! 4 ; ! 5 g; f! 3 g
>
;
:| {z00 } | {z00 } | {z00 } | {z00 }>
8
>
<
A1
A2
is not a partition of since [4i=1 Ai =
is a strict subset of .
A3
A4
n f! 7 g, i.e. the union of all (A00j )’s
De nition 23 Comparison between partitions: ner and coarser
partitions. Given two partitions A and A0 , we say that A0 is ner than
A if every element of A0 is contained in an element of A: Formally, given
A0 = fA01 ; :::; A0m g and A = fA1 ; :::; Al g, we say that A0 is ner than A if
for all A0j 2 A0 there exists Aj 2 A such that A0j Aj .
Example 24 Given
= f! 1 ; ! 2 ; :::; ! 7 g ; let us consider the following par-
6.1. INFORMATION STRUCTURES
63
titions:
A = ff! 1 g ; f! 2 ; ! 6 g ; f! 4 ; ! 5 ; ! 7 g ; f! 3 gg
A0 = ff! 1 g ; f! 2 g ; f! 6 g ; f! 4 g ; f! 5 ; ! 7 g ; f! 3 gg
00
A = ff! 1 ; ! 2 ; ! 6 g ; f! 4; ! 5 ; ! 7 g ; f! 3 gg
000
A = ff! 1 g ; f! 2 g ; f! 6 g ; f! 4 ; ! 5 ; ! 7 ; ! 3 gg
00
According to De nition 23, we have that A0 is ner than A; A is coarser
00
000
than A (put another way, A is ner than A ), while A is neither ner nor
000
coarser than A; i.e. A and A cannot be compared by “ neness”.
De nition 25 Information Structure. Given the set of dates T =
f0; 1; 2; :::T 0 1; T g and the set of states = f! 1 ; :::; ! K g, we call information structure on T and a family P = fPt gTt=0 of partitions of satisfying
the following three properties:
1. P0 = f g ;
2. Pt+1 is ner than Pt for all t = 0; 1; :::; T 0 1;
3. PT = ff! 1 g ; f! 2 g ; :::; f! K gg
Intuitively, an information structure can be described as follows. At time 0,
the information is simply made of a list of possible scenarios at time T . At
the intermediate dates t = 1; 2; :::; T 0 1, the ‡ow of information gathered
enables to focus on a subset of possible scenarios at time T , considering
those falling out of such subset as impossible. Finally, at time T we get to
know the scenario actually revealed as true.
Example 26 Let T = f0; 1; 2g,
= f! 1 ; ! 2 ; :::; ! 7 g and
P0 = f g
P1 = ff! 1 ; ! 2 ; ! 3 g ; f! 4 ; ! 5 g ; f! 6 ; ! 7 gg
P2 = ff! 1 g ; f! 2 g ; f! 3 g ; f! 4 g ; f! 5 g ; f! 6 g ; f! 7 gg
The collection P = fP0 ; P1 ; P2 g clearly satis es properties 1. and 3. in
De nition 25, and satis es property 2. as well, since according to De nition
64
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
23, P1 is ner than P0 ; and hence it is an information structure on T =
f0; 1; 2g, = f! 1 ; ! 2 ; :::; ! 7 g.
Example 27 Let T = f0; 1; 2; 3g,
= f! 1 ; ! 2 ; :::; ! 7 g and
P0 = f g
P1 = ff! 1 ; ! 2 ; ! 3 g ; f! 4 ; ! 5 g ; f! 6 ; ! 7 gg
P2 = ff! 1 ; ! 2 ; ! 3 g ; f! 4 ; ! 5 ; ! 6 ; ! 7 gg
P2 = ff! 1 g ; f! 2 g ; f! 3 g ; f! 4 g ; f! 5 g ; f! 6 g ; f! 7 gg
The collection P = fP0 ; P1 ; P2 ; P3 g is not an information structure on T =
f0; 1; 2; 3g, = f! 1 ; ! 2 ; :::; ! 7 g because P2 is not ner than P1 ; and hence
property 2. in De nition 25 is not satis ed for all t. In t = 2; in particular,
one can recognize a “memory loss”, since P2 is coarser than P1 :
6.1.1
“Event-Tree”Representation
It is commonly useful to employ an “event-tree” representation of a given
information structure. Indeed, by adopting this approach in a formal and
rigorous way, the de nition of information structure could be based on
graph theory. Here, however, we limit ourselves to providing a simple example to hint at the powerful interpretative strength of such representation. To this end, we take as given the sets T = f0; 1; 2; 3g of dates
and = f! 1 ; ! 2 ; :::; ! 7 g of possible nal states, and consider the collection
P = fP0 ; P1 ; P2 ; P3 g of partitions of de ned in the following way:
P0 = f g
P1 = ff! 1 ; ! 2 ; ! 3 g ; f! 4 ; ! 5 g ; f! 6 ; ! 7 gg
P2 = ff! 1 ; ! 2 g ; f! 3 g ; f! 4 ; ! 5 g ; f! 6 g ; f! 7 gg
P2 = ff! 1 g ; f! 2 g ; f! 3 g ; f! 4 g ; f! 5 g ; f! 6 g ; f! 7 gg
The collection P clearly satis es the de nition of information structure on
T , . The “event-tree” representation of P has the following graphical form:
6.1. INFORMATION STRUCTURES
t=0
t=1
f! 1 ; ! 2 ; ! 3 g
f g
%
0!
&
65
t=2
%
&
t=3
f! 1 g
f! 1 ; ! 2 g
%
&
f! 3 g
0!
f! 3 g
f! 2 g
f! 4 ; ! 5 g
0!
f! 4 ; ! 5 g
%
&
f! 4 g
f! 6 ; ! 7 g
%
&
f! 6 g
0!
f! 6 g
f! 7 g
0!
f! 7 g
f! 5 g
In such graphical description, the root of the tree represents the information
status at time 0, which only resides in knowing that at most one state among
! 1 ; ! 2 ; :::; ! 7 will be revealed as true at the nal date t = 3. At time t = 1,
the information obtained will enable either to exclude that at time t = 3
scenarios ! 4 ; ! 5 ; ! 6 ; ! 7 will happen (“higher” branch at time 1), either to
exclude that scenarios ! 1 ; ! 2 ; ! 3 ; ! 6 ; ! 7 will happen (“intermediate” branch)
or to exclude that scenarios ! 1 ; ! 2 ; ! 3 ; ! 4 ; ! 5 (“lower” branch) will happen.
At time t = 2, assuming that at time t = 1 the states ! 4 ; ! 5 ; ! 6 ; ! 7 have
been excluded, the information is of two (obviously incompatible) types: it
enables either to exclude scenario ! 3 , besides scenarios ! 4 ; ! 5 ; ! 6 ; ! 7 , thus
restricting the possible scenarios to ! 1 ; ! 2 , or to exclude both scenarios ! 1
and ! 2 ;, so that the only possible scenario left at time t = 3 is ! 3 .
We do not expand on the description of all other possible paths on the
branches of the event-tree representation, since we are sure that the example
just given is more than enough to provide a clear intuition. We conclude
by noting that the concept of information structure describes a learning
process regarding the information on the state in which the world will be at
the nal date t = 3, information that once gathered cannot be lost, so that
the learning process we are considering is “with memory”.
6.1.2
Information Structure Generated by Market Data
In several nancial applications, it is useful to build up an information structure based on observable quantities, in particular on security prices. In order
to describe the intuition upon which such procedure is based, we employ the
66
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
following simple example. We consider a market is which the evolution of a
security price S(t) at dates t = 0; 1; 2 can be described as follows:
t=0
10
t=1
t=2
11
%
&
9
%
&
%
&
14
10
10
7
Intuitively, the price today is 10, the price tomorrow can be 11 or 9, while
the price the day after tomorrow will be either 14 or 10, if the price tomorrow
is 11, either 10 or 7 if the price tomorrow is 9.
What is the information structure describing the evolution of the security
price? In order to construct it, we rst have to describe a set of possible
scenarios at the terminal date t = 2. To this end, let us identify the possible
scenarios at time t = 2 with the possible historical evolution of prices from
time t = 0 to time t = 2. The scenarios taking into account the whole
history of the security are the following:
! 1 = fS (0) = 10; S (1) = 11; S (2) = 14g
! 2 = fS (0) = 10; S (1) = 11; S (2) = 10g
! 3 = fS (0) = 10; S (1) = 9; S (2) = 10g
! 4 = fS (0) = 10; S (1) = 9; S (2) = 7g
In this case, the set of possible states at time t = 2 is the set = f! 1 ; ! 2 ; ! 3 ; ! 4 g
whose elements are the four scenarios just written above. We now need to
construct a partition P1 describing the information at time t = 1, date at
which the prices at dates t = 0 and t = 1 are available. Formally, at time
t = 1 we will have one of the following two alternative information sets:
fS (0) = 10; S (1) = 11; S (2) 2 f10; 14gg = fS (0) = 10; S (1) = 11; S (2) = 14g [
[ fS (0) = 10; S (1) = 11; S (2) = 10g
= f! 1 ; ! 2 g
6.2. ADAPTED STOCHASTIC PROCESSES
67
or
fS (0) = 10; S (1) = 9; S (2) 2 f7; 10gg = fS (0) = 10; S (1) = 9; S (2) = 7g [
[ fS (0) = 10; S (1) = 9; S (2) = 10g
= f! 3 ; ! 4 g
from which we see that the partition we are actually looking for is P1 =
ff! 1 ; ! 2 g ; f! 3 ; ! 4 gg :
At time t = 0, only the initial price, S(0) = 10, is known, and hence
the four scenarios described by the four alternative price evolutions are all
possible, so that P0 = f g = ff! 1 ; ! 2 ; ! 3 ; ! 4 gg.
Summing up, once the scenarios ! 1 ; ! 2 ; ! 3 ; ! 4 have been de ned on the
basis of the four di¤erent possible price evolutions, the collection of partitions P0 = f g, P1 = ff! 1 ; ! 2 g ; f! 3 ; ! 4 gg and P0 = f g = ff! 1 g ; f! 2 g ; f! 3 g ; f! 4 gg
is the information structure generated by the evolution of the security price
at times 0; 1; 2.
6.2
Adapted Stochastic Processes
In the next chapters, we will focus on the evolution over time of asset prices
and dynamic strategies. Intuitively, it should be possible to infer price levels
and strategy compositions at each time from the information available at
that time. The sense in which they can be inferred by investors is formalized by the concept of stochastic process adapted to a given information
structure. We provide the de nition below, right after the de nition of
measurability of a discrete random variable. In order to fully understand
these de nitions, we focus on the generic partition Pt of the information
structure P and denote by st the number of cells (or nodes of the event-tree
at time t) that make up Pt . We observe that, according to the de nition of
information structure, we have in any case s0 = 1 and sT = K, where K is
the number of possible scenarios at the nal date T . We then denote by fht ;
for h = 1; :::; st , the generic cell or element of the partition Pt .
De nition 28 Measurable Random Variable. The function X(t): 0!
<, is measurable with respect to Pt if and only if, for any xed cell fht of the
partition Pt we have X(t)(! 0 ) = X(t)(! 00 ) for any ! 0 , ! 00 2 fht . Therefore,
the random variable X(t) is measurable with respect to Pt if and only if X(t)
is a map from Pt to <.
68
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
De nition 29 Adapted Stochastic Process. Given a set T = f0; 1; 2; :::T 0 1; T g
of dates, a set = f! 1 ; :::; ! K g of states of the world at the nal date T , and
an information structure P = fPt gTt=0 based on T and , we call stochastic
process adapted to the information structure P a sequence X = fX (t)gTt=0
of random variables such that X (t) is measurable with respect to Pt for all
t. t = 0; 1; :::; T .
The variable random variable X(t) associated with X is then completely described by the values X(t)(fht ) taken by X(t) on the cells fht ;
P
for h = 1; :::; st . We nally denote by L = Tt=0 st the number of cells in a
given information structure, so that the whole stochastic process X can be
described by a vector in < L with coordinates X(t)(fht ) for h = 1; :::; st and
t = 0; 1; :::; T .
We can additionally verify the de nition and the notation just introduced
by going back to example 26 considered in the rst section of this chapter. By
employing the “event-tree” representation and the notation just introduced,
that example takes now the following form:
t=0
t=1
%
0!
&
f12 = f! 1 g
f22 = f! 2 g
f32 = f! 3 g
f21 = f! 4 ; ! 5 g
%
&
f42 = f! 4 g
f52 = f! 5 g
f31 = f! 6 ; ! 7 g
%
&
f62 = f! 6 g
f72 = f! 7 g
f11
f10 = f g
%
0!
&
t=2
= f! 1 ; ! 2 ; ! 3 g
In this case we have s0 = 1; s1 = 3 and s2 = 7, so that L = s0 + s1 +
s2 = 11. Hence, given any stochastic process X = fX (0) ; X(1); X(2)g
adapted to such information structure, the variable X(0) obviously takes the
unique value X(0)(f10 ), the variable X(1) can take the three values X(1)(f11 );
X(1)(f21 ) and X(1)(f31 ), while the variable X(2) can take the seven values
X(2)(! 1 ); :::; X(2)(! 7 ). The overall process X can then be identi ed with
the following vector in <11 :
(X(0)(f10 ); X(1)(f11 ); X(1)(f21 ); X(1)(f31 ); X(2)(! 1 ); :::; X(2)(! 7 ))T
We conclude this section by introducing an alternative, but equivalent,
representation of adapted stochastic process, which will be used in the third
6.3.
CONDITIONAL EXPECTATIONS AND MARTINGALES
69
part of this work to motivate and understand the passage from discrete-time
to continuous-time models.
Remark 30 Given the set of dates T, the set of terminal scenarios Q =
{w1,...,wK} and an information structure P on T and Q, a stochastic process X adapted to P is a map X:T xQ—§ such that for any fi of the
information structure P we have X(t)(w’) = X(t)(w”) for any w', w" € ff.
To conclude we observe that, for a fixed terminal scenario w, the sequence
X(0)(w), X(1)(w), ...., X(T —1)(w), X(T)(w) is called trajectory (or sample
path, realization) of the adapted stochastic process X, and represents the
“path” leading to scenario w at the final date T. Such path is clearly unique,
because of the properties satisfied by an information structure, and on the
basis of the above remark we can affirm that a stochastic process can be
interpreted as the set of “paths” that lead to the final possible scenarios.
We will exploit such interpretation to model financial markets in continuous
time.
6.3
Conditional Expectations
and Martingales
The conditional expected value of a random variable represents the best
prediction we can provide conditionally on the information available. Hence,
the conditional expectation acts on a random variable, but depends on the
information and on the probability with respect to which the expectation is
computed. If we describe the information available through the information
structure P = {Pr} 9; we can state the following definition:
Definition 31 Conditional Expectation.
Let P = {P:}t_, be an in-
formation structure and Q a probability on Q. Let X (s) be measurable with
respect to Ps, i.e. X(s): Ps + R.the function X : TxQ —R, whose
values are denoted by X(t)(w), describes a stochastic process adapted to the
information structure P if and only if, for any fied cell fi of the information
structure, and given any w', w” € fi, we have X(t)(w’) = X(t)(w").
The conditional expected value of X (s), given Py and under the probability Q, ts
1. E®[X(s)| Pil (fh) = Leet X (8) (£7)
2. E®[X(s)| Pe] (fp) = X (s) (fa) ft = 8;
Qlfi]
Q [fi] ift<s;
70
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
0 1
0 1
3. EQ [ X(s)j Pt ] fht = X (s) fht if t > s:
The random variable
E1Q [ X(s)j Pt ] associating with each fht 2 Pt the
0
value EQ [ X(s)j Pt ] fht is the conditional expected value of X (s)
given Pt and under the probability Q.
9T
8
For t ranging in f0; :::; T g, the collection EQ [ X(s)j Pt ] t=0 of conditional expected values with respect to P = fPt gTt=0 de nes the process
conditional expectation of X(s) with respect to P, which is adapted to
fPt gTt=0 :
In order to understand the de nition given, we comment it case by case.
1. Given the probability Q on
de ned thorugh Q
8 (!9k ) = qk for k =
1; :::; K; the probabilities of the events ffls g and fht are given by
Q [fls ] =
and hence the ratio
X
!2fls
X
2 3
Q [!] and Q fht =
Q [!]
!2fht
P
Q [fls ]
!2f s Q [!]
2 t3 = P l
Q fh
!2f t Q [!]
h
represents the probability that fls is revealed true at time s, conditional
s
on fht revealing true at time t, i.e. the conditional
2 s t 3 probability of fl
t
given fh at time t, which is denoted by Q fl j fh :
0 1
Hence, EQ [ X(s)j Pt ] fht de ned in 1: is an average of the possible
realizations of X(s) given the event fht at time t. The average is
obtained by weighing the realizations of X(s) in the nodes that at time
s come from fht with the conditional
probabilities of such nodes. As
0 1
a consequence, EQ [ X(s)j Pt ] fht yields a prediction (expected value)
of X (s) conditionally on the event fht at time t.
2. In case t = s, the formula in 1: immediately reduces to
EQ [ X(s)j Pt ] (fhs ) = X (s) (fhs )
Q [fhs ]
2 3
Q fhs
giving the formula in 2: This is consistent with our intuition: X (s) is
known at time s and hence the best possible predictor of the variable
is nothing else than the variable X(s) itself, which we already know.
6.3. CONDITIONAL EXPECTATIONS AND MARTINGALES
71
3. In case t > s, the variable X (s), already known at time s, remains
known at time t. Hence, taken fht 2 Pt , we have
0 1
0 1
EQ [ X(s)j Pt ] fht = X (s) fht
0 1
This means that the best possible predictor of X (s) given fht is X (s) fht
itself. The variable X (s) depends on the events fls that revealed true
s
X(s)
at time s: But in the nodes fht following
0 t 1 fl at time st, the variable
takes always the same value X (s) fh = X (s) (fl ), with fls fht . It
thus follows that
0 1
0 1
EQ [ X(s)j Pt ] fht = X (s) fht = X (s) (fls ) with fls fht
Remark 32 Under our assumptions, for t = 0 the P0 -conditional expected
value simply reduces to the expected value of X(s):
X
0 1
X (s) (fls ) Q [fls ] = EQ [X (s)]
EQ [ X(s)j P0 ] f10 =
fls Ps
Proof. The proof is straightforward. We use
1 in the de nition
9
2 3 of
8 point
conditional expectation: since P0 = f g = f10 , we have that Q f10 =
Q [ ] = 1: Furthermore, from the unique node f10 at time t = 0 follow all
nodes fls 2 Ps whose summation at point 1: is a summation over all nodes
of Ps : Hence, the conditional expected value of X (s) given P0 is simply the
average of X (s) under Q:4
We collect in the following proposition some important properties of the
conditional expectation. The rst one clari es the meaning of iterated conditional expected value, the second states that multiplicative variables that
are measurable with respect to Pt can be taken out of the Pt -conditional
expectation. The result is intuitive: a variable that is known in t; being
measurable with respect to Pt ; is the best possible predictor of itself given
the information available at time t. The last property extends the property
of linearity satis ed by (unconditional) expected values, which are additive
and for which constants can be ‘taken out’ of expectations, to conditional
expected values. In this case, however, we can exploit the second property and say that linearity can be expressed for conditional expectations in
terms of linear combinations with coe¢cients that are Pt -measurable random variables. In the sequel, we will use repeatedly the results stated in the
following proposition, since they allow to simplify computations in several
applications.
72
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
Proposition 33 Properties of Conditional Expectation
1. The law of iterated expectations.
Given 0 t1 t2 s, let us consider the random variables EQ [ X(s)j Pt2 ] :
Pt2 ! < and compute the conditional expectation given Pt1 . Then, the
following holds:
2
3
EQ EQ [ X (s)j Pt2 ] Pt1 = EQ [ X (s)j Pt1 ] :
2. Pt 0measurable random variables can be "taken out" of Pt -conditional
expectations.
Let s > t and let X (s) be measurable with respect to Ps and h (t) be
measurable with respect to Pt : We than have that
EQ [ h (t) X (s)j Pt ] = h (t) EQ [ X (s)j Pt ]
that is
0 1
0 1
0 1
EQ [ h (t) X (s)j Pt ] fht = h (t) fht EQ [ X (s)j Pt ] fht
for h = 1; :::; st :
3. Linearity of Conditional Expectation.
If X (s) ; Y (s) : Ps ! < and h (t) ; k (t) : Pt ! <, then
EQ [ h (t) X (s) + k (t) Y (s)j Pt ] = h (t) EQ [ X (s)j Pt ]+k (t) EQ [ Y (s)j Pt ]
The law of iterated expectations is a time-consistency property of conditional expectations. Indeed, it says that if we try to obtain a prediction of the variable X(s) by computing two predictions at times t1 <
t2 ; i.e. rst by computing node by node the prediction in t2 ; given by
EQ [ X(s)j
2 Pt2 ] ; and then by3taking the best prediction in t1 , i.e. by computing EQ EQ [ X (s)j Pt2 ] Pt1 in t1 ; what we obtain is just the best prediction
of X(s) in t1 ; i.e. EQ [ X (s)j Pt1 ] : Hence, the intermediate conditioning at
time t2 has no e¤ect on the conditional expectation given the information
available at time t1 ; which is poorer than that carried by Pt2 .
We can similarly justify properties 2. and 3. by exploiting the useful
interpretation of conditional expectations in terms of predictors.
A simple example will help in understanding how the results described
can be actually employ.
6.3. CONDITIONAL EXPECTATIONS AND MARTINGALES
73
Example 34 Let us go back to the previous example and suppose that a put
option is written on the security S with strike price K = 12: Its payo¤ at
time T = 2 is
8
0 in ! 1
>
>
<
2 in ! 2
X (2) =
2 in ! 3
>
>
:
5 in ! 4
Now, set Q [! 1 ] = 0:1458; Q [! 2 ] = 0:3942; Q [! 3 ] = 0:3312; Q [! 4 ] = 0:1288:
What is the conditional expected value of X (2) given P1 under Q?
The information structure at time t = 1 is given by P1 =
8 1Solution.
9
1
f1 ; f2 , with f11 = f! 1 ; ! 2 g and f21 = f! 3 ; ! 4 g : The probabilities of such
two events are given by
2 3
Q f11 = Q [! 1 ] + Q [! 2 ] = 0:54
and
2 3
Q f21 = Q [! 3 ] + Q [! 2 ] = 0:46
Hence
0 1
Q [! 1 ]
Q [! 2 ]
EQ [ X (2)j P1 ] f11 = X (2) (! 1 ) 2 1 3 + X (2) (! 2 ) 2 1 3 =
Q f1
Q f1
0:1458
0:3942
= 01
+21
= 1:46
0:54
0:54
0 1
Q [! 3 ]
Q [! 4 ]
EQ [ X (2)j P1 ] f21 = X (2) (! 3 ) 2 1 3 + X (2) (! 4 ) 2 1 3 =
Q f2
Q f2
0:3312
0:1288
= 21
+51
= 2:84
0:46
0:46
The random variable EQ [ X (2)j P1 ] : P1 ! < can then take the following
values:
2 3
1:46 on the event f11 ; with probability Q2 f131 = 0:54
EQ [ X (2)j P1 ] =
2:84 on the event f21 with probability Q f21 = 0:46
Its expected value can then be written as:
2
3
EQ EQ [ X (2)j P1 ] = 1:46 1 0:54 + 2:84 1 0:46 = 2:0948
74
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
which is equal to
EQ [X (2)] =
4
X
X (2) (! k ) Q [! k ] = 2:0948
k=1
as guaranteed by the law of iterated expectations, since
3
2
EQ EQ [ X (2)j P1 ] = EQ [X (2)] 4
If the random variable X(s) is independent from the information Pt for
t < s with respect to Q; then its prediction based on Pt ; EQ [ X(s)j Pt ] ;
coincides with the poorest possible prediction based on P0 : EQ [ X(s)j Pt ] =
EQ [X(s)] = EQ [ X(s)j P0 ].
Formally, a random variable X(s) with values x1 ; :::; xM on
is indipendent from the information Pt with respect to the probability Q if
2 3
3
2
Q f! : X(s)(!) = xm g \ fht = Q [f! : X(s)(!) = xm g] 1 Q fht
for all fht 2 Pt and for all m = 1; :::; M:
Such de nition requires that f! : X(s)(!) = xm g\fht 6= ? for all fht 2 Pt
and for all m = 1; :::; M:
In fact, 2if f! : X(s)(!) = xm g \ f3ht = ? for some xm and fht ; then
2 03 =
Q [?] = Q f! : X(s)(!) = xm g \ fht = Q [f! : X(s)(!) = xm g] 1 Q fht 6=
0; because
2 both
3 factors in the last product are non-zero: Q [f! : X(s)(!) = xm g] 6=
0 and Q fht 6= 0:
Hence, if X(s) is indipendent from Pt ; in any cell fht 2 Pt the random
variable X(s) takes all the possible values x1 ; :::; xM at least once.
Therefore, it can be proved that
EQ [ X(s)j Pt ] (fht ) = EQ [X(s)]
for all fht 2 Pt : In fact
EQ [ X(s)j Pt ] (fht ) =
X
m=1;:::;M
2
3
xm 1 Q f! : X(s)(!) = xm gj fht
3
2
Q f! : X(s)(!) = xm g \ fht
2 3
=
xm 1
Q fht
m=1;:::;M
2 3
X
Q [f! : X(s)(!) = xm g] 1 Q fht
2 t3
=
xm 1
Q
fh
m=1;:::;M
X
=
xm 1 Q [f! : X(s)(!) = xm g] = EQ [X(s)]
X
m=1;:::;M
6.3. CONDITIONAL EXPECTATIONS AND MARTINGALES
75
8
9
As an example, let
= f! 1 ; ! 2 ; ! 3 ; ! 4 g and P1 = f11 ; f21 with f11 =
f! 1 ; ! 2 g and ; f21 = f! 3 ; ! 4 g : Suppose that Q is uniform on ; that is
Q [! k ] =
Let X be a random variable on
1
4
k = 1; :::; 4:
de ned as follows:
X(! 1 ) = X(! 4 ) = a
X(! 2 ) = X(! 3 ) = b
where a and b are real constants. Then X is indipendent from P1 with
respect to Q and it follows that
EQ [ Xj P1 ] (fh1 ) = EQ [X]
for all fh1 2 P1 :
Having introduced the concept of conditional expectation, we are now
ready to de ne a class of processes, the so called martingales, which were
hinted at when discussing the features of the discounted gain process under
the risk-neutral probability in one-period nancial markets.
De nition 35 (Martingale) Let M = fM (t)gTt=0 be a process adapted to
P = fPt gTt=0 : The process M is a martingale under the probability Q and
with respect to the information structure P if, for any pair of dates t1 < t2
with t1 ; t2 2 f0; :::; T g, we have
EQ [ M (t2 )j Pt1 ] = M (t1 )
(6.1)
Property (6:1) ensures that, given any two dates t1 < t2 ; the best predictor of the process M at time t2 ; i.e. of M (t2 ) conditional on the information available at time t1 ; is given by M (t1 ) ; the time-t1 value of the process
M itself. From expression (6:1) we deduce also that a martingale is constant
on average, as we see in the next proposition.
Proposition 36 Let M = fM (t)gTt=0 be a martingale with respect to P =
fPt gTt=0 and under Q: Then, for all t 2 f1; :::; T g we have that
EQ [M (t)] = M (0)
76
CHAPTER 6. STOCHASTIC PROCESSES IN DISCRETE TIME
Proof. The proof is simple: we just need to consider (6:1) at times t1 = 0
and t2 = t and recall that the conditional expected value given P0 coincides
with the unconditional expected value:
M (0) = M (t1 ) = EQ [ M (t2 )j Pt1 ] = EQ [ M (t)j P0 ] = EQ [M (t)] :4
In the discrete-time setup, martingales can be equivalently de ned recursively. Such characterization will be useful in further applications, we
nd it convenient to present it:
Proposition 37 The process M = fM (t)gTt=0 is a martingale with respect
to P = fPt gTt=0 under Q if and only if for all t 2 f0; :::; T 0 1g we have
EQ [ M (t + 1)j Pt ] = M (t)
(6.2)
Proof. If M is a martingale, by employing (6:1) with t1 = t and t2 = t+1
we see immediately that (6:2) is satis ed. To show the converse, we exploit
the law of iterated expectations. Let us consider two generic dates t1 < t2
belonging to the set f0; :::; T g and de ne n = t2 0 t1 : From (6:2), by the an
iterated-expectations argument we obtain
M (t1 ) = EQ [ M (t1 + 1)j Pt1 ] =
= EQ [EQ [ M (t1 + 2)j Pt1 +1 ]jPt1 ] =
|
{z
}
M (t1 +1)
= EQ [ M (t1 + 2)j Pt1 ] =
= EQ [EQ [ M (t1 + 3)j Pt1 +2 ]jPt1 ] =
|
{z
}
M (t1 +2)
Q
= E [ M (t1 + 3)j Pt1 ] =
111
= EQ [ M (t1 + n)j Pt1 ] = EQ [ M (t2 )j Pt1 ] 4
Chapter 7
Multi-period Markets: Basic
Notions
7.1
Price Processes
In this second part we consider a multi-period discrete time nancial market
in which N + 1 securities, indexed by j = 0; 1; :::; N , are traded at dates
t = 0; 1; :::; T . At the nal date T , the market can be in one and only one
of K possible scenarios, ! 1 ; :::; ! K . The unfolding of experience and the
resolution of uncertainty about the nal scenario are described through an
information structure P = fPt gTt=0 given on the set of dates T = f0; 1; :::; T g
and the set = f! 1 ; :::; ! K g of possible scenarios at the nal date T . The
evolution of the prices of each of the N +1 securities is formally described by
a stochastic process adapted to the information structure P. In particular,
we will refer to the risky securities through the index j = 1; 2; :::; N , and
will denote by Sj = fSj (t)gTt=0 the adapted price process of the generic
j-th risky security. For xed date t and node fht (with h = 1; :::; st ) of the
information structure at time t, we will then denote by Sj (t) (fht ) the price
of the j-th risky security in that node.
The security indexed by j = 0 deserves a separate treatment. We denote by B = fB (t)gTt=0 the adapted stochastic process describing its price
evolution over time, with the normalization B (0) = 1; so as to have unitary
price at time 0. For t = 0; 1; :::; T 0 1, we de ne the quantity r(t) as follows
r (t) =
B (t + 1) 0 B (t)
B (t)
(7.1)
The quantity r (t) represents the net return obtained by buying one unit
77
78
CHAPTER 7. MULTI-PERIOD MARKETS: BASIC NOTIONS
of the security at time t and reselling it at time t + 1. The fundamental
01
is that it is a
assumption we make on the sequence of returns fr (t)gTt=0
stochastic process adapted to the information structure P of our multiperiod nancial market. According to the de nition of adapted stochastic
process, that means r(t) is known at time t. What does it mean from the
nancial point of view? That the security indexed by j = 0 is risk-free over
the unit time interval [t; t + 1], given the information available at time t,
i.e. given the node of the information structure at which we are at time t.
One has to pay particular attention to the interpretation of what we have
just said: indeed, we are not assuming that the security j = 0 is globally
risk-free, i.e. that its return on each unitary subtree is known at time 0;
we are instead requiring the security to be locally risk-free, in the sense
that the return between t and t + 1 becomes known at time t once the
information available at that time is known. For this reason, we call r(t)
computed according to expression (7.1) locally risk-free interest rate of our
multi-period market.
We provide an additional remark on the key concept just described.
The assumption that r(t) is adapted, hence known in t once the node of the
information structure becomes available, is equivalent (see 7.1) to saying that
B(t + 1) is already known at time t, once the node is known. Obviously,
the same requirement is not imposed on the risky securities, the value of
Sj (t + 1) being unknown at time t even having knowledge of the node of
the information structure. This is actually what distinguishes the security
indexed by j = 0 from the others and makes it locally risk-free.
We now observe that (7.1) can be rewritten as follows:
B (t + 1) = B (t) (1 + r (t)) ;
t = 0; 1; :::; T 0 1
Solving the equation (a so called di¤erence equation) “backwards”, it is easy
to see that
B (t + 1) = B (t) (1 + r (t))
= B(t 0 1)(1 + r(t 0 1))(1 + r(t))
B(t 0 2)(1 + r(t 0 2))(1 + r(t 0 1))(1 + r(t))
= ::::
= ::::
= B(0)
t
Y
=0
(1 + r( ))
7.2. DYNAMIC INVESTMENT STRATEGIES
79
and hence, recalling the normalization B(0) = 1; we get
B (t + 1) =
t
Y
(1 + r( ))
(7.2)
=0
Q
where
stands for repeated multiplication.
Expression (7.2) yields an additional interesting interpretation of the
price of the locally risk-free security in terms of accumulation factors. Indeed, we see that B(t + 1) represents the money obtained by investing 1 unit
of money at time 0 and reinvesting (rolling-over) the proceeds at each date.
The reciprocal of B(t + 1) then represents a discount factor, providing an
interpretation that will be used frequently in what follows. We conclude this
section by noting that in the particular case when the security j = 0 is not
only locally, but also globally, risk-free, then we are back to the well-known
quantities of the “classical ” nancial mathematics. Indeed, the security is
globally risk-free when r(t) is independent of dates and nodes, i.e. when
r(t)(fht ) = r for all h = 1; :::; st , t = 0; 1; :::; T 0 1. Under such simplifying
assumption, expression (7.2) reduces to B(t + 1) = (1 + r)t+1 , from which
1
we obtain
= (1 + r)0(t+1) , i.e. the “classical”accumulation and
B(t + 1)
discount factors of the compounded interest regime.
7.2
Dynamic Investment Strategies
We are now ready to describe the way investors develop their investment
decisions in the multi-period market described in the previous section. Investors will exploit the information structure available and will hence be able
to recalibrate their portfolio allocations at each node fht reached at time t,
based on the information then available. In order to formalize this, we let
the positions taken on the N + 1 securities be described by N + 1 adapted
01
stochastic processes denoted by #j = f#j (t)gTt=0
, with j = 0; 1; :::; N . It is
now important to clarify the interpretation that the variables #j (t) are to
be given. Speci cally, #j (t) represents the total position on the j-th security
at time t, after any portfolio rebalancing has been performed. As a result,
the value of #j (t) provides the total number of units of the j-th security
held (if positive) or sold short (if negative) at time t. Since our market is by
assumption open only at integer valued dates, such position will be held over
the whole time interval [t; t + 1[, where the right endpoint of the interval is
open because the position will be fully liquidated at time t + 1, followed
by a new allocation #j (t + 1). The message is that #j (t) represents the
80
CHAPTER
7.
MULTI-PERIOD MARKETS: BASIC NOTIONS
overall position on the j-th security and not, as might be misunderstood,
the increment/decrement in the position previously held. Consistently with
what just said, at the final date t = T the investor liquidates its portfolio
and has no future allocation to decide anymore, so that 3; = {¥; (t) an
is
consistently defined up to time t = T — 1.
From now onwards we will denote by J = (Uo, 01,...,Un) the set of
strategies on the N + 1 securities, and say that 0 is a dynamic investment
strategy for our discrete time multi-period market.
With each dynamic
strategy 0, we associate three adapted stochastic processes characterizing
the following fundamental financial objects: the value process, the cashflow
process and the discounted gain process.
Definition 38
Value
Process, multi-period
case.
For any given dy-
namic strategy 0, we call value process of 0, denoted by Vs = {Vo (t)}e-9)
the adapted stochastic process defined by
)
Vo
,
(t) =
penne
Jo (T-1)B(T)
+L, 9;,T-1 S(T),
t=0,1,..,T7—-1
+t=T
(7.3)
The definition extends to the multi-period setup the one provided in the
one-period case, as can be easily verified by setting T = 1 in expression
(7.3) and comparing with Definition (2) in the first chapter. Intuitively,
for t = 0,1,...,T —1 the value Vy (t) of the strategy 0 represents the cost
(possibly negative, since short selling is allowed) to be incurred at time t
to buy Vo (t) , 01 (t) ,..., 8y (t) units of the N +1 securities available on the
market.
At the final date T, the value process Vy (T’) provides instead the
final liquidation value of the strategy 1.
Definition 39 Cashflow Process. Given a dynamic strategy 3, we call
cashflow process of 0, denoted by Cy = {Cg (t)} 0, the adapted stochastic
process defined by
Cog (t) =
—Vz (0)
t=0
00 (t-1) B(t) + DL, 9} (t—1)S;(t)-Vo(t)
t=1,..,T-1
Vo (T)
t=T
7.2. DYNAMIC INVESTMENT STRATEGIES
81
The cash‡ow process C# is somewhat new with respect to the oneperiod framework, and is needed to describe the cash‡ows involved by the
strategy # when the portfolio allocation may be subject to intermediate
rebalancing. At time t = 0, the cash‡ow is simply an out‡ow equal to
the cost V# (0) incurred to set up the initial positions on the N + 1 securities (since short selling is allowed, 0V# (0) may be positive, thus leading
to an overall in‡ow if the proceeds from short sales are greater than the
buying costs incurred). At the intermediate dates t = 0; 1; :::; T 0 1 the
cash‡ow is generated by two components. The rst one, #0 (t 0 1) B (t) +
P
N
j=1 #j (t 0 1) Sj (t) ; represents the liquidation value of the overall investment strategy #0 (t 0 1) ; #1 (t 0 1) ; :::; #N (t 0 1) set up at time t 0 1. From
such liquidation value we then subtract the cost (which, once again, may be
negative, hence an in‡ow) V# (t) of the strategy #0 (t) ; #1 (t) ; :::; #N (t) set
up at time t and to be liquidated in t + 1. At time t = T , nally, the investor
cannot modify the positions anymore and simply receives the liquidation
value V# (T ) at time T of the portfolio #0 (T 0 1) ; #1 (T 0 1) ; :::; #N (T 0 1)
arranged at time T 0 1.
In order to get further nancial insights into the intuition upon which
the concept of cash‡ow is based, we nd it useful to substitute into (7.4),
for t = 1; ::; T 0 1, the expression V# (t) de ned by (7.3) in De nition 38.
In this way, we can rewrite the cash‡ow generated by a strategy # at the
intermediate dates t = 1; :::; T 0 1 as follows:
C# (t) = #0 (t 0 1) B (t) +
0 #0 (t) B (t) +
PN
j=1 #j
PN
j=1 #j
(t 0 1) Sj (t)
(t) Sj (t)
Grouping terms , we immediately get the following equivalent expression for
C# (t):
C# (t) = [#0 (t 0 1) 0 #0 (t)] B(t) +
N
X
[#j (t 0 1) 0 #j (t)] Sj (t)
(7.5)
j=1
The above expression allows us to interpret the cash‡ow as if they were generated by the sum of rebalancing costs (or in‡ows) incurred at time t on the
di¤erent securities. The message from (7.5) is that, under the fundamental
assumption of absence of transaction costs (e.g. bid-ask spreads), we obtain
the very same level of cash‡ow by adopting two possible rebalancing procedures. The rst is the one described before, based on the total liquidation at
time t of the positions held since t 0 1 and on setting up the new positions to
82
CHAPTER 7. MULTI-PERIOD MARKETS: BASIC NOTIONS
be held until t + 1. According to such approach, for instance, if the position
established at time t 0 1 on the security j = 1 is worth 10 units, and only 8
units are to be carried until t + 1, at time t the whole 10 units are sold and
then 8 are bought. Expression (7.5) suggests instead an approach based on
the incremental (decremental) rebalancing of the position. In the example
just given, by following such approach at time t, only 2 units of security j
would be sold (instead of 10) followed by the purchase of 8 units. The key
point of (7.5) is that, with no transaction costs, the to approaches lead to
the same result.
A particular class of dynamic strategies is of major interest in the valuation of European-type derivative securities. These are the so called selfnancing strategies, de ned below.
De nition 40 Self- nancing Dynamic Strategies. A dynamic strategy
# is said to be self- nancing if
C# (t) = 0,
t = 1; :::; T 0 1
(7.6)
By exploiting expression (7.4), we can immediately see that condition
(7.6) can be rewritten as follows:
#0 (t 0 1) B (t) +
N
X
#j (t 0 1) Sj (t) = V# (t) ,
t = 0; 1; :::; T 0 1
j=1
A self- nancing strategy is hence characterized
by the fact that the timeP
t liquidation value #0 (t 0 1) B (t) + N
#
(t
0
1) Sj (t) of the portfolio
j=1 j
#0 (t 0 1) ; #1 (t 0 1) ; :::; #N (t 0 1) set up at time t 0 1 is exactly enough to
cover for the cost V# (t) of the positions #0 (t) ; #1 (t) ; :::; #N (t) to be established at time t and to be held until time t+1. Put another way, for any such
strategy the liquidation value from an allocation exactly nances the cost of
the following allocation: hence, the strategy nances itself, in the sense that
no positive cash‡ow is generated for consumption, nor additional capital
in‡ows are required in excess of the liquidation value at any intermediate
dates t = 1; :::; T 0 1.
We conclude this section by de ning the third and last process associated
with an investment strategy, the discounted gain process.
De nition 41 Discounted Gain Process, multi-period case.
For
any given strategy #, we call discounted gain process of #, denoted by G# =
7.2.
DYNAMIC INVESTMENT STRATEGIES
{Gz (t) an
83
the adapted stochastic process defined by
VoBUH(t) +>
,
9 (t)t) =
»
yr=0
CsBln)’
(7)
t=0,1,..,T—1
;
(7.7)
Ca(t)
B(r)’
t=T
Similarly to what said for the value process, the discounted gain process
is the direct extension to the multi-period case of the one-period framework.
Indeed, recalling from expression (7.4) that for t = 0 we have Cy(0) =
—Vog (0), by setting t = 0 in (7.7) and recalling the normalization B (0) = 1,
we get
— Vo(0) | Co(0) _
Gg (0) = BO + Bio) > Vg (0) — Vo (0) =_ 0
(7.8)
Recalling further that, for t = T, from (7.4) we have Cy(T) = Vg (T),
and that the price of the locally risk-free security satisfies B(1) = B(0)(1+
r(0)) =1+7(0), we see that if we set T = 1 in (7.7), i.e. we are back to the
one-period case, the following holds
Co(1)
Gp (1) = B(1)
_ Co(0) Vel) _
Vo (0)
B(0) 1+7r(0)
(7.9)
By comparing expressions (7.8) and (7.9) with (1.5) and (1.6) in Definition
3 of the first chapter, we see immediately that the multi-period definition of
discounted gain encompasses the one-period case as well.
More generally, the discounted gain process associated with a dynamic
strategy represents the accumulated value of the cashflows generated up to
a given date plus the liquidation value of the strategy at that date. In order
to sum quantities that are financially consistent, we need to discount them
back to time 0. Indeed, recalling from the previous section that B(t) =
(1 +r(7T)), we see that any quantity appearing in (7.7) is discounted
at time 0 through the discount factor
BY)
[Eo(1 + r(7))
obtained by
combining the one-period locally risk-free rates available on the multi-period
market. In the next chapter, we will show that in a multi-period market
the absence of arbitrage is characterized by the existence of a probability
measure under which the discounted gain process is a martingale. Before
getting to that, however, in the following section we extend to the multiperiod setting the definition of violations of the law of one price and of
arbitrage opportunities of the first and second type.
84
7.3
CHAPTER 7. MULTI-PERIOD MARKETS: BASIC NOTIONS
Multi-period Arbitrage Opportunities
We now describe the situations that must be prevented in the multi-period
model developed so far. Symmetrically to what has been done in the oneperiod case, we group such situations in three categories, namely violations
of the law of one price, arbitrage opportunities of the rst type and arbitrage
opportunities of the second type.
De nition 42 Violations of the Law of One Price. A multi-period
nancial market gives rise to violations of the law of one price if there exist
two dynamic investment 0strategies
# and
#0 generating the same cash‡ows
1
1
0
in the future, i.e. C# (t) fht = C#0 (t) fht for all h = 1; :::; st , t = 1; :::; T ,
but having di¤erent initial values, i.e. V# (0) 6= V#0 (0).
In words, in a multi-period nancial market there are violations of the
law of one price when one can set up two dynamic strategies that, despite
generating the very same cash‡ows in any future node of the information
structure, have di¤erent cost at time 0. The key consequence of violations
of the law of one price is that we cannot de ne univocally security prices
anymore, and hence the model becomes useless from the point of view of
applications.
De nition 43 Arbitrage Opportunities of the 1st type. A multi-period
nancial market gives rise to arbitrage opportunities of the rst type if there
exists a dynamic strategy # having the following properties:
V# (0) 0
0 1
C# (t) fht 0,
for all h = 1; :::; st , t = 1; :::; T
C# ( ) (fl ) > 0,
for some 2 f1; :::; T g ; l 2 f1; :::; s g
Thus, a multi-period market leads to arbitrages of the rst type when we
can set up a strategy satisfying three conditions. The rst is that the initial
cost of such strategy is non positive, i.e. no out‡ows are required at time
0. The second condition is that the cash‡ow generated by such strategy is
nonnegative in every future node of the information structure. The third
condition states that there exists at least one node of the information structure in which the cash‡ow generated by the dynamic strategy is strictly
positive.
Similarly to the one-period case, every investor can exploit an arbitrage oppoprtunity of the rst type: its initial cost is null or even negative,
7.3. MULTI-PERIOD ARBITRAGE OPPORTUNITIES
85
hence no initial wealth is needed to enter the strategy. Furthermore, every
non-satiated investor (in the sense that more wealth is preferred to less)
would exploit the arbitrage opportunity spotted in the market. With what
consequences? That the demand for securities leading to arbitrages would
become in nite (if the dynamic strategy # leads to an arbitrage of the rst
type, so does the strategy #, for any > 0::::), so that the market being
modeled would never reach an equilibrium!
De nition 44 Arbitrage Opportunities of the 2nd type. A multiperiod nancial market gives rise to arbitrage opportunities of the second
type if there exists a strategy # such that
V# (0) < 0
0 1
C# (t) fht 0,
for all h = 1; :::; st , t = 1; :::; T
In this case, a dynamic strategy leads to an arbitrage of the second type
if it has a strictly negative initial cost (i.e. it yields a strictly positive in‡ow
at time 0), and generates a nonnegative cash‡ow in every future node of
the information structure. From the economic/ nancial point of view, such
opportunities are to be prevented for the same reasons described with regard
to arbitrage opportunities of the rst type.
In the multi-period case, the relation between the law of one price and noarbitrage is the same as in the one-period case. In particular, the following
result holds:
Proposition 45 In a multi-period nancial market, the absence of arbitrage
opportunities of the second type implies the law of one price.
We do not provide the proof here, since it is pretty much the same as that
provided in the one-period case. We invite the reader to convince his/herself
of that by proving the result as an exercise.
We are now ready to give the key de nition of this introductory chapter
on multi-period nancial markets.
De nition 46 No Arbitrage, multi-period case. A multi-period nancial market satis es the no-arbitrage condition if no arbitrage opportunities
of the rst nor of the second type are available. Furthermore, because of the
Proposition above, every multi-period arbitrage-free nancial market satis es
the law of one price.
86
CHAPTER 7. MULTI-PERIOD MARKETS: BASIC NOTIONS
In the next chapter we will provide characterizations of the absence of
arbitrage, stating a suitable extension of The First Fundamental Theorem
of Asset Pricing already discussed in the one-period case. As we will see in
the sequel, such result is based on the extension to the multi-period case
of the concept of state-price vector or, equivalently, of equivalent martingale measure, the multi-period counterpart of the risk-neutral probabilities
employed in the one-period case.
Chapter
8
No-arbitrage in Multi-period
Markets: the
Characterization
8.1
State Price Vectors in the Multi-period
Case
Exactly as in the one-period case, the concept of state-price vector is a
cornerstone in the theory of finance and of its applications to multi-period
markets.
The definition of state-price vectors in a multi-period setting is
given below. In order to fully understand it, we recall that L denotes the
overall number of nodes of the tree representing the information structure,
sz denotes the number of nodes at time ¢ and ff the generic node at time t.
At the initial date t = 0, in particular, we have so = 1 and f? = {Q}, while
at the final date t = T we have sp = K (the overall number of possible
scenarios), and ff = {wp} with h = 1,..., K.
Definition 47
State price
Vector,
multi-period
case.
For any given
vector w in RY, denoted by
b= (v (fo) 0 (Ft)
(f8) eB (FR) ee BAT) 1B (FR)
we say that w is a state-price vector for the discrete time multi-period market
if
>> 0, ie. w is strictly positive, p (f?) = 1, i.e. the first coordinate is
unitary, and if the following conditions are satisfied
a
1+r(t) (ff)
a et)
ey
= 1.58,
bff)”
87
re
t=1.,T-1
1)
88
NO-ARBITRAGE IN MULTI-PERIOD MARKETS
and
w
S(t) (fi) =
dS
(fi*1)
3 (ff
j
8; (t¢+1) (ff*),
wy
VN
(8.2)
t=1,...,T
h
fics
= 1,
h=1,...,8;
It is now useful to compare the definition just given with that provided
in the one-period case. We first note that in t = 0, by exploiting the fact
that ~ (f?) = 1 and setting r (0) (fP) =r, expressions (8.1) and (8.2) reduce
to
1
1l-+r
= 30 (f});
l=1
$; (0) = Ly (AS),
8.2
Equivalent
galeN
Martingale Measures
In the one-period model, the second building block of the setup was repres-
ented by the so called risk-neutral probabilities. The generalization of such
concept to the multi-period
case are the
equivalent martingale
Martingale
Measure
measures,
which we now formally define.
Definition 48 Equivalent
(EMM).
For a dis-
crete time multi-period financial market, an equivalent martingale measure
is a strictly positive probability, Q, on the possible scenarios at the final
date T, i.e. Q(wz) > 0 for all k = 1,...,K, such that for every security
j=1,...,N the following holds:
5; (t+ 1)
5; (t) = E°
1+r(t)
Pal,
t=0,1,..,T-1
(8.3)
Denoting now by 5; (t) the present value of the time t price of security
4, that is
5;(t) =
5; (t)
Bit)
fort = 0,...,T
we can restate the above definition in terms of the martingale property of
the process 5;. Observe in fact that from (8.3) we have
5; (t) = Fe
Bit)
1
sa
8; (t+ 1)
1+r(t) Pe ,
t=0,1,..,T7-1.
8.2.
EQUIVALENT MARTINGALE MEASURES
89
since B(t) is by assumption measurable with respect to the information P;
available at time t. Recalling now that B(t+1) = (1+7(¢)) B(t) we have
S,(t) = B? [Se +1)|%],
t=0,1,..,T7-1
which shows that 8; is a martingale under Q.
We now recall that by assumption both S;(¢) and r(t) are adapted to
the information ?; available at time t (i.e. they are P;—measurable), and
1
thus itr®
can be “taken out” of, while BO
can be “taken into”, the
conditional expectation in (8.3) (in this regard, see Proposition 33 in Chapter
6). From this, it follows that expression (8.3) can be equivalently rewritten
E2
Sj (t+ 1) — $5) P, =r(t),
S53)
t=0,1,..,T-1
XXX
Remark
49
Given a probability,
Q,
strictly positive on the possible scen-
arios at the final date T, the following statements are equivalent:
1. Q is an equivalent martingale measure;
2. the value and cashflow processes of every dynamic strategy 9 satisfy
the following condition:
Re
Vo (( +1) + Cg (+1)
1+ r (t)
Vo (t)=
:
Co (T)
ES le
1)
Pra] ;
P|
t=0,1,...,T—2
t=T-1
(8.4)
3. the discounted gain process of every dynamic strategy 0 is a martingale
under Q, that is:
Go (t) = E? (Go (t+1|P],
t=0,1,...,7-1,
(8.5)
Proof. 1.2. Let Q be an equivalent martingale measure. Then, for any
fixed t = 0,1,...,7’— 2, because of (7.4) in the Definition 39 of cashflow of a
dynamcial strategy given in the previous chapter, we have
Vo (t+1)+Co(t+1)
=Ve(t+1)4+00() B(t+1)+
+
Ly 9; (t) $5 (t+.1) — Vo (t+ 1)
= 99 (t) BE +1) +N, 8; 8 (+1)
90
NO-ARBITRAGE IN MULTI-PERIOD MARKETS
By multiplying both sides of such equations by ESO}
and recalling from
Chapter 6 that B(t+1) = B(t)[1+r(t)], we get
Vo (t+ 1) + Cy (t+1)
icr®
(t) Sj (E+1)
= 9p (t) B(t) + LV1+r(t)
By taking conditional expectations on both sides of the last equation, and
exploiting the properties of conditional expectation stated in Proposition 33
in Chapter 6, we then obtain
N
Pi] = 90) B+j=l 95 (927 |
| py
Since by assumption Q is an equivalent martingale measure,
according to
Vg (t +1)
+ Cy (t+ 1)
£9|
°
Ta7®)
(8.6)
(8.3), we can replace E® ae
|»
with S;(t) in (8.6), so that the ex-
pression becomes
Vo (t + 1)
+ Cg (t +1)
=|
.
ior@)
Pi = 0 (t) B +d
(t) S;(t)
thus yielding (8.4), for t = 0,1,...,T — 2, since by definition Vp (t) B(t) +
yr, Bi (t) 5; (t) = Vo (t). The proof of (8.4) for t = T — 1 goes along the
same lines.
2.1. In this case we just have to consider an investment strategy 0
that involves buying the security 5; at time 0 and holding it in portfolio up
to time T (buy-and-hold strategy), i.e.
aya
JL
t=)
wo) ={ 0, ixj
for i =0,1,...,N and for t= 0,1,...,7’—1. By exploiting the definitions of
value process and cashflow process, it is immediate to verify that for such a
strategy we have
Vo(t)
Cs (t)
=
S$) (6);
— S55 (0),
0,
t=0
t=1,.,T—-1.
8.2.
EQUIVALENT MARTINGALE MEASURES
91
As a consequence, since (8.4) now holds by assumption, we have
1) + Cy
5;()= Vol) = £9| Voot(t+
(t+1)
_
S; (+1)
a] = 20 |S
Pi
’
and so Q satisfies the definition of equivalent martingale measure.
2.3. We have to verify that Gy is a martingale under Q.
holds, we deduce that for t = 0,1,...,T — 2
t+1)
E2 (Gs (t+)|Pi]
=
Vo
(t + 1)
EP?
E@
t#Oo
~
B(t+1)
Vo (t +1)
B(t+1)
Since 2.
a(8)| 5 | _
B(s)
Cg(s)
Cg (t+1)
» BCs) + BETD
mi
But, since yo oe) is adapted to P;, by the properties of conditional
expectations we get the following
E®? [Gog (t+ 1)| Pi]
B(t+1)
-~ 5a [ery
scotty
B(t)(1+r(t))
P| +>
CySE(s) _=
s=0
Pl
+
2
Vo (t +1)
aa | eee
—_
w
~
* Il
_
)
=
- ViBe +EC, -a,
where the last two equalities are due to 2. and to the definition of discounted
gain respectively. A similar reasoning yields the result for the case of t =
T-1.
3.=>2. In this case we have to verify that, for t = 0,1,...,T — 2, we have
Vo (t) = E@ |
Gg(t)
tr)l+r(t
+ Co (trl | P|.
=
But we know by 3. that
E°(Gst+)iPl=
Vo(t+1),
Cy ()|
°| Beep
Vo(t+1)
E®
ets
+d Be”
Co (t+1)
+ BE+I)
*. Co (s)
, +h
Bs
92
NO-ARBITRAGE IN MULTI-PERIOD MARKETS
Since
Go (t)=
Cog ot
Z ie
the result follows by equating the two previous expressions and multiplying
both sides by B (t).@
In point 2. of the previous Proposition: under Q the one-period conditional expected return of any strategy is equal to that of the locally risk-free
security B: indeed, for t = 0,1,...,T — 2 we have
po [WeDo N—VOlp) 59,
while for t = T — 1 we have
XXX
In point 3.
of the previous Remark:
since the discounted gain process
is a martingale under Q, and martingales are processes constant on average
(recall Proposition 36 in Chapter 6), having G(0) = 0 for all 0 implies that
for every investment strategy we have
E° ([G9(t)} =0,
t=1,...,T
This means that for any date t at which the strategy @ is liquidated, the
discounted gain obtained up to ¢ is null on average, such average being of
course computed under an equivalent martingale measure.
We are now ready for the next section, in which the First Fundamental
Theorem of Asset Pricing is stated and proved for the multi-period case.
8.3.
The First Fundamental
cing in the Multi-period
Theorem
of Asset Pri-
Case
We now state and prove the First Fundamental Theorem of Asset Pricing.
Theorem
50
First Fundamental
Theorem
of Asset Pricing,
multi-
period case. In a discrete time multi-period financial market the following
statements
are equivalent:
1. the market is arbitrage-free;
THE FIRST FUNDAMENTAL
THEOREM OF ASSET PRICING
93
2. there exists a state-price vector w;
3. there exists a risk-neutral probability measure Q.
Proof. In the multi-period version of the theorem, we adopt the following approach: we first show the equivalence between 1. and 3., then the
equivalence between 2. and 3.
1.3. We first show that the multi-period absence of arbitrage implies
that the market is arbitrage-free on every one-period subtree. In order
to show this, we consider the subtree with “root” ff € P;, which can be
represented as follows:
t
where the nodes fit Cc fi represent the nodes immediately reachable at
time t+ 1 from the “root” node fi. Let us suppose that in such one-period
subtree there exist arbitrage opportunities, which means that there exists
an investment strategy (0, 01,
Nn)
€ R%+1 such that V5 (é) (fi) <0
and C3 (f+ 1) ( fit) > 0 for all fir a fi , with at least a strict inequality.
We then show that in this case we are able to set up a multi-period arbitrage
opportunity.
To this end, starting from the one-period strategy (8, 81, ...,9 nv) operating on the subtree considered, we define the following strategy {v (t)}/_9:
3; (8 (#) =0;, for all j =0,1,...,N
0; (£) (fi)
= 0, for allj and for alk #hA
9; (t) = 0, for allj and for all t 4 Z,
By applying the definition of cashflow process,
it is easy to see that the
cashflow generated by ¥@ is zero on all nodes not belonging to the subtree
considered above, and strictly positive in at least one of the nodes of the
94
NO-ARBITRAGE IN MULTI-PERIOD MARKETS
subtree (including the root ft). Consequently, our #@ generates a multiperiod arbitrage. We have then shown that an arbitrage opportunity in a
subtree implies the existence of multi-period arbitrages, so that the absence
of multi-period arbitrages ensures that every one-period subtree is itself
arbitrage-free.
We now proceed in the proof by adopting the following simple information structure:
\
/
\
{wi}
{we}
{ws}
N\
{
7
{ws}
\.
{w7}
—
w
4}
{we}
The proof of the result for a more general information structure would just
require more cumbersome notation, without yielding any additional insights.
In the following, we will denote by Mo,
t=1, ie.
t=0
fe
the subtree between t = 0 and
t=1
7
\
ft
f
f3
and by M11, M1,2 and My; the sub-trees between t = 1 and t = 2, given
by:
fi
7S
a
{wi}
>
ff
{wa}
;
fs
/
>
{ws}
{we}
respectively.
As we have shown above, every one-period subtree is arbitrage-free.
,
By
the one-period First Fundamental Theorem of Asset Pricing, we can then
say that:
THE FIRST FUNDAMENTAL
THEOREM OF ASSET PRICING
e there exists a risk-neutral probability Qoi1 in Mo,
ie.
95
Qo.
is a
strictly positive probability on {f/, f2, f3} such that for all j we have
.
8 (0) = >>
1
Oto,
(fi)
fhEPi
e there exists a risk-neutral probability Qii in Mii, ie. Qi is a
strictly positive probability on {w,w2} such that for all 7 we have
(I) (ft) = So
oye
(wn)
wrest
e there exists a risk-neutral probability Qi2 in M12, ie. Qi is a
strictly positive probability on {w3,w4} such that for all 7 we have
5; . (1) (f2) _= x
55 (2) (wa)
) Qi,2 W (wa)
T+r(f
(fd)
e there exists a risk-neutral probability Qi3 in M13, ie. Qi3 is a
strictly positive probability on {ws,wg,w7} such that for all 7 we have
$; (1) (f3) = So eye
(wn)
wrefd
Starting from the probabilities Qoi1, Qi1, Qi,2 and Qi,.3 we construct
an equivalent martingale measure for our multi-period tree. To this end, let
us define the probabilities of the seven final nodes of our multi-period tree:
Q (wn) = Qo,1 (ft) Q1,1 (wa), per h = 1,2
Q (wn) = Qo,1 (f2) Qi,2 (wn), per h = 3,4
(8.7)
Q (wn) = Qo,1 (f3) Qi,3 (wa), per h = 5, 6,7
We first prove that the probability Q so defined is strictly positive probab-
ility on Q = {w,...,w7}. First, we have that
2
Q (fi) =
re) (wn) = Qo, (f2) (Q1,1 1) + Qi,1 (w2)) = Qo, (ff) ;
h=1
96
NO-ARBITRAGE IN MULTI-PERIOD MARKETS
since Q:,1 is a probability on {wi,w2}.
Analogously:
Q (fh) = Qo1 (fi)
for h = 2,3.
Hence, in general
3
3
h=1
h=1
Q (wn) = >5Q (Fa) =
Qo, (fa) = 1,
since Qo,1 is a probability on { fj, fz, fz}. Finally, our Q is a strictly positive
probability because so are Qo,1, Qi1, Qi,2, Qi3.
Furthermore, from the definition of Q we also have, for h = 1, 2:
Q [wal fil=
Q (wn)_ Qo (Fi) Qi1 (wa)
Q (ft)
Qo, (fi)
(wp) .
= Qi
Similarly for h = 3, 4,5, 6, 7.
We are now left with the proof that Q is an equivalent martingale measure for the multi-period market.
° $3 (1) | —
1+7r (0)
By definition of Q, it follows that
yr) Va)
SQ
oan
(fh)
» Oe)
— iid 01 (fa)= $3 0),
where the last equality holds since Qo,1 is a risk-neutral probability for the
one-period submarket Mo,1. Moreover, we have
55 (2)
| alm]
S; (2)
_
= Baia
Pi (ft)
1
53 (2) (wa)
», rr (0) (fay 2 len fl
_
55 (2) wn)
Xie1l+r (1) (ft)
and the same holds for fd and f+.
Qi1
(wn)
= Sn (1) (fi)
We conclude that the probability Q
defined in (8.7) is actually an equivalent martingale measure and that the
Qin’s, with h = 1, 2,3, are the conditional probabilities of Q, given fi, f3, fd.
THE FIRST FUNDAMENTAL
THEOREM OF ASSET PRICING
97
3.=>1. We now assume that there exists an equivalent martingale measure Q and show that the multi-period market is arbitrage-free. We proceed
by absurd,
and suppose that a multi-period arbitrage opportunity can be
found. In particular, let 0 be a strategy
definitions given in the previous chapter,
Indeed, since B (t) > 0, saying that J is an
that the cashflow process satisfies Cy (t) >
leading to arbitrage. From the
it follows that
eo 30 > 0.
arbitrage is equivalent to saying
0 for all t and is strictly positive
in at least one node. We then obtain
(t)
Co
E® (Gs (T)]= E®@ 57 208
t=0
Bit)
Co) (Fh)h) 7?
) ) Bog
=r da
8
because the equivalent martingale measure Q is strictly positive by definition. Moreover, according to point 3. in Remark 49, the process Gy is
a martingale under Q for any ¥, and by definition Gy (0) = 0, so that
E® [G5 (T)| = Gp» (0) = 0. Since such results contradicts (8.8), the proof is
complete.
3.2. Let Q be an equivalent martingale measure.
4 =1,...,N and t = 0,1 we have
S; (t) = E®
By definition, for all
S; (+1)
| 1+r(t) P| |
In particular, for t = 0 the following holds
S; (1
80) =F 0/2
|=
$5 (1) (fa) 5 741
a “1+r(0) Q (fa)
Pi
while for t = 1 and all f} € Py (with h = 1,2,3) we have
say(a) = £2 SE l pc)
55 (2) We)
»,
1+r(1)
op)
Ay? |
fl
nl fal
.
Let us define w = (1,0 (ft) ,v (f8) ob (78) ,¥ W1),-..,P (w7))” as follows:
Bias,
betas
Q [wel fi] =o
wp € fis h=1,2,3
1+r(l) (fi) OS
(8.9)
98
NO-ARBITRAGE IN MULTI-PERIOD MARKETS
Since Q is an equivalent martingale measure, then w satisfies the following
properties:
e w is a vector with all strictly positive components;
1
e w satisfies the condition }> flePy v (fh) = T+rO)'
e the following holds
wwe)
Q[wel fal
_
Away > dio
1
Tay
yo
1)
el] —
Lal fal
1
1+r (1) (ft)’
.
e for all 7 =1,...,.N, we have
§;(0)=
DO v(fh) $5) (FA)
fKEPi
SOUS
oR)
As a consequence, ~ is indeed a state-price vector.
2.>3. Finally, let ~ be a state-price vector. Let us then use ~ to define
a probability Q as in expression (8.9). We obtain:
h=1,2,3
Q [wel ft] = (1 +7 (2) (fA) oan
we € fi; h=1,2,3
&
Q (fa) = (1 +97 (0)) d (fA)
that is
Q (we) = Q (ff) -Q [wel fa] = (A +7(0)) (1+
7 (1) (fa) v (wr)
for all w, € fi and h = 1, 2,3.
Let us now verify that the probability
Q just defined is risk-neutral.
From ~ >> 0 it follows immediately that Q is strictly positive on the final
THE FIRST FUNDAMENTAL
states W1,...,W7.
THEOREM OF ASSET PRICING
99
Furthermore, since
Y Qed] = DOr) Se
wees}
wees}
=
h
(1+ r r( (1) i)(fh
» (wr)
Xe)
= 0+ OM aay
= 1
and
d) Qf) =(+rO) SS vf) =1,
fpEPi
fr€Pa
then
>) Qs)
=
WREQ
SO YS Q(fi
Q [erl)ff]
FREPI weefp
=
=
> Q(fi) SS Q [welff]
fAEPi
1
wRESR
Thus, Q is a strictly positive probability measure on the final states. By the
very definition of Q in (8.9), it is immediate to verify that for all j = 1,...,N
and for
t = 0,1
5; (t) =
(¢+1)
| S;1+r(t)
P,
As a result, Q is an equivalent martingale measure.
Chapter
9
Dynamically Complete
Multi-period Markets
9.1
Dynamic
Completeness:
the Notion
Similarly to what done in the one-period case, we now focus on the char-
acterization of markets allowing investors to obtain any future cashflow. In
the multi-period case, this is achieved by exploiting the possibility of rebalancing the position at the intermediate dates t = 1,2,...,7’—1, on the basis
of the information P; available at those dates.
In order to formalize this intuition, we start by defining the concept of
contingent claim in a multi-period setting. Namely, in this case we call con-
tingent claim any sequence X = {X (t)}, of random variables adapted
to the given information structure P ={P,}/_,.
Hence, X(t): B > R
represents the cashflow generated by the contingent claim X in the nodes
identifying the information available at time t. We then say that the contin-
gent claim X = {X (t)}7, is attainable in the multi-period financial market
with securities B, S},..., Sy if there exists a dynamic investment strategy 3
such that
Co(th=X(t),
t=1,...,T
In other words, a contingent claim is attainable if there exists a dynamic
strategy whose cashflow process is, in every future node of the information
structure, equal to that of the contingent claim considered.
Example
51
A European
call option with maturity T on the security Sy
101
102
COMPLETE MARKETS
with strike price E generates the following contingent claim:
0;
t = 1; :::; T 0 1
X (t) =
t=T
max [S1 (T ) 0 E; 0]
Such contingent claim is attainable if there exists a strategy # such that
0
t = 1; :::; T 0 1
C# (t) =
max [S1 (T ) 0 K; 0]
t=T
Example 52 A portfolio made of
0 the European call option of the previous example
0 a European put option on SN with maturity T 0 1 and strike price E 0
0 a forward contract on S1 with maturity 1 and delivery price F
generates the following contingent claim:
8
0 + 0 + [S1 (1) 0 F ]
>
>
<
0+0+0
X (t) =
0 + max [E 0 0 SN (T 0 1) ; 0] + 0
>
>
:
max [S1 (T ) 0 E; 0] + 0 + 0
t=1
t = 2; :::; T 0 2
t=T 01
t=T
Such contingent claim is attainable if there is a dynamic strategy # such that
8
S1 (1) 0 F
t=1
>
>
<
0
t = 2; :::; T 0 2
C# (t) =
0
max [E 0 SN (T 0 1)]
t=T 01
>
>
:
max [S1 (T ) 0 E; 0]
t=T
In this chapter, we focus on multi-period markets in which every contingent claim is attainable. We provide a formal de nition below.
De nition 53 Dynamically Complete Multi-period Financial Market. We say that a discrete time multi-period nancial market is dynamically complete if every contingent claim X = fX (t)gTt=0 is attainable.
Di¤erently from the one-period case, in the multi-period case we associate the adverb “dynamically” with the concept of completeness, meaning
that the replication of contingent claims can be achieved by exploiting the
intermediate trading dates to rebalance the portfolio positions. What conditions must be satis ed by a multi-period market to be dynamically complete?
We provide the answer in the next section.
9.2. THE CHARACTERIZATION OF DYNAMIC COMPLETENESS103
9.2
The Characterization of Dynamic Completeness
In the one-period case, we have shown that a market is complete if and only
if the number of linearly independent tradeable securities is equal to the
number of possible nal scenarios. The situation is more articulated in the
multi-period case: we will see that, in general, many less securities can be
taken into account, but every one-period submarket must satisfy the law of
one price.
Proposition 54 A discrete time multi-period nancial market is dynamically complete if every one-period submarket is complete. Conversely, if a
multi-period market is dynamically complete and every one-period submarket
satis es the law of one price, then every one-period submarket is complete.
Proof. We prove the result for the information structure represented by the
following tree:
t=0
f10
t=1
%
0!
&
t=2
f11
%
&
f! 1 g
f! 2 g
f21
%
&
f! 3 g
f! 4 g
f31
%
0!
&
f! 5 g
f! 6 g
f! 7 g
In particular, let us denote by M0;1 the following one-period subtree
t=0
f10
t=1
%
0!
&
f11
f21
;
f31
and by M1;1 ; M1;2 and M1;3 the one-period sub-trees
f11
%
&
f! 1 g
;
f! 2 g
f21
%
&
f! 3 g
;
f! 4 g
f31
%
!
&
f! 5 g
f! 6 g ;
f! 7 g
104
COMPLETE MARKETS
respectively.
Let us rst show that if each of the sub-trees above is complete, then
dynamic completeness holds, i.e. every contingent claim X = fX (t)gTt=1
is attainable. To show this, let X be a generic contingent claim in the
multi-period market, and let us set up a dynamic strategy # such that
X (t) = C# (t) for t = 1; 2.
Recall that, for t = 2, we have:
2
6
6
6
6
X (2) = 6
6
6
6
4
X (2) (! 1 )
X (2) (! 2 )
X (2) (! 3 )
X (2) (! 4 )
X (2) (! 5 )
X (2) (! 6 )
X (2) (! 7 )
3
7
7
7
7
7,
7
7
7
5
In particular, we interpret the vector
X (2) (! 1 )
X (2) (! 2 )
as the cash‡ow of a contingent claim in the one-period market M1;1 . Similarly, we interpret the vector
X (2) (! 3 )
X (2) (! 4 )
as the cash‡ow of a contingent claim in the one-period market M1;2 . Finally,
we interpret the vector
2
3
X (2) (! 5 )
4 X (2) (! 6 ) 5
X (2) (! 7 )
as the cash‡ow of a contingent claim in the one-period market M1;3 .
By assumption, we know that the one-period markets M1;1 , M1;2 and
M1;3 are complete. As a result:
from the completeness of M1;1 follows the existence of a strategy #1;1 2
RN +1 such that
X (2) (! 1 )
V#1;1 (2) =
;
X (2) (! 2 )
9.2. THE CHARACTERIZATION OF DYNAMIC COMPLETENESS105
from the completeness of M1;2 follows the existence of a strategy #1;2 2
RN +1 such that
X (2) (! 3 )
V#1;2 (2) =
;
X (2) (! 4 )
from the completeness of M1;3 follows the existence of a strategy #1;3 2
RN +1 such that
2
3
X (2) (! 5 )
V#1;3 (2) = 4 X (2) (! 6 ) 5 :
X (2) (! 7 )
Let us now construct the dynamic strategy #(0); # (1) that replicates the
contingent claim X at dates t = 2 and t = 1. As far as t = 2 is concerned,
we de ne # (1) as
0 1
# (1) f11 = #1;1
0 1
# (1) f21 = #1;2
0 1
# (1) f31 = #1;3 :
so that, by what said before, we deduce that # (1) replicates the contingent
claim X at time t = 2.
We still have to show that X can be replicated at time t = 1 as well. We
then look for a strategy # (0) such that C# (1) = X (1). Let us recall that
!
N
N
X
X
C# (1) = #0 (0) B (1) +
#n (0) Sn (1) 0 #0 (1) B (1) +
#n (1) Sn (1) ;
n=1
n=1
where the sum of the rst two terms represents the liquidation value of # (0)
at t = 1 and the sum of the other two terms is equal to V# (1), i.e. to the
cost of # (1). We then have to nd # (0) satisfying the following equation:
#0 (0) B (1) +
N
X
#n (0) Sn (1) = X (1) + #0 (1) B (1) +
n=1
N
X
#n (1) Sn (1) ,
n=1
where #0 (0), #1 (0), ..., #N (0) are the unknowns, while the quantities on
the right-hand side are known terms (X (1) is given, #0 (1), #1 (1), ..., #N (1)
have been obtained in the previous step, from t = 1 to t = 2).
The previous problem then reduces to searching the one-period market
M0;1 for the one-period strategy (#0 (0) ; #1 (0) ; :::; #N (0))T 2 RN +1 with
payo¤ at maturity (t = 1)
X (1) + #0 (1) B (1) +
N
X
n=1
#n (1) Sn (1) ;
(9.1)
106
COMPLETE MARKETS
i.e. equal to the cash‡ow of the contingent claim X at t = 1 plus the
cost incurred for setting up the strategy # (1). Since the one-period market
M0;1 is complete by assumption, it follows that the quantity in (9.1) is
attainable and consequently a strategy (#0 (0) ; #1 (0) ; :::; #N (0))T 2 RN +1
exists, whose liquidation value in t = 1 exactly replicates it. As a result: the
dynamic strategy # = f# (t)gt=0;1 , with # (0) and # (1) determined above,
replicates the cash‡ow of X = fX (t)gt=1;2 . Since the contingent claim X
has been arbitrarily chosen, we obtain the completeness of the multi-period
market.
We now prove the second part of the proposition, i.e. that if the multiperiod market is dynamically complete and if the law of one price holds, then
every one-period subtree is complete. To do this, let us rst consider the
one-period market M1;2 and, for any contingent claim X1;2 de ned on it, let
us show that if M1;2 satis es the law of one price then X1;2 can be replicated
in M1;2 . We start from a claim X1;2 de ning the following contingent claim
X in the multi-period market:
X (1) , 0
X1;2 (!) ;
X (2) (!) ,
0;
if ! 2 f21
:
otherwise
As a consequence, X = fX (t)gt=1;2 is a contingent claim in the multi-period
market. Hence, from the dynamic completeness of such market follows that
X is attainable: there exists a strategy # = f# (t)gt=0;1 such that
C# (t) = X (t) ;
i.e. such that C# (1) = 0 and
C# (2) (! h ) =
for t = 1; 2;
X1;2 (!) ;
0;
if ! 2 f21
otherwise
In particular, # (1) satis es
C# (2) = V# (2) = X (2) ;
hence we have
V# (2) (! h ) = X1;2 (! h ) ;
for h = 3; 4:
By the de nition of value process, we get
#0 (1) (f11 )B (2) (! h ) +
N
X
n=1
#n (1) (f11 )Sn (2) (! h ) = X1;2 (! h ) ;
for h = 3; 4:
9.2. THE CHARACTERIZATION OF DYNAMIC COMPLETENESS107
1T
0
so that #0 (1) (f11 ); #1 (1) (f11 ); :::; #N (1) (f11 ) 2 RN +1 is the strategy replicating the contingent claim X1;2 in M1;2 . Since the contingent claim X1;2
has been arbitrarily chosen, we deduce the completeness of the one-period
market M1;2 . One can similarly prove the completeness of M1;1 and M1;3 .
Let us note that in the three previous one-period markets it was easy to
prove completeness because the investment strategy # = f# (t)gt=0;1 was
liquidated in t = 2.
We are now left with proving the completeness of the one-period market
M0;1 . We have to show that every contingent claim X0;1 is attainable in
the one-period market M0;1 . As before, we start from X0;1 and de ne the
following contingent claim X in the multi-period market:
X (1) , X0;1
X (2) , 0:
Because of the dynamic completeness of the multi-period market, such X is
attainable. As such, there exists a dynamic strategy # = f# (t)gt=0;1 such
that
C# (t) = X (t) ; for t = 1; 2;
i.e.
C# (1) = X (1)
C# (2) = V# (2) = X (2) = 0:
We now ask ourselves whether the strategy # (0) replicates X0;1 in M0;1 .
We know that
C# (1) = X (1) ;
hence
#0 (0) B (1) +
N
X
#n (0) Sn (1) 0 V# (1) = X (1) .
(9.2)
n=1
We now show that V# (1) = 0. This certainly holds, because # (1) satis es
C# (2) = V# (2) = X (2) = 0:
But #3 (1) = 0 is a strategy with null payo¤ in t = 2. By assumption, M1;1 ,
M1;2 and M1;3 satisfy the law of one price, from which follows
V# (1) = V#3 (1) = 0.
108
COMPLETE MARKETS
Because V# (1) = 0, from (9.2) we have that for all fh1 2 P1 the following
holds
N
0 1
0 1
0 11 X
0 1
#n (0) Sn (1) fh1 0 0 = X (1) fh1 = X0;1 fh1 ;
#0 (0) B (1) fh +
n=1
i.e. #0 (0) ; #1 (0) ; :::; #N (0) replicates X0;1 in the one-period market M0;1 .
We have thus proved completeness for every one-period submarket.4
The possibility of dynamically rebalancing the strategies increases the
replication opportunities. Re-trading is thus a way of completing the market
with a lower number of securities than that required in the one-period case.
We provide an example of what just said by considering the following oneperiod market:
%
f! 1 g
0!
f! 2 g
f10
f! 3 g
0!
&
f! 4 g
t=0
t=2
In order to complete the market, we need in this case four linearly independent securities. We now allow the market to be open at t = 1 and consider
the following information structure
f10
%
&
f11 = f! 1 ; ! 2 g
f21 = f! 3 ; ! 4 g
t=0
%
&
%
&
t=1
f! 1 g
f! 2 g
f! 3 g
f! 4 g
t=2
In this case, the market can be completed with the use of only two securities.
Example 55 Let us consider the multi-period market where traded are the
risk-free asset B, yielding a constant interest r = 5%, and the risky security
S, taking the following values:
S (0) = 10
S (1) =
S (2) =
8
>
>
<
>
>
:
12 on f11 = f! 1 ; ! 2 g
8 on f21 = f! 3 ; ! 4 g
15:6 on ! 1
8:4 on ! 2
9:6 on ! 3
6:4 on ! 4
9.2. THE CHARACTERIZATION OF DYNAMIC COMPLETENESS109
If trading is allowed only at dates t = 0 and t = 2, the market is clearly of
one-period type. The matrix
2
1:1025
6 1:1025
A=6
4 1:1025
1:1025
3
15:6
8:4 7
7
9:6 5
6:4
has rank equal to 2 and the market is incomplete.
Exercise 56 If the market is also open in t = 1, investors can revise their
strategy at time t = 1. Indeed, in this case an investor can choose
# (0) 2
0 1 13 in t = 0
0 11
in t = 1
# (1) = # (1) f1 ; # (1) f2
so that for a generic contingent claim X = [X (1) ; X (2)] one has
V# (2) = X (2)
C# (1) = X (1)
()
()
V# (2) (!
0 k1)1 k = 1; :::; 4
0 k1)1 = X (2) (!
C# (1) fh = X (1) fh h = 1; 2
The system has thus six0 equations
1
0(four
1 for t = 2 and two for t = 1) with six
unknowns (# (0) ; # (1) f11 ; # (1) f21 2 R2 ). V# (2) = X (2) is equivalent
to #0 (1) B (2) + #1 (1) S (2) i.e. for f11 2 Pt
0 1
0 1
#0 (1) 0f11 1 B (2) + #1 (1) 0f11 1 S (2) (! 1 ) = X (2) (! 1 )
#0 (1) f11 B (2) + #1 (1) f11 S (2) (! 2 ) = X (2) (! 2 )
which has one and only one solution in R2 ;
0 1
0 13
0 1 2
# (1) f11 = #0 (1) f11 ; #1 (1) f21
since
B (2) S (2) (! 1 )
1:1025 15:6
=
B (2) S (2) (! 2 )
1:1025 8:4
2
0 13
is nonsingular, since det A (1) f11 = 07:938:
Analogously, V# (2) = X (2) for the other node f21 2 Pt is equivalent to
the system
0 1
0 1
#0 (1) 0f21 1 B (2) + #1 (1) 0f21 1 S (2) (! 3 ) = X (2) (! 3 )
#0 (1) f21 B (2) + #1 (1) f21 S (2) (! 4 ) = X (2) (!4) ;
0 1
A (1) f11 =
110
COMPLETE MARKETS
0 1
which has one and only one solution # (1) f21 2 R2 , since
2
0 13
det A (1) f21
= det
B (2) S (2) (! 3 )
B (2) S (2) (! 4 )
= 03:528
= det
1:1025 9:6
1:1025 6:4
It thus remains univocally determined the vector
0 13
2
0 1
# (1) = # (1) f11 ; # (1) f21 :
We also need to impose C# (1) = X (1), i.e.
C# (1) = #0 (0) B (1) + #1 (0) S (1) 0 V# (1) = X (1)
with
V# (1) = #0 (1) B (1) + #1 (1) S (1)
already determined. In other words, we come up with the system:
0 1
0 1
0 1
#0 (0) B (1) + #1 (0) S (1) 0f11 1 = X (1) 0f11 1 + V# (1) 0f11 1
#0 (0) B (1) + #1 (0) S (1) f21 = X (1) f21 + V# (1) f21 ;
which admits one and only one solution (#0 (0) ; #1 (0)) because
A (0) =
0 1 B (1) S (1) 0f11 1
1:05 12
=
B (1) S (1) f21
1:05 8
is nonsingular, since det [A (0)] = 04:2 6= 0:
The risk-free asset B, the one yielding the constant rate of 5% over
unitary periods, and the risky security S are thus enough to replicate not
only every European contingent claim with maturity T = 2, but also every
contingent claim generating a cash‡ow in t = 1:
We conclude this section with a remark on a very speci c, yet widely
used, class of multi-period markets, the so called multinomial markets. Let
us rst mention that an information structure is said multinomial when
every one of its nodes has the same number of immediate successors (we
denote by n such number). A multi-period market is said to be multinomial
if n = N + 1, i.e. if the number of securities is equal to the common number
of following nodes. For multinomial models, dynamic completeness can be
9.3.
THE SECOND FUNDAMENTAL THEOREM OF ASSET PRICING111
characterized in terms of a simple necessary and sufficient condition.
We
first associate with every node f} the matrix A (f}) defined as follows:
B(t+1)(
A (ff) =
where
B(t+1)
(t+ (itt)
fit"
..
Si(t+1)(fi3°)
ft
B(t+1)(ff$ees) Si(¢#1) (fa)
( it) , ( th).,- -( fikis)
sweaty
(fp
Sw (t+)
ts
Sw (O42) (fhe)
€ Pz4i1 are the N + 1 nodes stemming
from the node ff. By exploiting Proposition 54, it is then easy to verify that
a multinomial market is dynamically complete if and only if the following
condition is satisfied:
rank [A (fz)
=N+1,
h=1,...,%,¢=0,1,..,7-1
We will analyze in detail the most meaningful example of multinomial market, namely the binomial market, with n = N+1=2.
9.3.
The
Second
Fundamental
Theorem
of Asset
Pricing
We conclude by observing that, as in the one-period case, the following result
holds.
Theorem 57 The Second Fundamental Theorem of Asset Pricing,
multi-period case. In a discrete time multi-period financial market the
following statements are equivalent:
1. the market is both arbitrage-free and dynamically complete;
2. there exists one and only one multi-period state-price vector;
3. there exists one and only one martingale measure.
Chapter 10
No-arbitrage Valuation in
the Multi-period Case
As in the one-period case, we now move one to the issue of no-arbitrage
pricing of nancial securities. The problem has been introduced in the rst
section of this chapter. In an arbitrage-free multi-period market (initial
market), a new security is introduced. We look for conditions on the price
of the new security such that extended market is arbitrage-free as well.
More precisely, since the new security will be available for trading at the
intermediate dates, we look for conditions on the price process of the new
security ensuring that the extended market is arbitrage-free.
The answer to the above question depends on whether the new security
can be replicated or not. In the second section, we focus on the redundant
case and provide two alternative necessary and su¢cient conditions on the
new security price process for the extended market to be arbitrage-free. The
rst condition requires the price process to be equal to the value process of
any strategy replicating the new security. The second condition requires the
price process to be equal, at each date, to the expected value of the discounted future cash‡ows generated by the new security, with the expectation
taken under any risk-neutral probability of the initial market.
In the third and concluding section, we consider the case of a nonredundant new security. As we will see, the key di¤erence with the attainable case resides in the fact that the price process ensuring that the
extended market is arbitrage-free is not unique anymore.
113
114CHAPTER 10. NO-ARBITRAGE VALUATION IN THE MULTI-PERIOD CASE
10.1
The Framework
We take as input a multi-period market in which traded are the locally
risk-free asset B (to which we will refer by employing the index j = 0,
i.e. B = S0 ) and N risky securities with indexes j = 1; :::; N . As in the
one-period case, from now onwards we call such market initial market and
we assume it to be arbitrage-free throughout the chapter. Let us now introduce a new security yielding a a cash‡ow fX(t)gTt=1 and available for
trading at dates t = 0; 1; ::; T at prices fSX (t)gTt=0 : Since at the nal date
T all positions must be liquidated, we assume that the cash‡ow generated
by the new security coincides with its liquidation value: X(T ) = SX (T ):
We call extended market the multi-period market obtained by adding the
new security to the initial market. The problem of no-arbitrage pricing is
nding conditions under which the price process of the new security can be
determined on the basis of the pre-existing security prices, either univocally
or in terms of price range.
The problem is then the following: how and under what conditions can
we determine fSX (t)gTt=0 starting from the price processes of the initial
market security prices? As we will see, the answer to the question depends
on whether the new security is redundant or not. We discuss the case of
a redundant new security in the next section, the case of a non-redundant
security in Section 10.3.
10.2
The Case of Redundant Securities
Let us suppose that the initial market is arbitrage-free and that the new
security X is redundant, i.e. there exists a dynamic investment strategy
8
9T 01
#X = #X (t) t=0 such that
C#X (t) = X (t)
for t = 1; :::; T: Based on such assumptions, we provide two necessary and
su¢cient conditions on the new security price process, SX = fSX (t)gTt=0 ,
for the extended market to be arbitrage-free.
Proposition 58 If the new security is redundant, the following three conditions are equivalent:
1. the extended market is arbitrage-free;
10.2. THE CASE OF REDUNDANT SECURITIES
115
2. for every strategy #X replicating the new security, we have
SX (t) = V#X (t)
(10.1)
for t = 0; 1; :::; T ;
3. for every risk-neutral probability measure, Q, of the initial market, we
have
i
h
X (t+1)
SX (t) = EQ X(t+1)+S
P
t
1+r(t)
for t = 0; 1; :::; T 0 2 and
h
i
X(T )
Q
SX (T 0 1) = E
1+r(T 01) PT 01
(10.2)
for t = T 0 1:
Proof. The proof goes along the same lines of the proof of Proposition
19 in the one-period case.
By observing that B(t + 1) = B(t)(1 + r(t)) is Pt 0measurable, we can
rewrite equation (10:2) in terms of discounted prices, i.e. as
h
i
SX (t)
Q X(t+1)+SX (t+1) P
=
E
t
B(t)
B(t+1)
for t = 0; 1; :::; T 0 2 and
h
i
SX (T 01)
Q X(T ) P
=
E
T
01
B(T 01)
B(T )
(10.3)
for t = T 0 1:
Equation (10:2) can be written in the equivalent meaningful form
"
SX (t) = EQ
#
T
X
B(t)
X( ) Pt ;
B( )
(10.4)
=t+1
for t = 0; :::; T 0 1: Equation (10:4) says that the time-t price of the security
X is equal to the Pt 0conditional expected value of the future cash‡ows
B(t)
discounted at time t: Indeed, the ratio B(
) is actually the discount factor
relative to the time interval [t; [; since
Y
01
B( )
=
(1 + r(s))
B(t)
s=t
116CHAPTER 10. NO-ARBITRAGE VALUATION IN THE MULTI-PERIOD CASE
represents the proceeds at time from investing 1 euro at time t at the
locally risk-free rate r(t), reinvesting in t + 1 at the rate r(t + 1) and so
on up to an instant before . The conditional expected value of the future
cash‡ows generated by the new security, computed under any risk-neutral
probability Q of the initial market, yields a unique no-arbitrage price SX (t)
at time t:
We now show how to derive expression (10:4) from (10:2): The proof is
based on backward induction. For t = T 0 1, expression (10:2) is clearly
equivalent to
X(T )
Q
PT 01 =
SX (T 0 1) = E
1 + r(T 0 1)
Q B(T 0 1)
= E
X(T ) PT 01
B(T )
Let us see what happens for t = T 0 2: Because of (10:2), we have
1
Q
(X(T 0 1) + SX (T 0 1)) PT 02 =
SX (T 0 2) = E
1 + r(T 0 2)
X(T )
1
Q
Q
= E
PT 02 =
X(T 0 1) + E
PT 01
1 + r(T 0 2)
1 + r(T 0 1)
1
X(T )
Q
= E
X(T 0 1) +
PT 02 =
1 + r(T 0 2)
1 + r(T 0 1)
B(T 0 2)
Q B(T 0 2)
= E
X(T 0 1) +
X(T ) PT 02
B(T 0 1)
B(T )
where the second last equality comes from the law of iterated expectations
and the last one from the properties of the locally risk-free asset B:
Iterating from t = T 0 3 back to t = 0 we obtain expression (10:4) for
all dates t = 0; :::; T 0 1: 4
Proposition 58 provides a way to determine the no-arbitrage price process of a redundant security. Let us suppose that a nancial institution
synthesizes and issues a speci c derivative security yielding the cash‡ow X
to the buyer. What is the no-arbitrage price process of the derivative? Since
that security can be synthesized, i.e. it can be replicated by suitably investing in the initial market, we know that the price process of the derivative is
univocally determined. Point 2. in Proposition 58 says that the price process of the derivative security is equal to the value process of any strategy
replicating the new security, as can be seen from equation (10:1). In order
to follow this way, we rst need to set up a dynamic strategy replicating the
10.3. THE CASE OF NON-REDUNDANT SECURITIES
117
security, i.e. a hedging stategy for the cash‡ow X, and then to determine
the value process.
However, the price of the new security at every date t can also be obtained as conditional expected value of the sum of cash‡ows generated by
the security at the next date and of its price (liquidation value) at the next
date, as can be seen from (10:2) for t < T 0 1. For t = T 0 1, the price of the
new security coincides with the conditional expected value of the discounted nal liquidation value. In both cases, such conditional expected value
is univocally determined if the security is redundant, independently of the
risk-neutral measure of the initial market employed. Let us now adopt the
perspective of the issuer of the derivative at time t. At the end of the period,
i.e. in t + 1, we must guarantee the cash‡ow X(t + 1) to the buyer of the
derivative and, before maturity, we must be ready to deliver the derivative,
whose price will be given by SX (t + 1): Hence, the numerator of the ratio
inside the expectation in equation (10:2) represents the liquidation value,
period by period, of the positions held by the writer of the derivative. By
discounting such liquidation value and taking conditional expectation under
any risk-neutral measure of the initial market, we then get the no-arbitrage
price of the security.
Equation (10:4) switches the focus on the nal date, looking at the whole
stream of future cash‡ows to be guaranteed by the writer of the derivative,
instead of focusing on each subperiod endpoint. On the basis of equation
(10:4), the no-arbitrage price of the new security is equal to the conditional
expected value of the discounted future cash‡ows under any risk-neutral
measure of the initial market.
Equation (10:2) and equation (10:4) enable to compute the price of the
new security as conditional expected value under a risk-neutral probability
measure, an approach in general faster than the determination of a dynamic
strategy replicating the new security. As in the one-period case, by following
the way suggested by Proposition 58 at point 3, the price is more readily
computable, but no information is gathered on how to hedge the derivative
(as in Proposition 58, point 2.)
10.3
The Case of Non-redundant Securities
We now assume that the new security cannot be replicated, i.e. there is no
dynamic strategy available in the initial market with cash‡ow process equal
to the one generated by the new security X: What can we say about the
new security price in this case? Here is the answer:
118CHAPTER 10. NO-ARBITRAGE VALUATION IN THE MULTI-PERIOD CASE
Proposition 59 If the new security cannot be replicated, the following three
conditions are equivalent:
1. the extended market is arbitrage-free;
2. for all t = 0; 1; :::; T 0 1 and for all fht 2 Pt
)
(
N
X
SX (t)(fht ) <
min
#n Sn (t)(fht )
#0 B(t)(fht ) +
#22X (fht )
where
2X (fht ) =
n
(10.5)
n=1
#
S
(t
+
1)
(flt+1 ) n=1 n n
9
X(t + 1)(flt+1 ) + SX (t + 1)(flt+1 ) for all flt+1 fht
# 2 <N +1
#0 B(t + 1) +
PN
for t = 0; :::; T 0 2, while for t = T 0 1
n
P
2X (fhT 01 ) = # 2 <N +1 #0 B(T ) + N
#
S
(T
)
(flT ) n=1 n n
o
X(T )(flT ) for all flT fhT 01
Furthermore, for all t = 0; 1; :::; T 0 1 and for all fht 2 Pt
(
SX (t)(fht )
where
20X (fht ) =
n
>
max
#220X (fht )
0
#0 B(t)(fht )
+
N
X
n=1
!)
#n Sn (t)(fht )
(10.6)
#
S
(t
+
1)
(flt+1 ) n=1 n n
9
0 (X(t + 1) + SX (t + 1)) (flt+1 ) for all flt+1 fht
# 2 <N +1
#0 B(t + 1) +
PN
for t = 0; :::; T 0 2, while for t = T 0 1
n
P
20X (fhT 01 ) = # 2 <N +1 #0 B(T ) + N
#
S
(T
)
(flT ) n
n
n=1
o
0X(T )(flT ) for all flT fhT 01
3. For t = T 0 1, we have that
X(T )
SX (T 0 1) < sup E
PT 01
1 + r(T 0 1)
Q
X(T )
Q
PT 01
SX (T 0 1) > inf E
Q
1 + r(T 0 1)
Q
10.3. THE CASE OF NON-REDUNDANT SECURITIES
119
and for all t = 0; :::; T 0 2
SX (t) < sup E
Q
Q
SX (t) > inf E
Q
Q
X(t + 1) + SX (t + 1)
Pt
1 + r(t)
X(t + 1) + SX (t + 1)
Pt
1 + r(t)
(10.7)
(10.8)
with the supremum and in mum computed over the set of all riskneutral measures of the initial market.
Proof. Omitted.4
We discuss the meaning of Proposition 59 starting from point 3., where a
lower and an upper bound are provided for the no-arbitrage price of the new
security over all time periods. In particular, in t = T 0 1 the price belongs
to the interval with endpoints the in mum and supremum of all conditional
expected values of the discounted cash‡ows generated at maturity by X
(computed over all risk-neutral probabilities of the initial market). The
cash‡ow X(T ) is what has to be guaranteed by the writer of the derivative
(i.e. the owner of a short position on the security), exactly as in the oneperiod case. In t T 0 2, the owner of the short position must not only
provide for the cash‡ow X(t + 1) generated at the next date, but also for
the following cash‡ows, possibly liquidating his/her position by buying the
new security at a price SX (t + 1): Equations (10:8) and (10:7) tell us that
the time-t no-arbitrage price of the new security must belong to the interval
with extremes the in mum and supremum of the conditional expectations
(computed over all possible risk-neutral probabilities of the initial market)
of the discounted liquidation value of the short position on the derivative at
the end of the period considered.
Point 2. of Proposition 59 gives a period-by-period interpretation of
the endpoints of the no-arbitrage price range in terms of super-replication
strategies. Expression (10:5) tells us (event by event) that the price of the
new security in fht is lower than the minimum super-replication cost of the
new security in the one-period market Mt;h with root in fht : Indeed, if such
inequality were not satis ed, the short side could pro t from an arbitrage in
Mt;h , by super-replicating its own position in t + 1 through the minimum
cost strategy, which could be nanced by in fht by employing the proceeds
from the sale of the new security, thus obtaining a strictly positive cash‡ow
in fht . Hence, inequality (10:5) prevents arbitrages based on short positions
on the new security.
120CHAPTER 10. NO-ARBITRAGE VALUATION IN THE MULTI-PERIOD CASE
Symmetrically, inequality (10:6) prevents arbitrage opportunities based
on long positions on the new security.If such inequality were not to hold in
some fht , it would be possible to buy the new security by selling the strategy
achieving its maximum in 20X (fht ): The net cash‡ow obtained in fht would
be strictly positive, while in the immediate successors of fht at time t + 1, the
in‡ows would be enough to match the (negative) value of the strategy that
was used to nance the strategy, thanks to the cash‡ow X(t + 1) generated
by the new security and to the liquidation value SX (t + 1).
Despite the cumbersome notations, the meaning of the bounds given in
point 2. of Proposition 59 is the very same as that proved in Proposition 20
with regard to the one-period setting.
Chapter 11
The Multi-period Binomial
Model
11.1
Description of the Model
The multi-period binomial model involves two securities.
The rst one is the risk-free asset B yielding a constant one-period interest rate, i.e. r (t) = r > 0 for t = 0; :::; T 0 1, employing the notation
introduced in the previous chapters. The risk-free asset B at a generic time
t will then have price B (t) = (1 + r)t :
The second security is the risky stock S. Given S(t), the time-t price of
the security S can take only two values at the following date t + 1:
S (t)
t
%
&
S (t) u
S (t) d
with probability p
with probability 1 0 p
t+1
for t = 0; :::; T 0 1.
The up factor, u, and the down factor, d, are constant over time. There
are di¤erent binomial models in which such factors are (deterministically or
randomly) time-dependent. Here, we consider only the standard version of
such model, where the factors u; d and the interest rate r are constant.
We now try to understand the structure of the information describing
the evolution of our market. For simplicity, we limit ourselves to the timehorizon T = 3. Starting from the initial value S = S(0), the risky security
121
122
CHAPTER 11. THE MULTI-PERIOD BINOMIAL MODEL
price S evolves in the following way:
S
t=0
%
&
Su
%
&
Sd
%
&
t=1
Su2
Sud
Sdu
Sd2
%
&
%
&
%
&
%
&
t=2
Su3
Su2 d
Su2 d
Sud2
Sdu2
Sd2 u
Sd2 u
Sd3
t=3
In T = 3, we have K = 2T = 8 possible scenarios taking into account
the whole history of the process:
8
9
! 1 = S (1) = Su; S (2) = Su2 ; S (3) = Su3
8
9
! 2 = S (1) = Su; S (2) = Su2 ; S (3) = Su2 d
8
9
! 3 = S (1) = Su; S (2) = Sud; S (3) = Su2 d
8
9
! 4 = S (1) = Su; S (2) = Sud; S (3) = Sud2
8
9
! 5 = S (1) = Sd; S (2) = Sdu; S (3) = Sdu2
8
9
! 6 = S (1) = Sd; S (2) = Sdu; S (3) = Sd2 u
8
9
! 7 = S (1) = Sd; S (2) = Sd2 ; S (3) = Sd2 u
8
9
! 8 = S (1) = Sd; S (2) = Sd2 ; S (3) = Sd3
In t = 2, we can “condense”the scenarios into
f12 = f! 1 ; ! 2 g ; f22 = f! 3 ; ! 4 g
f32 = f! 5 ; ! 6 g ; f42 = f! 7 ; ! 8 g ;
so that
8
9
P2 = f12 ; :::; f42 ;
while in t = 1; P1 is made of the two elements
f11 = f12 [ f22 = f! 1 ; ! 2 ; ! 3 ; ! 4 g
f21 = f32 [ f42 = f! 5 ; ! 6 ; ! 7 ; ! 8 g ;
and in t = 0,
P0 = f g :
11.1. DESCRIPTION OF THE MODEL
123
Since at each date t the security S undergoes either a relative increase u
with probability p or a relative decrease d with probability 1 0 p, we are
left with the following probabilities assigned to the K = 8 scenarios at time
T = 3:
P [! 1 ] = p 1 p 1 p = p3
P [! 2 ] = p 1 p 1 (1 0 p) = p2 (1 0 p)
P [! 3 ] = p 1 (1 0 p) 1 p = p2 (1 0 p)
P [! 4 ] = p 1 (1 0 p) 1 (1 0 p) = p(1 0 p)2
P [! 5 ] = (1 0 p) 1 p 1 p = (1 0 p)p2
P [! 6 ] = (1 0 p) 1 p 1 (1 0 p) = (1 0 p)2 p
P [! 7 ] = (1 0 p) 1 (1 0 p) 1 p = (1 0 p)2 p
P [! 8 ] = (1 0 p) 1 (1 0 p) 1 (1 0 p) = (1 0 p)3
However, to describe the future evolution of the risky security, we do not
need to keep track of the whole past, but only of the current security value
S(t), since we know that only two situations are possible in t + 1, each with
probability
P [S (t + 1) = S (t) u j S (t)] = p
and
P [S (t + 1) = S (t) d j S (t)] = 1 0 p
for t = 0; :::; T 0 1:
The risky security price S is Markovian. That means, in order to know
the future, i.e. the possible values taken by S(t + 1); S(t + 2); etc., we just
need the present, i.e. S(t); and not the past, i.e. the values S(t01); S(t02); :::
For this reason, if our focus is limited to the description of S, it is enough to
adopt an event-tree representation simpler than that described above. Such
representation just keeps track of the current value of the risky security S :
S
t=0
%
&
Su
Sd
t=1
%
&
%
&
Su2
Sud
Sd2
t=2
%
&
%
&
%
&
Su3
Su2 d
Sud2
Sd3
t=3
124
CHAPTER 11. THE MULTI-PERIOD BINOMIAL MODEL
Thus, at time t = 2 we have condensed the two nodes f22 ; f32 in which S (2)
is Sud, and at time t = 3 we have condensed f! 2 ; ! 3 ; ! 5 g, the states in
which S (3) is Su2 d, and f! 4 ; ! 6 ; ! 7 g, the states in which S (3) is Sud2 : The
number of elements making up the representation is much lower than that
of our original information structure. For every date t, there are indeed t + 1
elements, as opposed to the 2t elements involved bu Pt : The reduction is due
to the fact that the tree is recombining: if the security S undergoes rst an
increase and then a decrease, it reaches the same level as if it had undergone
rst a decrease and then an increase. Put another way, what actually matters is only the overall number of upward and downward movements, not
the order in which they occur. The event-tree representation given is quite
handy, but it is not a proper information structure. It is more than enough
when we just want to study the evolution of a derivative whose price only
depends on the current price of the underlying, but not enough when we
consider for example a path-dependent option, i.e. an option whose payo¤
depends on the whole path followed by the risky security up to maturity.
With this caveat in mind, we note that
3
2
P S (3) = Su3 = P [! 1 ] = p3
2
3
P S (3) = Su2 d = P [! 2 ] + P [! 3 ] + P [! 5 ] = 3p2 (1 0 p)
2
3
P S (3) = Sud2 = P [! 4 ] + P [! 6 ] + P [! 7 ] = 3p(1 0 p)2
2
3
P S (3) = Sd3 = P [! 8 ] = (1 0 p)3
The probability distribution obtained for S (3) is binomial. Indeed, for all
t = 0; :::; T we have
h
i t
k t0k
P S (t) = Su d
=
pk (1 0 p)t0k
k
t
for k = 0; 1; :::; t: The term
is the so called binomial coe¢cient, given
k
by
t!
t
=
k! (t 0 k)!
k
and counting the possible ways in which S(t) can reach the level Suk dt0k
starting from S at time t = 0, i.e. the number of cases in which we have k
upward and t 0 k downward movements.
Under P, the random variable S(t) is hence distributed as a binomial
random variable with parameters t and p: this is why the model is called
binomial.
11.2.
NO-ARBITRAGE AND DYNAMIC COMPLETENESS
125
We finally write the relative increment of S between t and t+ 1:
AS(t) _ S(t+1)—S(t)
S(t)
for all t =
i.e.
S(t)
_ f u-—1
with probability p
~ | d—1
with probability 1 — p
0,...,7 — 1. All increments
then have the same
they are identically distributed under P.
distribution,
They are further mutually
independent.
11.2
No-Arbitrage and Dynamic
Completeness
We start with the issue of market completeness.
Every node ff of the
information structure P; has two immediate successors in which the risk-
free asset B and the risky security S take the following values, grouped in
the one-period payoff matrix:
ty)
A(t)(fh) =
[| (+r)?
(1-4 ret
S@)(fp)-u
sth) a
Since u > d, the matrix A(t) has maximum rank (i.e. 2) in every node.
Every one-period submarket is hence complete and the multi-period binomial market is consequently dynamically complete.
With regard to the issue of no-arbitrage, we reduce the analysis to that
carried out in a one-period case.
Indeed, every one-period submarket is a
one-period binomial market.
We know that no-arbitrage holds in the one-period binomial model if:
d<l4+r<u,
as was shown in Chapter 5. Hence, we assume throghout thatd<1l+r<
u, so that we know again from Chapter 5 that there exists a risk-neutral
probability in every one-period submarket. In particular, the risk-neutral
probability of an up movement of S(t) between ¢ and t + 1 is:
r—dd
lirii
Q [s(t +1) =S(t)-uJ ==
U
_
while that of a down movement is
G9
-g=4 -(1
Q [S(t +1) = S(t) -d] =1
for all t = 0,..., 7’ — 1. Hence, these risk-neutral probabilities do not depend
on the date t nor on the specific node of the information structure, but are
126
CHAPTER 11. THE MULTI-PERIOD BINOMIAL MODEL
constant. As we have seen in the proof of the First Fundamental Theorem
of Asset Pricing, the risk-neutral probability of every one-period submarket coincides with the corresponding multi-period one, Q, conditional on
the root of the one-period tree considered. We can thus obtain the riskneutral probability of each scenario f! k g 2 PT by multiplying conditional
probabilities. E.g., for T = 3 we have:
(1 + r) 0 d 3
3
Q [! 1 ] = q =
u0d
(1 + r) 0 d 2 u 0 (1 + r)
2
Q [! 2 ] = Q [! 3 ] = Q [! 5 ] = q (1 0 q) =
u0d u 0 d u
0
(1 + r) 2
(1
+
r)
0
d
Q [! 4 ] = Q [! 6 ] = Q [! 7 ] = q(1 0 q)2 =
u0d
u0d
3
u
0
(1
+
r)
Q [! 8 ] = (1 0 q)3 =
u0d
Looking at the outcomes of S(3) we have that:
3
2
Q 2S (3) = Su3 3= Q [! 1 ] = q 3
Q 2S (3) = Su2 d3 = Q [! 2 ] + Q [! 3 ] + Q [! 5 ] = 3q 2 (1 0 q)
Q 2S (3) = Sud32 = Q [! 4 ] + Q [! 6 ] + Q [! 7 ] = 3q(1 0 q)2
Q S (3) = Sd3 = Q [! 8 ] = (1 0 q)3
Hence, S (3) is binomially distributed under Q as well, but with parameters
T = 3 and q.
For a generic date t, we get:
i t
h
k t0k
=
q k (1 0 q)t0k
Q S (t) = Su d
k
for t = 1; :::; T and k = 0; :::; t:
The reason why S(t) is binomially distributed also under Q is that the
one-period risk-neutral probabilities of up/down movements are constant,
namely
Q [S (t + 1) = S (t) u j S (t)] = q
and hence independent of t and of the speci c node of the information structure.
(1 + r) 0 d
Recalling that q =
, we can verify that the conditional expecu0d
ted return on S under Q is equal to that of the risk-free asset, r. We have
to compute
0 1
Q S (t + 1) 0 S (t)
Pt fkt
E
S (t)
11.3. OPTION PRICING IN THE MULTI-PERIOD BINOMIAL MODEL127
Given fkt 2 Pt , we can write
0 1
S (t) 0fkt 1 u with risk-neutral probab. q
S (t + 1) =
S (t) fkt d with risk-neutral probab.1 0 q
from which EQ
h
S(t+1)0S(t)
S(t)
Pt
i0 1
fkt =
S(t)(fkt )u0S(t)(fkt )
S(t)(
fkt
)
q+
S(t)(fkt )d0S(t)(fkt )
S(t)(fkt )
(1 0 q) =
(1 + r) 0 d
+d01 = 1+r0d+d01 = r: Under Q,
u0d
the expected return on the risky stock is then equal to that of the risk-free
asset.
(u 0 d) q +d01 = (u 0 d)
11.3
Option Pricing in the Multi-period Binomial
Model
We know that if d < 1 + r < u, the binomial market is complete and
arbitrage-free. We now introduce a derivative security in the market, which
can be replicated by suitably investing in the risk-free asset and in the risky
security S, because the market is complete. According to Proposition 58,
there is a unique price process consistent with an arbitrage-free extended
market. Such price process can be obtained from the value process of the
strategy replicating the new security (Proposition 58, point 2) or by computing the conditional expected value under the risk-neutral measure Q of
the discounted future cash‡ows generated by the new security (Proposition
58, point 3).
11.3.1
Valuation of a Call Option via Replication
We follow both approaches to price a European call option on the security
S with strike price K and maturity T . The cash‡ow generated by a long
position on the call is
0
t = 1; :::; T 0 1
X (t) =
max [S (t) 0 K; 0]
t=T
We look for a strategy 3 replicating X, i.e. such that V3 (t) = X(t) for
t = 1; :::; T: The no-arbitrage price of the call option is then given by
c(t) = SX (t) = V3 (t)
For simplicity, we limit ourselves to the case of T = 3.
128
CHAPTER 11. THE MULTI-PERIOD BINOMIAL MODEL
The strategy replicating X can be found by backward induction, starting
from the terminal condition
V3 (T ) = max [S (T ) 0 K; 0]
(11.1)
Since V3 (3) is determined by 3 (2), we can rewrite (11.1) in every node
fk2 2 P2 as follows:
3 2
0 1
0 1 0 2 1 0 (2) fk2
max 2S(2) 0fk2 1 u 0 K; 03
1
0
A(2) fk
=
1 (2) fk2
max S(2) fk2 d 0 K; 0
where
0 1
A(2) fk2 =
0 1 (1 + r)3 S (2) 0fk2 1 u
(1 + r)3 S (2) fk2 d
By solving the system, we get:
8
d max [S(2)u 0 K; 0] 0 u max [S(2)d 0 K; 0]
>
3
>
< 0 (2) =
(1 + r)3 (d 0 u)
max [S(2)d 0 K; 0] 0 max [S(2)u 0 K; 0]
>
>
: 31 (2) =
S(2) (d 0 u)
where for convenience we omitted the dependence of S(2) (and, as a consequence, of 30 (2) and 31 (2)) on the node fk2 of the information structure at
time t = 2: We thus have
V3 (2) = 30 (2) (1 + r)2 + 31 (2)S(2) =
max [S(2)u 0 K; 0] (1 + r) 0 d max [S(2)d 0 K; 0] u 0 (1 + r)
+
=
=
1+r
u0d
1+r
u0d
= c(2) [S(2)]
The value of the replicating strategy one period before maturity is thus a
function of the current value of the underlying S(2): Only such dependence
accounts for the randomness of V3 (2) ; which does not depend on the speci c
node of the information structure, but only on the price of the underlying
at time t = 2. We can stress such dependence by making it explicit in the
price c(2); the no-arbitrage price of the call at time t = 2; which by point 2.
in Proposition 58 is given by
c(2) [S(2)] = V3 (2) :
In order to determine 3 (1), we note that
C3 (2) = X(2) = 0;
11.3. OPTION PRICING IN THE MULTI-PERIOD BINOMIAL MODEL129
from which we get
30 (1) (1 + r)2 + 31 (1)S(2) = V3 (2) = c(2):
0 1
Hence, in every node fh1 2 P1 we look for 31 (1) fh1 such that
2
0 1 3 0 11 3 0 11
c(2) 2S(1) 0fk1 1 u3
A(1) fk (1) fk =
;
c(2) S(1) fk1 d
with
A(1)
0
fk1
1
=
0 1 (1 + r)2 S(1) 0fk1 1 u
:
(1 + r)2 S(1) fk1 d
Solving the system, we get
8
uc(2) [S(1)d] 0 dc(2) [S(1)u]
>
3
>
< 0 (1) =
(1 + r)2 (u 0 d)
c(2) [S(1)u] 0 c(2) [S(1)d]
>
>
: 31 (1) =
S(1) (u 0 d)
from which
V3 (1) = 30 (1) (1 + r) + 31 (1)S(1)
c(2) [S(1)u] (1 + r) 0 d c(2) [S(1)d] u 0 (1 + r)
+
=
1+r
u0d
1+r
u0d
= c(1) [S(1)] :
Finally, we determine the vector
3 (0) = [30 (0); 31 (0)]
such that
that is
C3 (1) = X(1) = 0;
30 (0) (1 + r) + 31 (1)S(1) = V3 (1) = c(1) [S(1)] :
We can rewrite the above equality (between random variables), exactly as
done before, by employing the one-period payo¤ matrix
(1 + r) Su
A(0) =
;
(1 + r) Sd
to obtain
3
A(0) (0) =
c(1) [Su]
c(1) [Sd]
:
130
CHAPTER 11. THE MULTI-PERIOD BINOMIAL MODEL
By solving the system we now get
8
uc(1) [S(0)d] 0 dc(1) [S(0)u]
>
>
< 30 (0) =
(u 0 d) (1 + r)
c(1)
[S(0)u]
0 c(1) [S(0)d]
>
>
: 31 (0) =
S(0) (u 0 d)
from which
V3 (0) = 30 (0)1 + 31 (0)S(0)
c(1) [S(0)u] (1 + r) 0 d c(1) [S(0)d] u 0 (1 + r)
+
=
1+r
u0d
1+r
u0d
We note that for t = 0; 1; 2 the value of the strategy replicating the call
option, i.e. the no-arbitrage price of the call, is given by
c (t + 1) [S (t) u] (1 + r) 0 d c(t + 1) [S(t)d] u 0 (1 + r)
+
1+r
u0d
1+r
u0d
(11.2)
By iterating the backward procedure we can obtain the same result for t =
0; 1; :::; T even when the maturity T is longer than 3. The weights in formula
(1 + r) 0 d
u 0 (1 + r)
(11.2),
and
, are respectively Q [S (t + 1) = S (t) u j P (t)]
u0d
u0d
and Q [S (t + 1) = S (t) d j P (t)].
Formula (11.2) can be rewritten in shorter form as
Q c (t + 1)
Pt
c (t) = E
1+r
c (t) = V3 (t) =
for t = 0; 1; :::; T 0 1, i.e. as the risk-neutral pricing formula of point 3. in
Proposition 58.
11.3.2
Call Option Pricing via Risk-Neutral Valuation
We now want to price the same European call option on S with strike price
K and maturity T , by employing point 3. in Proposition 58. That result
guarantees that the unique no-arbitrage price process of the option is given
by the conditional expected value, under the risk-neutral measure, of the
discounted future cash‡ows of the option. In particular, from equation (10.4)
we get
" T
#
X
c (t) = SX (t) = EQ
(1 + r)0( 0t) X( ) Pt =
= E
Q
h
=t+1
0(T 0t)
(1 + r)
max(S(T ) 0 K; 0) Pt
i
11.3. OPTION PRICING IN THE MULTI-PERIOD BINOMIAL MODEL131
for t = 0; :::; T 0 1: In order to compute the conditional expectation, we need
the distribution of S(T ) under Q:
Recalling that S (T ) is binomially distributed with parameters T and
(1 + r) 0 d
, we compute c(t) for t = 0: The expected value we want to
q=
u0d
compute is the following:
2
3 T
X
max Suj dT 0j 0 K; 0
T
q j (1 0 q)T 0j
c (0) =
T
j
(1 + r)
j=0
T!
T
is the binomial coe¢cient.
where
=
j! (T 0 j)!
j
Since the time-T payo¤ of the call option is strictly positive only when
S (T ) > K (when the option is in the money), we let a be the minimum
number of upward movements of S, between 0 and T , such that the option
ends up in the money, i.e. S(0)uj dT 0j > K for j = a; a + 1; :::; T and
S(0)uj dT 0j K for j = 0; 1; ::; a 0 1:In this case:
T
X
S(0)uj dT 0j 0 K T
c (0) =
q j (1 0 q)T 0j
j
(1 + r)T
j=a
T T X
X
T
qu j (1 0 q) d T 0j
T
K
q j (1 0 q)T 0j
= S(0)
0
T
1+r
1+r
j
j
(1
+
r)
j=a
j=a
By setting
q0 =
we have that
1 0 q0 =
and
qu
;
1+r
(1 0 q) d
;
1+r
q 0 ; 1 0 q 0 2 ]0; 1[ :
The introduction of q 0 allows us to simplify the expression de ning c (0),
since the term
T T X
1T 0j
T
qu j (1 0 q) d T 0j X T 0 0 1j 0
1 0 q0
q
=
1+r
1+r
j
j
j=a
j=a
represents the probability that a binomial random variable with parameters
T and q 0 takes a value greater than a, i.e. the probability that with T
132
CHAPTER 11. THE MULTI-PERIOD BINOMIAL MODEL
trials there is a number of upward jumps greater than a (hence a number of
upward jumps equal to a or a + 1... or T ).
If we use the notation
1
0
8 a; T; q 0 = probability that a binomial r.v. with parametersT and q 0 is a
= complement to 1 of the distribution function computed in a 0 1
the call price in 0 can be rewritten as
0
1
c (0) = S8 a; T; q 0 0
K
(1 + r)T
8 (a; T; q)
The factor multiplying the discounted strike price, 8 (a; T; q), is the riskneutral probability that there are at least a upward movements, i.e. the
risk-neutral probability that the call option ends up in the money. We will
see that also in the Black-Scholes Model one obtains a formula for the call
option price involving the multiplication of the discounted strike price by
the risk-neutral probability of ending up in the money. If in the previous
formula we collect the discount factor, we can write
h
i
0
1
c (0) = (1 + r)0T S (1 + r)T 8 a; T; q 0 0 K8 (a; T; q)
and so the term
0
1
S (1 + r)T 8 a; T; q 0
is the risk-neutral expected value of a random variable equal to S(T ) if
S(T ) > K, to zero otherwise.
11.3.3
Put-call Parity
The so called Put-call Parity holds in the multi-period binomial model as
well. In order to prove this, we construct a portfolio replicating a put option
by investing in the risk-free asset, in the risky security S and in a call option
with same strike and maturity as the put considered. The strategy we need
is the following at time 0
8
K
K
>
~
>
< 0 (0) = B (T ) =
(1 + r)T
> ~1 (0) = 01
>
: ~
2 (0) = 1
and
~n (t) = ~n (0)
11.3. OPTION PRICING IN THE MULTI-PERIOD BINOMIAL MODEL133
for the subsequent dates t = 1; 2; :::; T 0 1: In other words, the strategy
K
~ requires buying at time 0 a number
of units of risk-free asset,
(1 + r)T
shortselling 1 unit of S and taking a long position in the call option. Such
positions are held until maturity (it is a buy-and-hold strategy). Since the
portofolio positions do not change, the strategy is clearly self- nancing. Furthermore, at maturity it is worth
K
V~ (T ) =
1 B(T ) 0 1 1 S (T ) + 1 1 max [S(T ) 0 K; 0]
(1 + r)T
= max [K 0 S(T ); 0]
i.e. exactly as much as the put option. As a result, since the strategy ~
and a long position on the put option provide the same cash‡ows C~ (t) for
t = 1; 2; :::; T , the market made of the risk-free asset, the risky security S,
the call option and the put option, is arbitrage-free if and only if the put
price, p(t), is equal to
p (0) = V~ (0) =
K
(1 + r)T
1 1 0 1 1 S + c (0)
at time 0, and in the following nodes fht 2 Pt
V~ (t) = p (t)
for t = 1; 2; :::; T:
Hence
K
(1 + r)T 0t
0 S (t) + c (t) = p (t)
i.e.
c (t) 0 p (t) = S (t) 0
K
(1 + r)T 0t
for t = 0; 1; 2; :::; T
which is actually the Put-call Parity.
We can exploit the Put-call Parity to derive the formula for p (0) starting
from the one obtained for c(0) :
p (0) = c (0) 0 S +
=
K
(1 + r)
T
K
(1 + r)T
11
0
0
(1 0 8 (a; T; q)) 0 S 1 1 0 8 a; T; q 0
134
CHAPTER
11.
THE MULTI-PERIOD BINOMIAL MODEL
The factor multiplying the discounted strike price, (1 — ®(a;T,q)), is the
risk-neutral probability that at most a—1 upward movements occur, i.e. the
risk-neutral probability that the put option will end up in/at the money. If
we collect the discount factor in the previous formula, we get
p(0) =(1+r)-7 [K(1- 8 (aT, g)) — $(1+r)7 (1-8 (@7,¢))|
and the term
S(1+r)7 (1-6 (a;T7,¢))
is the risk-neutral expected value of a random variable equal to S(T) if
S(T) < K, to zero otherwise.
Before concluding this section, we remark that the Put-call Parity ob-
tained in the multi-period binomial model holds in more general contexts.
Indeed, the portfolio replicating the put option can be set up every time
there are available the risk-free asset, the underlying asset (independently
of its random behaviour) and a call option on the same underlying, with
same
maturity
and same strike price.
We
then only need to impose the
absence of arbitrage (more precisely, the law of once price) in the extended
market.
Part III
Continuous-Time Financial
Markets
Chapter 12
Stochastic Processes in
Continuous Time
12.1
Trajectories and Measurability of Stochastic
Processes in Continuous Time 1
A continuous time stochastic process is a family of random variables indexed
by t 2 [0; T ] or t 2 <+ : In Chapter 6, we have dealt with discrete time
stochastic processes, for which the parameter t belongs to the index set
f0; 1; :::; T g: We also introduced the concept of information structure to
describe the ‡ow of information available in the market as time goes by.
We refer to Chapter 6 for the precise de nitions and simply recall that the
information structure is a collection of ner and ner partitions on the set
of states of the world, one of which will be revealed true at time T . In
order to describe the evolution of information in continuous time, we need a
tool more powerful than an information structure, i.e. the one of ltration.
Its de nition is in turn based on the notion of sigma-algebra, exactly as the
information structure was based on the notion of partition of . Let us
provide the following de nitions.
De nition 60 (0Algebra) A 0algebra (read “sigma-algebra”) of
a collection 6 of subsets of satisfying the following properties:
is
1. ? 2 6;
2. If A 2 6 then Ac 2 6, where Ac is the complement of A in
1
;
The symbol 3 will indicate the sections of the chapter that are not part of the syllabus.
137
138
CONTINUOUS-TIME STOCHASTIC PROCESSES
3. If An 2 6 for all n 2 N then also [+1
n=1 An 2 6
Property 2. says that a sigma-algebra 6 is closed with respect to complementation in
: if a set belongs to 6, then also its complement does.
Property 3. stands for closure with respect to countable unions. So, not
only nite unions of elements of 6 are again in 6; but also countable unions
are. Property 1. together with 2. yields that also the whole state space
is in 6:
Sigma-algebras on an in nite
replace the partitions we used in the
discrete time setup. The concept of measurability of a function with respect
to a sigma-algebra is formalized in the following
De nition 61 A function Y :
! < is measurable with respect to the
sigma-algebra 6 on (i.e. Y is a random variable on endowed with the
sigma-algebra 6) if for every open interval (a; b) < the following holds:
Y 01 ((a; b)) 2 6
The technical notion of measurability formally translates the idea of
knowing the random variable Y based on the information described by the
sigma-algebra 6: The counter-image (through Y ) of any interval of real
numbers in which Y takes values is indeed an element of the sigma-algebra
6:
We call F the sigma-algebra containing the information carried by all
functions we are going to consider on . Unless otherwise speci ed, in the
future a random variable will denote a F0measurable function on .
The sigma-algebra F, the ner available on , will represent the nal
information achievable by investors, who gradually gather information as
time goes by and re ne their knowledge about the market. In order to
better specify such concept, we introduce an ordering based on neness,
exactly as we did for partitions.
De nition 62 Given two sigma-algebras 6 and 60 on , we say that 60 is
ner than 6 if for any element A 2 6 we have that A 2 60 .
The de nition has the same interpretation of the one given in the case
of partitions. The ner a sigma-algebra, the more detailed the information
carried. In the ner sigma-algebra 60 we nd all the bricks that we need
to construct (with unions) the bigger elements of the the sigma-algebra 6:
Hence, if the set A is known based on the information entailed in 6; i.e.
SAMPLE PATHS AND MEASURABILITY
139
A 2 6, then the set A is still known based on the information entailed in
60 ; because A 2 60 :
We introduce a simple example to see how the de nitions we gave int he
discrete-time setup apply to the continuous-time framework.
Let P be the partition of = f! 1 ; ! 2 ; ! 3 g given by
P = ff! 1 g ; f! 2 ; ! 3 gg :
The sigma-algebra generated from the partition P; (P), is constituted by
the unions and the complements of all the subsets of that are the elements
of P :
(P) = f?; ; f! 1 g ; f! 2 ; ! 3 gg :
Consider now a new partition P 0 of
= f! 1 ; ! 2 ; ! 3 g
P 0 = ff! 1 g ; f! 2 g ; f! 3 gg :
P 0 is ner than P: The sigma-algebra generated from the partition P 0 is
0 1
P 0 = f?; ; f! 1 g ; f! 2 g ; f! 3 g ; f! 2 ; ! 3 g ; f! 1 ; ! 3 g ; f! 1 ; ! 2 gg :
The sigma-algebra (P 0 ) is ner than the sigma-algebra (P) following the
de nition for sigma-algebras given above.
We are now ready to introduce the concept of ltration, which will replace
that of information structure in the sequel and describe how information
evolves as time goes by.
De nition 63 Filtration A ltration of
is a family fF t gt2[0;T ] of increasingly ner sigma-algebras on . In particular, we will assume in what
follows that
F0 is the trivial sigma-algebra, i.e. F0 = f? ; g;
FT = F is the nest sigma-algebra on
for all t, Ft contains the subsets of
;
having null probability measure.
One usually refers to the entire family of sigma-algebras fF t gt2[0;T ] with
the symbol F. As in the discrete time case, we now provide the de nition of
stochastic process adapted to the ltration fF t gt2[0;T ] :
De nition 64 A stochastic process X = fX(t)gt2[0;T ] , with X(t) : ! <
for each t, is said to be adapted to the ltration F= fF t gt2[0;T ] if the random
variable X(t) is Ft 0measurable for all t 2 [0; T ].
140
CONTINUOUS-TIME STOCHASTIC PROCESSES
Hence a process X = fX(t)gt2[0;T ] is said to be adapted to the ltration
F= fF t gt2[0;T ] if, as in the discrete time setting, for all t 2 [0; T ] the random
variable X(t) is measurable with respect to Ft , i.e. it is known on the basis
of the information available at time t.
We will denote the dependence of X on t indi¤erently with X(t) or Xt :
The same applies when the time horizon is in nite, i.e. when t 2 <+ :
The concept of ltration will pervasively replace that of information
structure, so that for example we will take conditional expectations with
respect to ltrations instead of information structures.
As far as the computation of conditional expected values is concerned,
there is no easy recipe as in the discrete time case. Given a random variable
Y on and integrable with respect to the probability Q on , one can prove
that there exists an Ft 0measurable random variable, called Ft 0conditional
expected value of Y under Q, denoted by
E Q [ Y j Ft ] ;
that has the properties stated in Proposition 33. In particular, we recall
that if Y is Ft 0measurable then E Q [ Y j Ft ] = Y; that is the best forecast of
the Ft 0measurable random variable Y based on the information Ft is Y
itself.
What happens if the random variable Y is indipendent from the information Ft ? Formally, a random variable Y is independent from Ft under the
probability measure Q if
Q [(Y 2 (a; b)) \ A] = Q [Y 2 (a; b)] 1 Q [A]
for any (a; b) < and for any A 2 Ft :
In this case the conditional expectation collapses into the unconditional
expectation, as stated in the following:
Proposition 65 If Y : ! < is a measurable random variable independent
of Ft under Q, then E Q [ Y j Ft ] = E Q [Y ]
Intuitively, if a random variable is independent of the information carried
by Ft ; the events in Ft are of no use in improving our prediction of Y: Hence,
the Ft -conditional expected value does not add anything to the prediction
that can be computed on the basis of the poorer information, represented
by F0 : The message of the previous Proposition is that the two predictions
coincide.
SAMPLE PATHS AND MEASURABILITY
141
We will also speak of martingales with respect to a ltration: the de nition is the same as in the discrete time case, but with a ltration clearly
replacing the information structure.
For a xed ! 2 ; by observing the evolution of X(t)(!) as t varies in
[0; T ] we can draw the trajectory (or path, realization) of X on ! :
Figure 12.1: Trajectory of a continuous process.
In Figure 12.1 we see an example of continuous path (the function X(t)(!)
is continuous in t 2 [0; T ]). In the following gure instead (Figure 12.2), an
example of trajectory with jumps at times T1 ; :::; T5 is provided.
As was already mentioned at the beginning of this section, the fact that
the process X is adapted to the ltration F makes the realization of X(t)
known once the information Ft (i.e. the whole history up to, and including,
time t) is revealed. For continuous time processes additional measurability
properties are important. When for all t the realization of X(t) is determined
on the basis of the whole history up to, but not including, time t, the process
X is said predictable. Predictability is a key property when the trajectories
of the process may have discontinuities. The processes employed to describe
the dynamics of securities subject to the risk of default typically allows for
discontinuous trajectories. In the celebrated Black-Scholes model instead,
asset price trajectories are continuous. The continuity assumption enables
to consider the information available simply up to t; including or excluding
t. We will limit ourselves to the study of continuous processes (i.e. with
continuous trajectories) and hence will not delve into issues of measurability
beyond adaptedness.
142
CONTINUOUS-TIME STOCHASTIC PROCESSES
Figure 12.2: Trajectory of a process with jumps.
12.2
Wiener Processes
The process underlying our models is the (standard) Brownian Motion or
(standard) Wiener process, which we now de ne.
De nition 66 (Wiener Process) The process W = fW (t)gt2[0;T ] , adapted to the ltration F, is a standard Brownian Motion (B.M.), or standard
Wiener process under the probability measure P and with respect to the ltration F if the following properties are satis ed:
1. W (0) = 0;
2. W (1)(!) : [0; T ] ! < is a continuous function for all2 ! 2
process is continuous;
i.e. the
3. for any chosen times 0 = t0 < t1 < ::: < tn T , the increments
W (t1 ) 0 W (t0 ); :::; W (tn ) 0 W (tn01 ) are mutually independent;
P
4. for all 0 s < t T , the increment W (t) 0 W (s) v N (0; t 0 s)
(where N (m; v) denotes the distribution of a normally distributed random variable under P, with mean m and variance v):
For all s; t 0, the increment of the process W from t to t + s is distributed under P as a normal random variable with zero mean and variance
2
One should more correctly say almost surely, i.e. apart from a set of null measure,
that is for all ! 2 nA with P (A) = 0:
12.2. WIENER PROCESSES
143
equal to the length of the time interval considered, i.e. W (t + s) 0 W (t) v
N (0; s) : Furthermore, such increment W (t + s) 0 W (t) is independent of Ft
under P.
The adjective standard in the previous de nition refers to the fact that in
the standard Brownian motion the increments are distributed as a Normal
random variable with mean zero and variance equal to the length of the time
P
P
interval considered, i.e. W (t) v N (0; t) ; W (t + s) 0 W (t) v N (0; s) :
We now provide some brief comments on the properties enjoyed by the
standard Wiener process:
Figure 12.3: Trajectories of the process W and standard deviation of W (t)
as a function of t
1. Property 1. says that at the starting time t = 0 the value W (0) is
known and equal to zero.
2. Property 2. requires the trajectories to be continuous. In Figure
12.3 we have drawn some trajectories of the process W: They are all
continuous, but looking at them closely we notice that they are pretty
144
CONTINUOUS-TIME STOCHASTIC PROCESSES
rough sequences of points: every trajectory switches from increasing
to decreasing trends, with such speed that even if we could observe it
more closely we would notice the same behavior. The trajectories of W
are indeed fractals. In particular, the trajectories of the process W are
not di¤erentiable at any time t: This aspect has serious consequences
in the development of the theory of stochastic integration, which will
be examined later.
P
3. Property 3. characterizes the distribution of W (t) v N (0; t) : When t
varies, so does the variance of W (t), which increases linearly with time.
Its standard deviation,
depicted in Figure 12.3 with the label “std”,
p
is thus equal to t. The linear growth with t of the variance explains
why the Wiener process can take any value, even huge, positive or
negative.
4. Property 4. and 3. yield the independence and stationarity of the increments: for any times t0 = 0 < t1 < :::: < tn , the random increments
W (ti+1 ) 0 W (ti ), with i = 0; :::; n 0 1, are mutually independent and
P p
distributed according to W (ti+1 ) 0 W (ti ) v ti+1 0 ti 1 N (0; 1) ( the
p
standard deviations of the increments are then equal to ti+1 0 ti ):
Remark 67 (3) It is possible to give a more “essential” de nition of Brownian
motion: given the probability space ( ; F; P ); the process B = fB(t)gt2<+ is
a Brownian motion (B.m.), or Wiener process (with respect to F and under
the probability P ) if the following hold:
1. it is adapted to F;
2. it has continuous trajectories;
3. it has independent and stationary increments, i.e. for all s t the
increment Bt 0Bs is independent of Fs and further Bt 0Bs and Bt0s 0
B0 have the same3 distribution under P:
(3) The properties stated above induce the distribution of the increments
of the process. Indeed, one can show (not easily) that the generic increment
Bt 0 Bs is distributed under P as a Normal
N (m 1 (t 0 s); v 1 (t 0 s))
3
Hence, the increments are stationary when their distribution only depends on the
length of the time interval considered, and not on its endpoints.
12.2. WIENER PROCESSES
145
When m = 0 and v = 1, we are back to the standard Brownian motion.
The fact that normality (also called gaussianity) of the process is implied by the requirements of continuity, independence and stationarity of
the increments, sheds light upon the wide use of such process in nance. In
the Black-Scholes model, the continuously compounded returns of the stock
are independent, identically distributed as a Normal with mean and variance proportional to the length of the time interval. Actually, every time
the continuously compounded returns of a stock are continuous, indipendent
and stationary, they are automatically normal or Gaussian.
Remark 68 We note that the probability measure P enters only in the
Properties 3. and 4. of the de nition of W . By employing another probability measure, neither the independece nor the distribution of the random
variables considered (W (t) or the increments W (t) 0 W (s)) is in general
preserved. This remark is essential when looking at the changes of probability measure discussed in the multi-period case, which we will thoroughly
examined in the sequel.
De nition 69 Generalized Brownian Motion We call generalized Brownian
motion, or generalized Wiener process, the process B given by
p
Bt = mt + vWt ;
where W is a standard Brownian motion.
The process B has the following properties. The process is null at t = 0;
since W (0) = 0; the trajectories of B are continuous (because so are those of
p
W ) and the generic increment is given by Bt 0Bs = m(t0s)+ v(Wt 0Ws ):
p
P
P
Since Wt 0 Ws v N (0; t 0 s) ; the product v(Wt 0 Ws ) v N (0; v(t 0 s))
and hence by adding the constant m(t 0 s); we have that
P
Bt 0 Bs v N (m 1 (t 0 s); v 1 (t 0 s))
The Wiener process B is sometimes called “generalized” to distinguish it
from the standard Wiener process W: In what follows, unless otherwise
stated, W will be referred to simply as a Brownian motion or Wiener process,
dropping the adjective standard.
In the next chapter, a Wiener process W will be assumed to represent
the source of randomness a¤ecting the nancial market. Observing security
prices will then be equivalent to observing the Wiener process. As a consequence, the information available to investors will be simply described by
the ltration generated by W , which is de ned below for the curious reader.
146
CONTINUOUS-TIME STOCHASTIC PROCESSES
Example 70 (Filtration Generated by the Wiener Process) (3) The
ltration generated by the Wiener process W , indicated by
9
8
FW = FtW t2[0;T ]
is the smallest ltration satisfying the following properties:
1. For all t we have that Wt is FtW 0measurable.
2. For any t < t1 < :::: < tn , the random increments
W (t1 ) 0 W (t) ; W (t2 ) 0 W (t1 ) ; :::; W (tn ) 0 W (tn01 )
are independent of FtW :
Such ltration can be constructed as follow: for xed t, we put in FtW
all subsets of of the type
f! 2
: W (s)(!) 2 Ag
where A belongs to the open subsets of < and s 2 [0; t]: Then, we add the
other subsets of necessary for FtW to be a sigma-algebra.
In this way, FtW contains the overall information linked to the observation of W up to t:
From now onwards, we will always refer to the ltration generated by
W
the standard Wiener process W; which will be simply denoted by Ft , F t :
The standard Brownian motion satis es the following important property:
Proposition 71 Let W be a standard Wiener process under P: Then W is
a martingale under P with respect to its own generated ltration.
Proof. In order to show that W is a martingale, we need to verify that,
for all s < t
E P [ W (t)j Fs ] = W (s);
that is
E P [ W (t) 0 W (s)j Fs ] = 0;
where E P [1] indicates expectation under the measure P: Since W (t) 0 W (s)
is independent of Fs ; the previous conditional expected value reduces to
E P [W (t) 0 W (s)]
P
Since W (t) 0 W (s) v N (0; t 0 s) the above expected value is zero, thus
completing the proof. 4
12.3. DIFFUSIONS AND STOCHASTIC INTEGRATION
12.3
147
Di¤usions and Stochastic Integration
In the following chapters we will assume that the dynamics of asset prices
are described by di¤usions. Before giving the formal de nition of such class
of processes, we examine their properties. Given W , Brownian motion on
( ; F; P ); the dynamics of a di¤usion process X on ( ; F; P ) (adapted to F)
is described locally by the following expression:
1X(t) = X(t + 1t) 0 X(t) = a(t; X(t))1t + b(t; X(t))1W (t)
where 1W (t) = W (t + 1t) 0 W (t): The increment 1X(t) of X between t
and t + 1t is speci ed through two terms:
1. the rst one, a(t; X(t))1t; is an increment, known on the basis of
the information available at time t, that determines4 the value X(t);
a(t; X(t))1t is then locally deterministic, i.e. the time-t information
is enough to know its behavior over [t; t + 1t[.
2. the second one, b(t; X(t))1W (t); is an increment unknown on the
basis of the information available at time t: Indeed, the information
Ft available in t allows to determine b(t; X(t)) but not the increment
1W (t), since 1W (t) depends on W (t + 1t) and hence on the information carried by Ft+1t , which will become known at time t + 1t: The
coe¢cient b multiplying 1W (t) is called di¤usion coe¢cient.
If b = 0; the variation 1X(t) = a(t; X(t))1t leads for 1t ! 0 to the
following expression of X in di¤erential terms
dX(t) = a(t; X(t))dt;
where the conditional variation of X, given the information Ft ; is deterministic over the in nitesimal time interval (t; t + dt). By employing the usual
di¤erentiation rules, this can be equivalently expressed as:
dX(t)
= a(t; X(t))
dt
The last expression is a di¤erential equation. Knowing that the derivative
of X(t) with respect to t is equal to a(t; X(t)); we need to determine the
4
Here is a reason why we need to make sure that a process is adapted.
148
CONTINUOUS-TIME STOCHASTIC PROCESSES
unknown function X(t). The equation is simple to solve (at least formally),
because we just need to integrate both sides
Z Z dX(t)
dt =
a(t; X(t)) dt
dt
0
0
to get
Z
X( ) 0 X(0) =
0
a(t; X(t)) dt
(12.1)
R If, for example,R a(t; X(t)) is constant and equal to , X( ) = X(0) +
dt = : This is a simple example, where the integral
0 a(t; X(t)) dt = 0
can be explicitly solved, since the term a(t; X(t)) does not depend on X(t)
nor on t: The dependence on X(t) is the most problematic to deal with,
because in order to determine X( ) on the basis of (12:1) we need to know
the whole history of X up to ; so that we basically should already have
found the unknown function X(t): In the applications of interest, we will
employ changes of variable formulas to reduce the problem to equations of
the type
Z
X( ) 0 X(0) =
0
(t) dt;
in which a(t; X(t)) = (t) does not depend on X(t):
There is an additional problem that has not been disclosed so far, i.e. the
dependence of X on ! (we recall that X is a family of random variables). We
could overcome it by repeating the same steps seen above for every single !,
i.e. working on a path by path basis. That would be ne for the moment, but
we would be in big trouble as soon as b 6= 0, because the integral equation
Z Z Z dW (t)
dX(t)
dt =
dt
a(t; X(t)) dt +
b(t; X(t))
dt
dt
0
0
0
requires to compute the derivative of W with respect to t,
dW (t)
:
dt
Unfortunately, one can prove that such derivative does not exist. We could
see this immediately if we could look very closely to any sample path of
W : a continuous line made of an in nity of rough in nitesimal peaks, thus
never di¤erentiable on [0; T ]. We can overcome the problem by giving up
the search for path by path solutions, and going back to the original problem
dX(t) = a(t; X(t))dt + b(t; X(t))dW (t);
12.4. CONSTRUCTING THE STOCHASTIC INTEGRAL
Figure 12.4: The simple process
149
and its integral in dt:
trying to understand how to de ne the integral
Z (t)dW (t)
0
for processes = f (t)gt2<+ for which the dependence on ! (dropped only
notationally) satis es suitable regularity conditions and encompasses the
important case (t)(!) = b(t; X(t)(!)):
12.4
Constructing the Stochastic Integral
(3) We begin by considering the very simple case in which is piecewise
constant. We then suppose that there are times t0 = 0 < t1 < ::: < tn = such that (t)(!) = (tk )(!) for all t 2 [tk ; tk+1 [: Such a process is called
simple, and a typical
R trajectory is depicted in Figure 12.4. The integral
with respect to time 0 (t)dt is the area underlying the graph of the generic
sample path of : As can be seen from Figure 12.4, such area (lightly shaded)
is equal to
Z n01
X
(t)dt =
(tk ) 1 (tk+1 0 tk )
(12.2)
0
k=0
Such integral is in turn a random variable, because the above equality must
be read path by path, i.e.
Z
0
(t)(!) dt =
n01
X
k=0
(tk )(!) 1 (tk+1 0 tk )
150
CONTINUOUS-TIME STOCHASTIC PROCESSES
Figure 12.5: The simple process
and its integral in dW (t):
for each ! 2 : Going back to the problem of de ning an integral with
respect to W , we can do it for the simple process in the very same way as
in (12:2), by setting
Z
0
(t) dW (t) :=
n01
X
(tk ) 1 (W (tk+1 ) 0 W (tk ))
(12.3)
k=0
We now try to better understand the intuition behind de nition5 (12:3).
The
R area underlying the generic piecewise constant trajectory of , i.e.
0 (t)dt de ned by (12:2) ; is made up of the areas of rectangles with height
(tk ) andR base length equal to the timespan 1tk = tk+1 0 tk : In order to
compute 0 (t) dW (t), we need to employ a di¤erent clock: we assign the
measure W (t) to time t, as can be seen6 from Figure 12.5. According to
our clock, the length of the interval with endpoints tk and tk+1 is equal
to W (tk+1 ) 0 W (tk ): The process is constant and equal to (tk ) over the
same interval. The only di¤erence between (12:2) and (12:3) is then the base
length of each single rectangle, which is equal to tk+1 0 tk in the rst case,
to W (tk+1 ) 0 W (tk ) in the second case. By summing the areas obtained
(with the base measured with the clock W ) we obtain formula (12:3), which
is graphically represented by the shaded area in Figure 12.5.
5
It should be noted that the two (de ning) formulas (12:2) and (12:3) do not lead to
the same result!
6
Note that in Figure 12.5 we have depicted the very special case in which W (t1 ) <
W (t2 ) < ::: < W (t5 ): Since the paths of W are not monotone, it is not true in general
that t1 < t2 implies W (t1 ) < W (t2 ):
12.4. CONSTRUCTING THE STOCHASTIC INTEGRAL
151
Clearly, expression (12:3) must be read path by path:
Z
0
(t)(!) dW (t)(!) =
n01
X
(tk )(!) 1 (W (tk+1 )(!) 0 W (tk )(!))
k=0
so that we can notice that now ! a¤ects not only (tk ) (the rectangles
height), but also W (tk+1 ) 0 W (tk ) (the base length measured with W ):
We are then working path by path: where is then the dead end (mentioned
at the end of the previous section referring to the di¤erentiability of W with
respect to t) we have been trapped in by following the geometric intuition?
The problem is that, provided we work with a nite number of terms, i.e.
the integral translates into a nite summation, everything is ne. As a
consequence, considering simple processes poses no problem. But how about
processes that do not have piecewise constant trajectories? The solution
can be achieved in three steps:
1. we approximate the process
with simple processes
lim
n!1
2. for each n, we compute the integral
(12:3);
n
n;
so that
=
R
0
n (t)
dW (t) de ned by formula
3. we prove that the sequence of integrals computed in the previous step
converges and we de ne
Z Z (t) dW (t) := lim
n (t) dW (t):
n!1 0
0
Steps 1. and 3. hide a mathematical complication residing on the type
of limit that is used. Indeed, such limit is not taken path by path (as
already mentioned: we would end up in a dead end), but jointly considers
the dependence of on t and !: The three steps can be formally described
as follows:
Claim 72 (3) Let us consider the family of processes g = fg(t)gTt=0 adapted
to F and such that
Z E[g 2 (t)] dt < 1
0
Let us then denote by L2 [0; ] the set of such processes. We say that
if 2 L2 [0; ] for all > 0:
2 L2
152
CONTINUOUS-TIME STOCHASTIC PROCESSES
1. For each given L2 , one can show that there exists a sequence of
processes n 2 L2 approximating , in the sense that
Z h
i
E ( n (t) 0 (t))2 dt = 0:
lim
n!1 0
R
2. For each n , the stochastic integral In = 0 n (t) dW (t) is wellde ned and the sequence of the In ’s can be proved to converge. More precisely, there exists a random variable I such that
i
h
lim E (In 0 I)2 = 0:
n!1
3. One can show that the limit I does not depend on the chosen approximating sequence of processes n , so that we can set
Z (t) dW (t) := I
0
The integral de ned in this way enjoys some important properties described in the following proposition.
Proposition 73 Let
be an F0adapted process such that
Z T
E[ 2 (t)] dt < 1
0
The following properties are satis ed:
8R 9
1. the process 0 (t) dW (t) 2[0;T ] is F0adapted:
Z
0
(t) dW (t) is F 0 measurable
R
2. for all 2 [0; T ], the integral 0 (t) dW (t) has null expectation:
Z (t) dW (t) = 0
E
0
3. the above expectation being zero, the variance of
by
"Z
2 # Z 2
=
(t) dW (t)
E
E
0
0
R
0
2
(t) dW (t) is given
3
(t) dt
12.5. ITO PROCESSES
153
9
8R 4. the process
0 (t) dW (t) 2[0;T ] is an F0martingale, i.e. for all
0 s < T we have
Z Z s
E
(t) dW (t) Fs =
(t) dW (t)
0
0
5. RSince stochastic integrals
R have zero mean value, the covariance between
(t)
dW
(t)
and
0 1
0 2 (t) dW (t) is given by
Z Z Z =
E [ 1 (t) 1 2 (t)] dt
E
1 (t) dW (t) 1
2 (t) dW (t)
0
0
0
Property (4) is remarkably important: if the process satis es the assumptions in the previous proposition, its stochastic integral (i.e. with respect to W ) is a martingale. We will see that a kind of converse is also true:
under suitable integrability conditions, every continuous martingale can be
written as a stochastic integral with respect to W . Continuous martingales
are thus closely linked to stochastic integrals.
R
Remark 74 (3) The assumption 0 E[ 2 (t)] dt < 1 for all is not necessary for the stochastic Rintegral to be de ned, and can be weakened by
T
requiring for example that 0 2 (t)(!) dt < 1 for all ! 2 (possibly apart
from a set of null 8R
measure). Under
9 such weaker assumption, however, the
stochastic integral 0 (t) dW (t) 2[0;T ] is not guaranteed to be a martingale anymore, nor properties 2. and 3. are guaranteed to hold.
12.5
Ito Processes
Ito processes are a wide class of processes de ned on ( ; F; P ). Let us see
how they are de ned:
De nition 75 Let us consider the space ( ; F; P ) and let W be a Wiener
process on it. We then say that X = fX(t)gt2[0;T ] is an Ito process if it can
be represented as
Z t
Z t
X(t) = x0 +
(s) ds +
(s) dW (s)
(12.4)
0
0
where xo is a real number and ; are F0adapted processes such that
RT
1 e 0 j (s)(!)j2 ds < 1 for all ! 2 almost surely.
RT
0
j (s)(!)j ds <
154
CONTINUOUS-TIME STOCHASTIC PROCESSES
We stress that formula (12:4) is to be read path by path, i.e.
Z t
Z t
(s) ds (!) +
(s) dW (s) (!) =
X(t)(!) = x0 +
0
0
Z t
Z t
(s)(!) ds +
(s) dW (s) (!)
= x0 +
0
Rt
0
for each ! 2 : The stochastic integral 0 (s) dW (s) is a random variable
and hence depends on !; but since it is not constructed path by path (simple
processes apart)
we cannot separate the dependence on ! from that on t, as
Rt
we did for 0 (s)(!) ds:
Formula (12:4) is often referred to as decomposition of the Ito processX:
It is also common to replace the integral notation in (12:4) with the
shorthand di¤erential notation
dX(t) = (t) dt + (t) dW (t)
(12.5)
Formula (12:5) points directly to expression (12:4) : Actually, the process X
admits the stochastic di¤erential (12:5) if and only if X is of the
R tform (12:4) :
Formula (12:4) de nes X in terms of the path by path integral 0 (s) ds and
Rt
of the stochastic integral 0 (s) dW (s): We have seen that the impossibility
to di¤erentiate W (t) with respect to t on a path by path basis does not allow
to provide an immediate meaning to the di¤erential term (t) dW (t) and
it is for this reasonR that we had to go through the three step procedure in
t
order to construct 0 (s) dW (s): Hence, the di¤erential formula (12:5) has
the meaning of a shorter notation for the integral formula (12:4).
In the previous paragraph we have mentioned that the stochastic integral
is a martingale and we have hinted at the converse of such statement. Let
us now write this formally:
Proposition 76 Let X = fX(t)gt2[0;T ] be an Ito process with decomposition
(12:4) and suppose that it is a martingale. Then
Z t
X(t) = x0 +
(s) dW (s)
0
i.e.
(t) = 0 per t 2 [0; T ]
Under suitable integability conditions, if an Ito process X is a martingale,
then for its di¤erential we have necessarily
dX(t) = (t) dW (t)
12.6. ITO’S FORMULA(*)
155
which coincides with the stochastic integral.
We do not provide a thorough proof of the proposition, we rather give
an intuitive idea of how it works. We consider the increment between t and
t + 1t of the Ito process X :
1X(t) = (t) 1t + (t) 1W (t)
Because X is a martingale by assumption, we have
E [ 1X(t)j Ft ] = E [ X(t + 1t) 0 X(t)j Ft ] =
= E [ X(t + 1t)j Ft ] 0 E [ X(t)j Ft ] =
= X(t) 0 X(t) = 0
As a result,
0 = E [ 1X(t)j Ft ] = E [ (t) 1t + (t) 1W (t)j Ft ]
Since (t) 1 1t and (t) are Ft 0measurable, we have that
0 = E [ (t) 1tj Ft ] + E [ (t) 1W (t)j Ft ] =
=
(t) 1 1t + (t) 1 E [ 1W (t)j Ft ]
and recalling that W is an F0martingale under P , since E [ 1W (t)j Ft ] = 0
we obtain
(t) 1 1t = 0
from which (t) = 0 follows.
12.6
Ito’s Formula(*)
The stochastic integral has allowed us to de ne the Ito processes, i.e. a wide
class of processes on ( ; F; P ) with which however we are not very operative
yet. We stress once again that the di¤erential writing (12:5) of an Ito process
has the mathematical meaning of shorthand notation for the integral form
(12:4). If X is an Ito process and f : < ! < is a di¤erentiable function,
what can we say about the process Y (t) = f (X(t))? If = 0 in (12:4) ; we
are back to the deterministic case: by employing the chain di¤erentiation
rule, we can show that the di¤erential of Y is given by
dY (t) = f 0 (X(t))dX(t) =
= f 0 (X(t)) (t) dt
156
CONTINUOUS-TIME STOCHASTIC PROCESSES
where the equality is to be read path by path, i.e.
dY (t)(!) = f 0 (X(t)(!)) (t)(!) dt
for ! 2 : However, if 6= 0, the di¤erential of X involves a stochastic term
as well and hence cannot be solved on a path by path basis. The solution
is given by the celebrated Ito’s Formula, which we state for a slightly more
general case allowing for explicit dependence on time.
Lemma 77 (Ito’s Formula) Let fX (t)gt2[0;T ] be an Ito process given by
(12:4) and let ' : <+ 2 < ! < be di¤erentiable with continuity, once with
respect to the time variable, denoted by t, and twice with respect to the second
variable, denoted7 by x: Let
Y (t) = ' (t; X(t))
Then, Y (t) is an Ito process as well and its integral decomposition is
given by:
Zt "
Y (t) = Y (0) +
0
Zt
+
0
@' (s; x)
@s
1 @ 2 ' (s; x))
2
@x2
@' (s; x)
+
@x
(s;X(s))
2
(s;X(s))
Zt
(s) ds +
0
or, in di¤erential notation, by
"
@' (s; x)
@' (s; x)
dY (t) =
+
@s
@x
(t;X(t))
+
1 @ 2 ' (s; x))
2
@x2
2
(t;X(t))
#
(s;X(s))
1 (s) ds +
@' (s; x)
@x
(s;X(s))
(s) dW (s)
#
(t;X(t))
(t) dt +
1 (t)+ dt
@' (s; x)
@x
(t;X(t))
which will written for short as
1 @ 2 '(t; X(t))
@'(t; X(t)) @'(t; X(t))
+
1 (t) +
dY (t) =
@t
@x
2
@x2
@'(t; X(t))
(t)dW (t)
+
@x
7
(12.6)
(t)dW (t)
2
(t) dt +
A function continuously di¤erentiable once with respect to the rst variable and twice
with respect to the second is said to be of class C1;2 (<+ 2 <)
12.7. STOCHASTIC DIFFERENTIAL EQUATIONS
12.7
157
Stochastic Di¤erential Equations
At the beginning of this chapter we introduced the so called di¤usion processes. We now formally de ne them as solutions to suitable stochastic
di¤erential equations. Let us consider the functions a; b : [0; T ] 2 < ! <;
the real number x0 2 < and a Brownian motion W on the space ( ; F; P ).
Let us further consider the equation
Zt
X(t) = x0 +
Zt
a(s; X(s)) ds +
0
b(s; X(s)) dW (s)
(12.7)
0
for t < T: Equation (12:7) is a stochastic di¤erential equation (SDE).
The functions a and b are called coe¢cients of the SDE, the term x0 the
initial value.
We can rewrite the equation in di¤erential terms as
dX(t) = a(t; X(t)) dt + b(t; X(t)) dW (t)
(12.8)
X(0) = x0
For a process X on ( ; F; P ) to exist and to satisfy uniquely equation (12:7)
(or (12:8)), some regularity conditions must be satis ed by the coe¢cients
a and b. The following theorem provides conditions for the existence and
uniqueness of solutions to (12:7).
Theorem 78 (Existence and Uniqueness of SDE Solutions) (3) Let
a; b : <+ 2 < ! < be continuous functions and let us suppose that there
exists a constant K 2 < such that for all x; y 2 < and all t 2 [0; T ] the
following hold
ja(t; x) 0 a(t; y)j + jb(t; x) 0 b(t; y)j K 1 jx 0 yj
ja(t; x)j + jb(t; x)j K 1 (1 + jxj)
Then, there exists a unique process X = fX(t)gt2[0;T ] on ( ; F; P ) satisfying equation (12:7) : The process X is said to be a di¤usion.
From now onwards, we will assume that the assumptions of the above
theorem hold.
The solution X to the SDE (12:7) has some important features:
1. its trajectories are continuous;
2. it is Markovian;
158
CONTINUOUS-TIME STOCHASTIC PROCESSES
Intuitively, the process X is Markovian when the future behavior of Xs ,
for s > t, only depends on Xt and not upon the whole history of the process
up to t: More precisely, let X be a solution to (12:7) and x t 2 (0; T ):
At time t, the solution to the SDE is X(t). Thus, starting from t, let us
consider the SDE having the same coe¢cients a and b as (12:7), given the
starting value X(t): Let us denote by Y = fY (s)gs2[t;T ] the solution of such
SDE, i.e.
dY (s) = a(s; Y (s)) ds + b(s; Y (s)) dW (s)
Y (t) = X(t)
One can then show that X(s) = Y (s) for all s 2 [t; T ]. As a consequence,
we just need X(t) to determine the evolution of X over s 2 [t; T ], while
the whole past values fX(s)gs2[0;t] can be disregarded. It then follows that,
for a given function f : < ! <, the expected value E [ f (X(T ))j Ft ] only
depends on X(t): This is an important characteristic that will be used in
the analysis of the Black-Scholes model. We nd it useful to provide the
following proposition.
Proposition 79 (3) Under the assumptions of the Theorem of Existence
and Uniqueness, we x t 2 (0; T ) and consider the SDE
dY (s) = a(s; Y (s)) ds + b(s; Y (s)) dW (s)
(12.9)
Y (t) = x
whose initial value at time t is Y (t) = x 2 <: For any
(12:9)
8 x,t;xequation
9
t;x
admits a unique solution which we write as X
= X (s) s2[t;T ] : Let
8
us2consider a 3function f : < ! <: For each x, we can compute (x) =
E f (X t;x (T )) : The Markov property ensures that
E [ f (X(T ))j Ft ] = (X(t))
(12.10)
The Markov property expressed by formula (12:10) is important, because
it helps to compute Ft 0conditional expected values. The latter are by de nition Ft 0measurable random variables and expression (12:10) is actually a
computational recipe:
2
3
We compute the (real) function (x) = E f (X t;x (T )) 2: For xed x;
3
we just need to know the law of X t;x (T ) to compute E f (X t;x (T )) ,
which is a real number. We then obtain (x):
8
The function f must be such that the expectations written below are well de ned. It
is enough, for example, that f is bounded.
12.7. STOCHASTIC DIFFERENTIAL EQUATIONS
159
The conditional expected value E [ f (X(T ))j Ft ] is given by the (real)
function (x) computed at x = X(t): By exploiting the dependence
on !; we can rewrite (12:10) path by path, i.e.
E [ f (X(T ))j Ft ] (!) = (X(t)(!))
for each ! in
:
For our nancial applications we will mainly employ Ito processes that
are solutions to SDEs of the type:
Zt
X(t) = x0 +
Zt
a(s; X(s)) ds +
0
b(s; X(s)) dW (s)
0
with a and b satisfying the assumptions of Theorem 78 (existence and
uniqueness of SDE solutions). We specialize Ito’s formula to these processes,
because of their fundamental importance.
Lemma 80 (Ito’s Formula) Let fX (t)gt2[0;T ] be an Ito process given by
(12:7), i.e.
dX(s) = a(s; X(s)) ds + b(s; X(s)) dW (s)
X(0) = X0
with a and b satisfying the assumptions of Theorem 78. Let ' : [0; T ]2< ! <
be continuously di¤erentiable, once with respect to the rst variable, denoted
by t; twice with respect to the second, denoted by x: Let
Y (t) = ' (t; X(t))
Then Y (t) is itself an Ito process and its integral decomposition is given by:
Zt "
Y (t) = Y (0) +
0
Zt
+
0
Zt
+
0
@' (s; x)
@s
1 @ 2 ' (s; x))
2
@x2
@' (s; x)
@x
@' (s; x)
+
@x
(s;X(s))
(s;X(s))
(s;X(s))
1 b2 (s; x)
#
(s;X(s))
(s;X(s))
ds +
b (s; x)j(s;X(s)) dW (s)
1 a(s; x)j(s;X(s)) ds +
160
CONTINUOUS-TIME STOCHASTIC PROCESSES
which can be written for short as
h
+ @'(t;X(t))
1 a(t; X(t)) +
dY (t) = @'(t;X(t))
@t
@x
i
1 @ 2 '(t;X(t)) 2
b (t; X(t))
2
@x2
b(t; X(t))dW (t)
+ @'(t;X(t))
@x
dt+
(12.11)
Chapter 13
The Black-Scholes Model
In this chapter we study the Black-Scholes option pricing model. In the rst
section we introduce the securities tradeable in our reference market, i.e. a
risk-free asset and a risky stock. Our main concern will be the analysis of
the risky security, whose dynamics will be driven by a Wiener process or
Brownian motion representing the source of randomness in the market. We
will employ what explained in the previous chapter to analyze the behavior
of the risky security price, e.g. to compute its expectation and variance over
a given time-horizon.
In the second section, we brie‡y describe the way in which investors
operate in the market, being now allowed to buy and/or short sell continuously over time the securities available. We will then discuss self- nancing
strategies in continuous time, drawing parallels with the multi-period setting.
In the third section, we deal with the no-arbitrage analysis of our nancial market, o¤ering some general remarks on the formulation of the First
Fundamental Theorem of Asset Pricing in continuous time and with an innite state space. We conclude with the analysis of market completeness of
the Black-Scholes model, providing the full proof that every European-type
derivative security can be replicated by suitably employing a continuous
time strategy based on the risk-free asset and the risky stock.
13.1
The Basic Securities
In the Black-Scholes model two securities are considered: a risk-free asset
and a risky stock.
The risk-free asset B has unitary price at time t = 0 (i.e. B(0) = 1) and
161
162
CHAPTER 13. THE BLACK-SCHOLES MODEL
time-t price given by B (t) = et , where the constant 2 <+ stands for the
dB (t)
= et , the di¤erential
instantaneous risk-free rate of interest. Since
dt
of B is given by
dB (t) = B (t) dt;
which, together with the initial condition B(0) = 1, univocally identi es the
risk-free asset.
The risky security S is a di¤usion on ( ; F; P ) whose dynamics is described by the SDE
dS (t) = S (t) dt + S (t) dW (t)
(13.1)
S(0) = S0
with and real positive constants.
The coe¢cient in equation (13:1) is usually called drift, while the
coe¢cient is instead called volatility.
Recalling now the de nition of di¤usions, the processes solving our SDEs,
the coe¢cients a; b : [0; T ] 2 < ! < of (12:7) are given by
a (t; S(t)) = 1 S (t)
b (t; S(t)) = 1 S (t)
Equation (13:1) is a short-cut notation for the following integral equation:
Zt
Zt
S (t) = S0 + S (s) ds + S (s) dW (s)
(13.2)
0
0
We can apply the Theorem of Existence and Uniqueness of SDE solutions, since the coe¢cients a and b trivially satis es the assumptions of the
theorem:
ja(t; x) 0 a(t; y)j + jb(t; x) 0 b(t; y)j K 1 jx 0 yj
ja(t; x)j + jb(t; x)j K 1 (1 + jxj)
since
ja(t; x) 0 a(t; y)j = jx 0 yj
jb(t; x) 0 b(t; y)j = jx 0 yj
ja(t; x)j + jb(t; x)j = (jxj + jxj)
so that
ja(t; x) 0 a(t; y)j + jb(t; x) 0 b(t; y)j = ( + ) 1 jx 0 yj
ja(t; x)j + jb(t; x)j = ( + ) 1 jxj
13.1. THE BASIC SECURITIES
163
and we can just set K = + :
For each xed initial value S(0) = S0 of the risky security, the SDE
(13:1) admits a unique solution S. The risky security price process is hence
well-de ned (it does exist and is unique).
But can we write such process in explicit form? We try to solve the SDE
(13:1): It is clear that the coe¢cients a and b do not depend on t, although
they are functions of S: We look for a suitable change of variable allowing us
to get to an SDE with coe¢cients not depending on the solution anymore.
Let us set
Y (t) = ln S (t)
and compute the di¤erential of Y: Since Y is a function of a di¤usion process
(hence an Ito process), we can employ Ito’s Formula. With the notation
employed in Ito’s Lemma, we can write:
X(t) = S(t)
a (t; S(t)) = 1 S (t)
b (t; S(t)) = 1 S (t)
' (t; S(t)) = ln S (t)
To apply Ito’s formula, we need the derivatives1 of the function ' with
respect to t and x:
1
@'
1 @2'
@'
= 0;
= ; 2 = 0 2;
@t
@x
x @x
x
which will then be computed in (t; S(t)): By applying Ito’s Lemma (12:11),
we get
Zt Y (t) = Y (0) +
0
Zt 1
1
S (s) +
S (s)
2
1
0 2
S (s)
Zt
S (s) ds +
2 2
0
1
S (s) dW (s)
S (s)
Zt
1
0 2 ds + dW (s)
2
0
0
1 2
= Y (0) + 0 t + W (t)
2
= Y (0) +
1
The function ' : <+ 2 <+ ! < is continuously di¤erentiable once with respect to the
rst variable and twice with respect to the second variable on the whole domain <+ 2 <+ :
In this region we can then use Ito’s Formula.
164
CHAPTER 13. THE BLACK-SCHOLES MODEL
Now, recalling that Y (t) = ln S (t), we have Y (0) = ln S0 and hence
1 2
Y (t) = ln S0 + 0 t + W (t)
2
We thus have an explicit expression for Y (t) = ln S (t) : To obtain S(t), we
just need to note that
1
ln S (t) = ln S0 + 0 2 t + W (t)
2
and hence we can take exponentials on both sides of the equality
1 2
S (t)
= 0 t + W (t)
ln
S0
2
to get
1 2
S (t)
= e(0 2 )t+W (t)
S0
so that we nally obtain
1 2
S (t) = S0 1 e(0 2 )t+W (t)
Since
we have
(13.3)
0
1
P
W (t) N 0; 2 t
1
1
P
0 2 t + W (t) N
0 2 t; 2 t
2
2
We say that S (t) is lognormally distributed, because its logarithm is a normally distributed random variable.
We now see how to nd some estimators of the parameters and characterizing the behavior of S: We denote by
(t) = ln
S (t)
;
S0
the continuously compounded return on the security S from time 0 to time t:
As we have seen, if the stock S satis es equation (13:1); then its logarithm
is a normal random variable. In particular, we have that its continuously
compounded return is given by
1 2
(t) = 0 t + W (t)
2
13.1. THE BASIC SECURITIES
165
so that it is a generalized Brownian motion. This is why S(t) is also called
geometric Brownian motion. Because of the properties of the process , its
expected value under the physical measure P is hence proportional to t :
1
E P [ (t)] = 0 2 t
2
and so is its variance
#
1 2 2 2
t =
V ar [ (t)] = E (t) 0 0 2
1 2
P
2
= E 2 1 0 t 1 W (t) + (W (t)) = 2 t
2
P
"
P
2
Hence, the expected value of the continuously compounded returns on the
stock S under the physical
measure
P is proportional to t with proportional1
0
ity factor equal to 0 12 2 : The variance under the physical measure P is
also proportional to t but with proportionality factor equal to 2 : Moreover,
since a generalized Brownian motion is a process with independent and
identically distributed increments, we can easily nd estimators for the statistics above. Let us then x the length 1t of the time intervals over which we
sample the price process S: For given t0 < t1 < ::: < tn with 1t = ti+1 0 ti ;
the increments 1(t0 ) = (t1 ) 0 (t0 ); :::; 1(tn01 ) = (tn ) 0 (tn01 ) are
independent and identically distributed normal random variables under the
physical measure P:
1 2
P
2
0 1t; 1t
1(ti ) = (ti+1 ) 0 (ti ) N
2
Let us then set 1t = 1 and collect the observations for 1 by computing
1(ti ) = ln
S (ti+1 )
S (ti )
S (ti+1 )
0 ln
= ln
S0
S0
S (ti )
for i = 0; :::; n01: The sample mean of the 1(ti )’s is an unbiased estimator
for 0 12 2 ; while the sample variance provides us with an estimate for 2 :
Estimates for the parameters and are thus obtained.
The assumption on the dynamics of S based on the SDE (13:1) leads to a
Normal distribution for the continuously compounded returns on the stock
S or, more precisely, to a generalized Brownian motion. The converse is
clearly true: by assuming the continuously compounded return on the stock
166
CHAPTER 13. THE BLACK-SCHOLES MODEL
S to be a generalized Brownian motion, we get to a lognormal distribution
for S: Indeed, if
(t) = t + W (t)
with and real constants, we just need to write S as a function of to
obtain immediately
S (t) = S0 1 e(t) = S0 1 e
t+ W (t)
which is exactly (13:3) with
1
= 0 2
2
= As an exercise, we now use Ito’s Formula to derive the SDE solved by S
starting from the assumption that the continuously return on S is a generalized Brownian motion.
Example 81 If the continuously compounded return on S at time t, (t),
is such that
S (t) = S0 1 e(t)
with given by
(t) = t + W (t)
(13.4)
what is the SDE solved by S in this case?
Solution. We have to compute the (stochastic) di¤erential S on the basis
of S (t) = S0 1 e(t) , i.e. by knowing that S is a function of the process given by (13:4) : Expression (13:4) can be rewritten in di¤erential form as
d (t) = dt + dW (t) :
To compute the di¤erential of S (t) = S0 1 e(t) , we set in Ito’s Lemma
(12:11)
X(t) = (t)
a (t; (t)) =
b (t; (t)) =
' (t; (t)) = S0 1 e(t)
13.1. THE BASIC SECURITIES
167
By computing the partial derivatives of ' with respect to the rst variable
t and to the second variable , we get
@2'
@'
@'
= 0;
= S 0 1 e ; 2 = S0 1 e ;
@t
@
@
so that from (12:11) we get (in di¤erential form for ease of notation):
@'
@'
1 @2'
2
dS (t) = 0 +
(t; (t)) +
(t; (t)) dW (t) =
dt +
(t; (t))
2
@
2 @
@
1
(t)
(t)
2
=
S0 1 e
S0 1 e
dt + S0 1 e(t) dW (t) =
+
2
1 2
= S(t) 1
+
dt + dW (t)
2
We thus nd again the SDE (13:1) solved by S in the Black-Scholes
model with parameters
=
+
=
:
1
2
2
Hence, assuming S to be lognormal is equivalent to assuming that the
continuously compounded return on S is a non standard or generalized
Brownian motion (13:4) : The parameters characterizing the stochastic dynamics of the two processes are linked through the two equations just written. We end this section by deriving the expected value and the variance of
S(t) under the measure P.
Exercise 82 By employing (13:3), show that under P the expected value of
S(t) is
E P [S(t)] = S0 1 et ;
that the second moment of S(t) is given by
3
2
2
E P S 2 (t) = S02 1 e2t+ t
and hence that the variance of S(t) is
2
V arP [S(t)] = S02 1 e2t e t 0 1 :
168
CHAPTER 13. THE BLACK-SCHOLES MODEL
Solution. Since under P the process W at time t is distributed as
P
W (t) N (0; t) =
p
t 1 Z;
P
with Z N (0; 1) ; from (13:3) it follows that
1 2
P
S (t) S0 1 e(0 2 )t+
and thus
p
t1Z
h
i
p
1 2
E P [S(t)] = E P S0 1 e(0 2 )t+ t1Z =
h 1 2 p i
= S0 1 et 1 E P e0 2 t+ t1Z =
Z
p
1 2
= S0 1 et 1
e0 2 t+ t1z fZ (z) dz;
<
where fZ (z) is the density of a standard normal random variable, i.e.
1 2
1
fZ (z) = p e0 2 z :
2
As such, we have
P
E [S(t)] = S0 1 e
t
= S0 1 e
t
Z
1
1
= S0 1 et ;
because
Z
<
Z<
<
1
e0 2 2 t+
p
t1z
1 2
1
p e0 2 z dz =
2
p 2
1
1
p e0 2 (z0 t) dz =
2
p 2
1
1
p e0 2 (z0 t) dz = 1:
2
p
The latter equality can be veri ed by setting y = z 0 t and noting that
Z
Z
1 2
1 0 1 (z0pt)2
1
p e 2
p e0 2 y dy = 1;
dz =
2
2
<
<
since the integral in dy is the integral over < of the density of a standard
Normal, thus equal to 1: If we want to avoid any change of variables, we can
observe that
Z
h
i
p 2
p
1
1
p e0 2 (z0 t) dz = P N ( t; 1) 2 < = 1
2
<
13.2. INFORMATION AND INVESTMENT STRATEGIES
169
since the rst term is thepintegral over the whole real line < of the density
of a Normal with mean t and variance 1:
Similarly, to compute the second moment of S we have
h
i
p
3
2
1 2
E P S 2 (t) = E P S02 1 e2(0 2 )t+2 t1Z =
h p i
2
= S02 1 e2t0 t 1 E P e2 t1Z =
Z
p
2
= S02 1 e2t0 t 1
e2 t1z fZ (z) dz;
<
where fZ (z) is the density of a standard Normal random variable, i.e.
1 2
1
fZ (z) = p e0 2 z :
2
As a result, by completing the square in z in the exponent:
Z
p
2
3
1 2
1
2
e2 t1z p e0 2 z dz =
E P S 2 (t) = S02 1 e2t0 t 1
2
Z<
p 2
1
1
2
2
p e0 2 (z02 t) 1 e2 t dz =
= S02 1 e2t0 t 1
Z< 2
p 2
1
1
2
p e0 2 (z02 t) dz
= S02 1 e2t+ t 1
2
<
we obtain the result, since
Z
<
p 2
1
1
p e0 2 (z02 t) dz = 1:
2
p
The latter equality can be veri ed by setting y = z 0 2 t and observing
that
Z
Z
1 2
1 0 1 (z02pt)2
1
p e 2
p e0 2 y dy = 1
dz =
2
2
<
<
since the integral in dy is nothing else than the integral over < of the density
of a standard Normal, thus equal to 1:
2
3
The variance can now be obtained by computing V ar P [S(t)] = E P S 2 (t) 0
0 P
12
E [S(t)] :4
13.2
Information and Investment Strategies
The securities tradeable in our market are the risk-free asset B and the
risky stock S; whose evolution is random since it is driven by the Brownian
170
CHAPTER 13. THE BLACK-SCHOLES MODEL
motion W; the only source of randomness not directly observed by investors.
However, the observation of the price process S is equivalent to observing
W: Indeed, formula (13:3) ; expressing S as a function of W , can be inverted
to obtain W in terms of S:
1 2
S (t)
1
ln
0 0 t :
W (t) =
S0
2
In the Black-Scholes model it is postulated that investors observe the prices
of the securities B and S: Since B is not a¤ected by the risk source W;
investors do not gather any information on W by observing B: They can
instead know exactly W by observing S. Hence, the ltration generated
by W coincides with the ltration generated by S: As a result, taking as
reference ltration the one generated by the Brownian motion W does make
sense in this model.
We now examine the concept of dynamic investment strategy in continuous time, a concept that is analogous to that seen in the discrete time
setting. We will formalize in the sequel some integrability conditions by
de ning more precisely which strategies are admissible in the model. Such
conditions will enable to avoid dealing with several technical issues immediately arising when adventuring in the continuous time world.
De nition 83 An investment strategy # = f#(t)gt2[0;T ] is an F0adapted
process taking values in <2 :
For all t 2 [0; T ], we have #(t) = (#0 (t); #1 (t)), where
- #0 (t) represents the number of units of B held at time t;
- #1 (t) represents the number of units of stock S held at time t.
The value of the strategy at time t is then given by
V# (t) = #0 (t) B (t) + #1 (t)S (t) ;
The discounted value of the strategy at time t is given by
V#3 (t) =
where S 3 (t) =
time t.
S(t)
B(t)
V# (t)
= #0 (t) + #1 (t)S 3 (t)
B(t)
= S(t)e0t is the discounted value of the price of S at
Particular strategies are the so called simple strategies:
De nition 84 An investment strategy # = f#(t)gt2[0;T ] is simple if it is
bounded and if positions are rebalanced only at xed deterministic times
0 = t0 < t1 < ::: < tn = T:
13.2. INFORMATION AND INVESTMENT STRATEGIES
171
We note that a simple strategy is a discrete time strategy that can be
modi ed at the given dates 0 = t0 < t1 < ::: < tn = T of the simple
strategies considered. Recalling what seen for the multi-period framework,
the simple strategy is hence self- nancing if
C# (tj+1 ) = 0 for j = 0; :::; n 0 2;
i.e. if the cash‡ow generated at the intermediate dates t1 ; :::; tn01 is always
zero. As seen in the discrete time case, this condition can be written as
follows
V# (tj+1 ) = #0 (tj ) B (tj+1 ) + #1 (tj )S (tj+1 ) for j = 0; :::; n 0 2;
from which
V# (tj+1 ) 0 V# (tj ) = (#0 (tj ) B (tj+1 ) + #1 (tj )S (tj+1 ))0
0(#0 (tj ) B (tj ) + #1 (tj )S (tj ))
or
V# (tj+1 ) 0 V# (tj ) = #0 (tj ) (B (tj+1 ) 0 B (tj )) + #1 (tj )(S (tj+1 ) 0 S (tj ))
for j = 0; :::; n 0 1: Setting 1t = tj+1 0 tj , we see that the strategy is
self- nancing if its variation in value is such that
1V# (tj ) = V# (tj+1 ) 0 V# (tj ) = #0 (tj ) 1B(tj ) + #1 (tj )1S(tj )
(13.5)
for j = 0; :::; n 0 1:
Formula (13:5) suggests how to de ne the self- nancing property beyond
simple strategies, i.e. to strategies where rebalancing can occur continuously
in time: by considering the di¤erential expression for 1t ! 0:
De nition 85 An investment strategy # = f#(t)gt2[0;T ] is self- nancing2 if
dV# (t) = V# (t + dt) 0 V# (t) = #0 (t) dB(t) + #1 (t)dS(t)
(13.6)
for t 2 [0; T ]:
De nition (13:6) can be rephrased in terms of discounted value process.
2
For expression (13:6) to be mathematically correct, we need some integrability condiRT
RT
tions. For example, if 0 j#0 (s)(!)j ds < 1 and 0 j#1 (s)(!)j2 ds < 1 for each ! 2 ;
then all integrals - including the stochastic ones - are well de ned.
172
CHAPTER
13.
THE BLACK-SCHOLES MODEL
Proposition 86 An investment strategy 3 = {9(t) retary is self-financing
if and only if
(13.7)
dV5 (t) = 01(t)dS*(t)
for t € [0;T].
Proof. Let 0 be self-financing. By construction, Vs (t) = Vo (t)-e~® and
therefore
dV s(t) = [dV (t)] - et _ Vy (t) - (de~% dt) =
= [Bo (t) (6e% dt) + O1(t)dS@)] -e-** —
— (Bo (t) e + 84(£)S (t)) - (Se dt) =
=01(t) [dS(t) -e~% + S(t) - (—de7* dt)] =
= #,(t) 4s"(0)
since dS*(t) = d(S(t) -e~**) = dS(t) -e~* + S(t) - (—de~* dt).
The converse implication can be proved in a similar way.
The self-financing condition (13.7) can be expressed in the shorthand
differential form as follows
Vg (t) = Vs
t
+ | dVs (s) = Vs
t
+ | 91(s)dS*(s)
In the Black-Scholes model, the integral in dS*(s) is made of two terms, one
in ds and the other in dW(s)
v5) = v5 (0)+ | “y(s) (S*(s) (u — 8) ds + S*(s)oaW(s)),
and can then be linked to the stochastic integral defined in the previous
chapter. From the financial point of view, the self-financing condition (13.7)
tells us that the variations over infinitesimal time intervals of the discoun-
ted value of a self-financing strategy are only due to the variation of the
discounted value of the risky security within the time interval considered.
Everything works as if we froze the positions of the strategy in the infinitesimal interval [t;¢+ dt{ and kept them constant. The position invested in the
risk-free asset cannot lead to variations in the discounted value, hence the
increment (or decrement) of the discounted value of the strategy between t
and t + dt is only due to variations between t and t + dt of the discounted
value of the risky security, dS*(t).
13.3. NO-ARBITRAGE ANALYSIS
13.3
173
No-Arbitrage Analysis
In the context of discrete time (and nite states) nancial markets, we have
discussed the concept of arbitrage opportunities and we have shown in the
First Fundamental Theorem of Asset Pricing the equivalence between noarbitrage and the existence of an equivalent martingale measure or riskneutral probability measure.
We now want to get to the same result also for the Black-Scholes model,
on the lines of what we have learned in the discrete time case. To understand the way the First Fundamental Theorem of Asset Pricing works in
continuous time, we have to deal with the following key elements:
1. the no-arbitrage condition;
2. risk-neutral probability measures or equivalent martingale measures;
3. the equivalence between the absence of arbitrage and the existence of
equivalent martingale measures.
13.3.1
The No-Arbitrage Property
In the continuous time model we focus for simplicity on self- nancing strategies
only.
In this case, disregarding some integrability issues that will be treated
in the sequel, we say that an arbitrage opportunity is a self- nancing
strategy # = f#(t)gt2[0;T ] having zero or negative cost at time t = 0, i.e.
V# (0) 0;
and generating a positive payo¤ at time t = T , i.e.
V# (T ) 0 with P [V# (T ) > 0] > 0:
The absence of arbitrage opportunities is referred to as no-arbitrage condition, NA for short.
13.3.2
Equivalent Martingale Measures
In the discrete time case we have seen several examples of how an equivalent
martingale measure, Q, can be shown to exist and how it can be computed
explicitly (i.e. ! by !). In analogy with the discrete time case, we say that a
probability measure Q, equivalent to P, is a risk-neutral probability measure
174
CHAPTER 13. THE BLACK-SCHOLES MODEL
or equivalent martingale measure if the discounted value of the risky security,
S, is a martingale under Q, i.e. if for all s < t 2 [0; T ], we have
S(s)
Q S(t)
=E
Fs
B(s)
B(t)
The discounted price process of S is the best possible predictor of its future
realizations under Q and with respect to the information represented by the
ltration fFt gt2[0;T ] :
In order to derive Q in the Black-Scholes model, we analyze the dynamics
of the discounted price process of the stock S. Such process, denoted by
S 3 = fS 3 (t)gTt=0 (by de nition S 3 (t) = S(t) 1 e0t ) has dynamics described
by the following SDE
dS 3 (t) = S 3 (t) [( 0 )dt + dW (t)]
(13.8)
S(0) = So
which can be obtained by applying the (13:1)c to the equality dS 3 (t) =
d(S(t) 1 e0t ) = dS(t) 1 e0t + S (t) 1 (0e0t dt).
The presence of the drift makes it impossible for the process S 3 to be
a martingale under the physical measure P: So, what form has to take the
measure Q so as to make S 3 a martingale under Q? We have seen in the
previous chapter that a process with continuous sample paths is a martingale
when it can be expressed as a stochastic integral , i.e. an integral with
respect to a standard Brownian motion. After some manipulation in (13:8),
we can rewrite the di¤erential of S 3 as follows:
0
3
3
dt + dW (t) ;
dS (t) = S (t) with W a standard Brownian motion under P:
Let us set
0
dW 3 (t) :=
dt + dW (t)
with W 3 (0) = 0: It is immediate to integrate the di¤erential, thus obtaining
W 3 (t) =
0
t + W (t) :
The process W 3 is a non standard Brownian motion under P (if that were
not the case, S 3 would be a martingale under P). But there exists a probability measure, equivalent to P; under which the process W 3 is a standard
Brownian motion with respect to the ltration F. Such measure is given by
the following theorem:
13.3. NO-ARBITRAGE ANALYSIS
175
Theorem 87 (Girsanov) Employing the notation introduced so far, let us
set
(
)
1 0 2
0
T
W (T ) 0
L = exp 0
2
and denote by Q the probability equivalent to P having density3 L under P;
i.e. for every subset A 2 FT we have
Q(A) = E P [L 1 IA ]
where IA is the indicator function of the set A; i.e.
1 se ! 2 A
IA (!) =
0 se ! 2
=A
Then, under Q the process
W 3 (t) =
0
t + W (t)
is a standard Brownian motion with respect to the ltration F.
Girsanov’s Theorem provides us with the probability measure Q under
which S 3 is a martingale. Indeed, from
0
3
3
dt + dW (t) =
dS (t) = S (t) 1
= S 3 (t) 1 dW 3 (t)
we get the following integral expression
Z t
3
S (t) = S (0) +
S 3 (s) 1 dW 3 (s):
0
(13.9)
Since S 3 (t) is an integral with respect to dW 3 (s); a standard Brownian
motion under Q; then S 3 is a martingale under Q:
The martingale property of S 3 can also be proved directly, by employing
the explicit expression for S given by (13:3) : Solving the SDE (13:3) ; we
have found that
1 2
S (t) = S0 1 e(0 2 )t+W (t)
3
L de nes an equivalent measure because L is strictly positive.
176
CHAPTER 13. THE BLACK-SCHOLES MODEL
from which
1 2
S 3 (t) = S0 1 e(00 2 )t+W (t)
(13.10)
Since by construction of W 3 we have W (t) = W 3 (t) 0 0
t, the term in the
exponent can be expressed as
1
1
0
0 0 2 t + W (t) =
t =
0 0 2 t + W 3 (t) 0
2
2
1
= 0 2 t + W 3 (t)
2
and hence we obtain
1
S 3 (t) = S0 1 e0 2 2 t+W 3 (t)
(13.11)
Both formulas (13:10) and (13:11) yield for all t and for each ! 2 the
same value S 3 (t)(!).
Expression (13:10) is useful when working under the physical measure P;
since under P the process W appearing in (13:10) is a standard Brownian
motion, of which we know law and properties. Similarly, expression (13:11)
is useful when working under the measure Q; since under Q the process W 3
showing up in (13:11) is a standard Brownian motion. Since the martingale
property of S 3 under Q is key to the no-arbitrage analysis, we o¤er an
additional proof based on expression (13:11) : For this purpose, we need the
following remarkably important property of conditional expectations, which
will be used also in the pricing of derivative securities:
Proposition 88 Let X and Y be two random variables de ned on , taking
values in < and measurable with respect to F: Let X be Fs 0measurable
and Y be independent of Fs under Q: Then, for any well-behaved function4
g : < 2 < ! < we have that
E Q [ g(X; Y )j Fs ] = G(X)
where
G(x) := E Q [g(x; Y )]
Proposition 88 is of great importance, since it helps in computing conditional expected values, for which we do not have any simple recipe as in
the discrete time case. In particular, Proposition 88 shows the way to be
followed in the sequel.
4
To avoid integrability problems, we can assume g to be bounded.
13.3. NO-ARBITRAGE ANALYSIS
177
We identify two random variables: the rst one, X; is measurable with
respect to the sigma-algebra Fs ; the second one, Y; is independent of
the sigma-algebra Fs under the reference probability measure, Q in
this case.
We write the object of which we want to compute the Fs 0conditional
under Q as a deterministic function of the two variables X and Y , i.e.
as g(X; Y ) (with g real-valued function of two real variables).
We x x 2 < and compute G(x) := E Q [g(x; Y )] : Since Y is independent of Fs under the probability Q; computing its Fs -conditional
expectation or simply its unconditional expectation is the same thing.
Finally, the conditional expectation E Q [ g(X; Y )j Fs ] is nothing else
than the function G(x) where we replace x by the path by path value
X(!) of X, i.e.
E Q [ g(X; Y )j Fs ] (!) = G(X(!))
Now, by employing Proposition 88 we can solve the following exercise:
Exercise 89 Use (13:11) to prove that S 3 = fS 3 (t)gt2[0;T ] is an F0martingale
under Q, i.e. that for all 0 s < t T one has
E Q [ S 3 (t)j Fs ] = S 3 (s)
where E Q denotes expectation under Q:
get
Solution. Let us x s < t and derive S 3 (s) and S 3 (t) from (13:11) : We
1
S 3 (t) = S 3 (s) 1 e0 2 2 (t0s)+(W 3 (t)0W 3 (s))
;
where the following two factors appear:
1. S 3 (s), which is Fs 0measurable (it plays the role of X in Proposition
88);
1
2
3
3
2. e0 2 (t0s)+(W (t)0W (s)) , which is independent of Fs ; since W 3 is a
Brownian motion under Q and hence the increment W 3 (t) 0 W 3 (s) is
independent of Fs under Q (it plays the role of Y in Proposition 88).
178
CHAPTER 13. THE BLACK-SCHOLES MODEL
We now need to compute the conditional expectation E Q [ S 3 (t)j Fs ] :
The term S 3 (t) can be simply rewritten as
1
S 3 (s) 1 e0 2 2 (t0s)+(W 3 (t)0W 3 (s))
= X 1 Y;
so that, with the notation of Proposition 88, we have g(x; y) = x 1 y: Hence,
we have to compute the following conditional expectation:
E Q [ g(X; Y )j Fs ] = G(X);
with
G(x) = E Q [g(x; Y )] = E Q [x 1 Y ] = x 1 E Q [Y ] :
Because of the properties of the Brownian motion W 3 under Q, we have
that
h 1 2
i
3
3
E Q [Y ] = E Q e0 2 (t0s)+(W (t)0W (s)) = 1
To verify the latter equality, we recall that under Q we have W 3 (t) 0
Q
Q p
W 3 (s) t 0 s 1 Z, with Z N (0; 1) : As a consequence,
h 1 2
i
h 1 2
i
p
3
3
E Q e0 2 (t0s)+(W (t)0W (s)) = E Q e0 2 (t0s)+ t0s1Z =
Z
p
1 2
=
e0 2 (t0s)+ t0s1z fZ (z) dz;
<
where fZ (z) is the density of a standard normal random variable:
1 2
1
fZ (z) = p e0 2 z ;
2
We thus obtain
h
i
Z
p
2
1
1
p e0 2 (z0 t0s) dz =
E [Y ] = E e
=
2
<
Z
1 0 1 y2
p e 2 dy;
=
2
<
p
where we have set y = z 0 t 0 s: The latter integral is clearly 1, since it is
the integral over < of the density of a Normal random variable. Hence, the
conditional expectation of S 3 (t) with respect to Fs and under Q is given by
h 1 2
i
3
3
E Q [ S 3 (t)j Fs ] = S 3 (s) 1 E Q e0 2 (t0s)+(W (t)0W (s)) =
Q
Q
0 12 2 (t0s)+(W 3 (t)0W 3 (s))
= S 3 (s) 1 1 = S 3 (s)
and thus S 3 is a martingale under Q with respect to fFt gt2[0;T ] :4
Since the behavior of S is fundamental in our study, we nd it convenient
to collect in the following proposition the results seen so far.
13.3. NO-ARBITRAGE ANALYSIS
179
Proposition 90 (Distribution of S) The process S has dynamics given
by the (13:1), i.e.
dS (t) = S (t) [ dt + dW (t)]
S(0) = S0
with W a standard Brownian motion under P. By solving (13:1) we obtain
1 2
S (t) = S(0) 1 e(0 2 )t+W (t) ;
(13.12)
from which we can get the expected value of S(t) under the physical measure
P:
E P [S(t)] = S0 1 et :
Recalling that W 3 (t) =
0
t
+ W (t), by (13:1) we obtain
dS (t) = S (t) [ dt + dW 3 (t)]
S(0) = S0
with W 3 a standard Brownian motion under Q (where Q is given by Girsanov’s
Theorem). Solving the latter equation or substituting W (t) = W 3 (t) 0 0
t
into (13:12), we arrive to
1 2
3
S (t) = S(0) 1 e(0 2 )t+W (t) ;
which provides us with the expected value of S(t) under Q :
E Q [S(t)] = S0 1 et :
As can be gathered from the previous proposition, when switching from
the physical measure P to the equivalent martingale measure Q, the lognormal distribution of the process S is preserved, although its mean parameter
is changed. Put another way, the di¤usion coe¢cient is the same in both
cases, while the drift changes and is equal to under P, to the risk-free rate
under Q. We remark that the process S remains the same, in the sense
that its trajectories (the values taken by S on each !) are the same.
13.3.3
The Equivalence between No-arbitrage and the Existence of an Equivalent Martingale Measure
The First Fundamental Theorem of Asset Pricing, proved in the discrete
time case, states that
180
CHAPTER
13.
THE BLACK-SCHOLES MODEL
NA holds + there exists a probability measure Q equivalent
to P under which discounted security prices are martingales.
It is
Why
The
The
possible to prove a similar result also in the continuous time model.
similar and not the same?
theorem involves two implications.
implication => is hard to prove for general continuous time processes,
requiring the study of some topological properties of the set of payoffs attainable at maturity by adopting self-financing strategies. In continuous time
models more general than the Black-Scholes one, an assumption stronger
than NA must be made to ensure the existence of an equivalent martingale
measure Q. We refer the interested reader to the textbooks in bibliography
for details.
The opposite implication < can instead (almost) be tackled by employing the tools developed so far and can be instructive for understanding some
of the differences with the discrete time case, leading to a slightly different
formulation of the First Fundamental Theorem of Asset Pricing.
Hence, we assume that a probability measure equivalent to P exists
under which discounted security prices are martingales.
Is this enough to guarantee the absence of arbitrage opportunities?
We first focus on simple strategies. This is not a stringent assumption,
since in real life investors update their positions at specific points in time
0=to
< ty <... < tp, =T, which can be very close for n large enough.
If the strategy 0 is simple, for any updating date t; condition (13.5) leads
to
Vg (ty41) — Ve (tj) = AVG (tj) = 81 (tj) AS*(t;)
and hence
E2 [AV} (t;)| Fj]
= E® [01 (t)AS*(t;)Fay]
|
=
81 (t;)E? [AS*(t3)| Fe;]
=
01 (t;)
-0=0
where the last equality follows from the fact that S* is a martingale under
Q with respect to F.
This means that the discounted value process V$ of a simple self-financing
strategy satisfies the martingality property at the updating dates to < t1 <
... <ty. By exploiting the tower property of conditional expectations it can
13.3. NO-ARBITRAGE ANALYSIS
181
be proved that V#3 enjoys this property for any pair of instants s < t 2 [0; T ] ;
that is
E Q [ V#3 (t) 0 V#3 (s)j Fs ] = 0:
Thus, any self- nancing simple strategy # makes the discounted value
process V#3 a martingale under Q with respect to the ltration F: As a result,
if V#3 (T ) 0 and P [V#3 (T ) > 0] > 0, i.e. Q [V#3 (T ) > 0] > 0 (recall that P
and Q are equivalent), the following holds
V# (0) = V#3 (0) = E Q [V#3 (T )] > 0
The latter result means that we cannot set up a self- nancing simple strategy
yielding a positive (almost surely) cash‡ow at maturity with zero or negative
cost at time t = 0: Indeed, if the payo¤ at maturity were strictly positive,
the same would be true for its initial value.
How about a strategy # allowing continuous rebalancing over time then?
Recalling the de nition (13:7) of self- nancing strategy in continuous
time, we have in this case
dV#3 (t) = #1 (t) [dS 3 (t)] =
= #1 (t) [S 3 (t) 1 dW 3 (t)] ;
with W 3 a standard Brownian motion under Q: We note that
Z t
3
3
#1 (s) [S 3 (s) 1 dW 3 (s)] ;
V# (t) = V# (0) +
0
i.e. that V#3 (t) is an integral with respect to W 3 and hence (under suitable
integrability conditions) a martingale under Q. As a result, the generic selfnancing strategy # cannot lead to arbitrage: as for simple strategies, we
have V# (0) = V#3 (0) = E Q [V#3 (T )] > 0 whenever # guarantees a positive
payo¤ at maturity.
There is a (not only mathematical) caveat hiding behind our request of
suitable integrability conditions. The updating of the positions in continuous
time provides investors with a too large set of possible strategies, including
the so-called doubling strategies. By these we mean that investors can take
arbitrarily large long/short positions in the available nancial securities, B
and S. As a result, they can set up zero-cost self- nancing strategies leading
to positive payo¤s at maturity with probability 1.
We provide a simple example in the multi-period case with in nite timehorizon. Let us suppose we repeatedly toss a coin and gamble in the following
182
CHAPTER 13. THE BLACK-SCHOLES MODEL
way: we bet 1 euro at time t = 0 and win 1 euro in case the result is head; if
we lose, we borrow (at a zero rate) 2 euros and double the bet. In this way,
at time t = 1 we gain (2 0 1) euros if we see head, we borrow and double
the bet if we see tail. And so on until head shows up. If we can play in
perpetuity, sooner or later a head will result, with probability 1. Suppose
that happens at the n0th trial. We would then win 2n euros against a
cumulated out‡ow (debts to be paid back) of
n01
X
2j = 2n 0 1;
j=0
thus obtaining a net in‡ow of 1 euro.
The consequence is that, provided we can wait long enough (in nite
time-horizon) and borrow money for free (an overall amount of 2n 0 1 euros
against n 0 1 consecutive tails, for each n), it is possible to make a sure
pro t.
In the continuous time model, i.e. with t 2 [0; T ], we can perform an
in nite number of trades even if the time-horizon is nite, since we can
arbitrarily increase the frequency of trading dates.
We can exploit such arbitrage opportunity even with the simple BlackScholes model (with = = 0 and = 1), provided investors are allowed
to take arbitrarily large long/short positions on the securities B and S:
We can avoid the problem described in two ways:
1. we can impose bounds on the units of securities held in the portfolio:
Z T
2
Q
3
(#1 (s)S (s)) ds < 1
E
0
2. we can impose a lower bound on the value of the strategy, so that it
cannot become arbitrarily negative:
Z t
3
3
V# (t) 0 V# (0) =
#1 (s)dS 3 (s) 0a
0
for all t 2 [0; T ], with a a positive constant.
Both restrictions do make sense from the practical point of view. They
are also equivalent when investors are non-satiated, i.e. when they always
13.3. NO-ARBITRAGE ANALYSIS
183
prefer to have more to less. The second assumption is the one currently
adopted in the literature.
We conclude this section by stating the First Fundamental Theorem of
Asset Pricing. We rst focus on the assumptions we have to make on the
strategies and de ne the so called admissible strategies:
Notation 91 We let A denote the set of strategies # = f#(t)gt2[0;T ] having
the following properties:
1. the strategy # is self- nancing, i.e. its value process satis es expression
(13:6);
2. the discounted value process, V#3 , is nonnegative, i.e.
V#3 (t) = #0 (t) + #1 (t)S 3 (t) 0
for all t 2 [0; T ];
3. the process V#3 is such that
2
EQ 4
!2 3
sup V#3 (t)
t2[0;T ]
5 < 1:
The strategies in A are said to be admissible.
The integrability restrictions characterizing the admissible strategies are
actually strong, but they allow to ease the study of market completeness
and replication of derivative securities.
We can now state the First Fundamental Theorem of Asset Pricing in a
way similar to that followed in the discrete time case.
The assumptions made allow us to avoid the concept of No Free Lunch
with Vanishing Risk, essential to the formulation of the no-arbitrage property in more general continuous time models. The latter requires not only
the absence of arbitrage strategies (as in the de nition of NA) but also of
sequences of admissible strategies leading to arbitrages in the limit.
Theorem 92 (First Fundamental Theorem of Asset Pricing) Let
us consider the Black-Scholes market and the set of admissible strategies A.
Then, NA holds if and only if there exists a probability measure Q, equivalent
to P, under which the discounted price process of the risky security, S 3 , is
an F0martingale under Q:
184
CHAPTER 13. THE BLACK-SCHOLES MODEL
Corollary 93 NA holds in the Black-Scholes market.
Proof. By Girsanov’s Theorem, we have found an equivalent probability
measure Q under which the process S 3 is an F0martingale. By applying
the First Fundamental Theorem of Asset Pricing we can then conclude that
the market is arbitrage-free.4
13.4
Completeness of the Black-Scholes Model
The Black-Scholes market is complete. We will show in particular how to
attain any European-type contingent claim at time T , by suitably investing
in the securities B and S, provided the claim has nite variance under both
P and Q: The replication is possible (with just two securities) because we
can rebalance the admissible strategies continuously over time. The BlackScholes market is thus dynamically complete.
The proof of such property exploits the so called martingale representation theorem, which we provide without proof:
Theorem 94 (Kunita and Watanabe) Let M = fM (t)gt2[0;T ] be a martingale under Q and with respect
to ithe ltration generated by the Brownian
h
3
Q
motion W , such that E (M (t))2 < 1 for all t 2 [0; T ]: Then, there exhR
i
T
ists a (predictable) process h = fh(t)gt2[0;T ] satisfying E Q 0 (h(s))2 ds <
1 and such that
Z
M (t) = M (0) +
t
0
h(s)dW 3 (s)
for all t 2 [0; T ]:
Theorem 95 (Completeness) For
contingent claim X(T ),
h every European
i
2
Q
with maturity T and such that E (X(T )) < 1, there exists an admissible strategy # 2 A replicating it, i.e. such that
V# (T ) = X(T ):
Proof. Let X(T ) be a contingent claim satisfying the assumptions of our
theorem. We need to nd a strategy # 2 A replicating it, i.e. such that
V#3 (T ) = X 3 (T )
13.4. COMPLETENESS OF THE BLACK-SCHOLES MODEL
185
in terms of discounted value process. We have seen that for all # 2 A the
process V#3 is an F0martingale under Q: So, if a replicating strategy # exists,
it must be such that
V#3 (t) = E Q [ V 3 (T )j Ft ] = E Q [ X 3 (T )j Ft ]
for all t 2 [0; T ]: The process M := V#3 is an F0martingale under Q, such
that M (T ) = X(T ) and that its variance is nite under Q (because # is
admissible). The ltration F generated by the Brownian motion W (i.e. by
the price process S) coincides with the ltration generated by W 3 ; since
W 3 (t) = 0
t + W (t) : Thus, the (F; Q)0martingale M satis es the assumptions of the Kunita-Watanabe Theorem. We can then say that there
exists a process h such that
Z t
M (t) = M (0) +
h(s)dW 3 (s):
(13.13)
0
From expression (13:9), we can write dW 3 (s) as a function of dS 3 (s)
dW 3 (s) =
and, substituting into (13:13), we get
Z
M (t) = M (0) +
Now, setting5
#1 (s) =
t
0
dS 3 (s)
S 3 (s)
h(s)
dS 3 (s) :
S 3 (s)
h(s)
S 3 (s)
for all s 2 [0; T ], we obtain
V#3 (t)
= M (t) = M (0) +
Z
t
0
#1 (s) dS 3 (s) :
Noting that M (0) = E Q [M (T )] = E Q [X 3 (T )] ; we are left with verifying
whether the component of the strategy # concerning the investment is B:
This can be obtained by solving the equation
#0 (t) = V#3 (t) 0 #1 (t)S 3 (t)
for all t 2 [0; T ], starting from V#3 (t) = E Q [ X 3 (T )j Ft ] and #1 just computed.
5
We omit to prove that such strategy is admissible.
186
CHAPTER 13. THE BLACK-SCHOLES MODEL
Remark 96 The previous theorem only guarantees that a replicating strategy
exists, without providing a closed form expression for #, since no explicit
form is speci ed for the process h in equation (13:13).
The theorem proved shows that we can replicate European options by
suitably investing in B and S. Actually, the result can be extended to derivatives generating intermediate cash‡ows and to American options, those
which can be exercised before the maturity T: Since the Black-Scholes market is dynamically complete, if we introduce a new security in the market,
it will be redundant. Its price process must then be equal to the value process of the corresponding replicating strategy, in order to avoid arbitrage
opportunities. Hence:
Remark 97 In the Black-Scholes market, for every derivative security there
is one and only one no-arbitrage price.
Such no-arbitrage price can be determined by:
computing the value of the strategy replicating the derivative;
computing the expected value of the discounted payo¤ at maturity
under the unique martingale measure Q:
We have not gone into the details concerning the uniqueness of the equivalent martingale measure Q: Nevertheless, we can show that in the BlackScholes market there is a unique measure making the price process S de‡ated
by B a martingale. The density of Q with respect to P is given by Girsanov’s
Theorem.
In general, it is not easy to compute explicitly the strategy replicating a
contigent claim, hence one usually prefers the second approach when pricing
derivative securities, i.e. computing the conditional expectation of the discounted payo¤ under the equivalent martingale measure. We will deal with
this issue in the next section.
Chapter 14
No-Arbitrage Pricing in the
Black-Scholes Market
We mentioned in the previous chapter that for every derivative security in
the Black-Scholes market there is a unique no-arbitrage price. Such price
coincides
with the value of the strategy replicating the derivative;
with the conditional expected value (under the unique equivalent martingale measure Q) of the discounted payo¤ at maturity.
Employing the notation introduced in the previous chapter, we can say
that the no-arbitrage price process of a contingent claim with payo¤ X(T )
at T is the process SX given by1
3
SX
(t) =
SX (t)
= V#3 (t) = E Q [ V#3 (T )j Ft ] = E Q [ X 3 (T )j Ft ]
B(t)
In the next sections, we employ risk-neutral valuation to compute the price
SX of a European derivative with payo¤ at T depending only on the underlying security price S(T ):
14.1
No-arbitrage Valuation of European Derivatives in the Black-Scholes Model
In this section, we price European derivatives in the Black-Scholes market by
computing risk-neutral expectations of their discounted payo¤ at maturity.
1
The notation
3
applies to discounted values, as in the previous chapter.
187
188
PRICING IN THE BLACK-SCHOLES MARKET
Speci cally, we assume that we have to price a call option or a put option
written on the security S, with maturity T and strike price K: The payo¤
at maturity is is given by
X(T ) = max(S(T ) 0 K; 0)
for the call option and by
X(T ) = max(K 0 S(T ); 0)
for the put option.
The payo¤ X(T ) is a random variable, but its dependence on ! 2 is
not explicit: indeed, X(T ) depends on ! 2
because it is a function of
S(T ); which in turn depends on ! (in the case of a call option, for example,
X(T )(!) = max(S(T )(!) 0 K; 0)). Provided some integrability conditions
are satis ed, the market completeness theorem ensures that the derivative is
redundant and that there is a unique no-arbitrage price. As we have already
said, the discounted no-arbitrage price at time t of the derivative with payo¤
X(T ) at maturity is given by
3
(t) = E Q [ X 3 (T )j Ft ]
SX
3 (t) is hence an F 0measurable random variable.
The discounted value SX
t
We can show that the dependence on ! 2 is only implicit. In the following
3 (t)(!) is indeed
proposition, we use the Markov property of S to show that SX
just a function of the current security price S(t)(!):
Proposition 98 Let us consider a European derivative with payo¤ at maturity that is a function of S(T ):
X(T )(!) = f (S(T )(!))
h
i
for ! 2 , with f : <+ ! <. Let us further assume that E Q (X(T ))2 < 1:
The no-arbitrage price at time t of the derivative X is then a function of t
and of the value at time t of the underlying security. Formally, we have
SX (t) = F (t; S(t))
where the function F is given by
h
i
1 2
3
3
F (t; x) = E Q e0(T 0t) 1 f (e(0 2 )(T 0t)+(W (T )0W (t)) 1 x)
for all t 2 [0; T ] and all x 2 <+ :
RISK-NEUTRAL VALUATION
189
Proof. Since X(T ) satis es the assumptions of the market completeness
theorem, we know that it admits a replicating strategy. The discounted
value of such strategy is given by
3
(t) = E Q [ X 3 (T )j Ft ] =
SX
h
i
= E Q e0T 1 f (S(T )) Ft
and hence
3
SX (t) = et 1 SX
(t) =
h
i
Q
0(T 0t)
= E
e
1 f (S(T )) Ft :
From (13:11), we have that for all t the following holds:
1 2
3
S(t) = et 1 S 3 (t) = S0 1 e(0 2 )t+W (t) :
In particular, S(t) is a function of W 3 (t) and is Ft 0measurable. Furthermore, we have
S(T ) =
The rst term,
1 2
3
3
S(T )
1 S (t) = e(0 2 )(T 0t)+(W (T )0W (t)) 1 S (t)
S(t)
1 2
3
3
e(0 2 )(T 0t)+(W (T )0W (t)) ;
is a function of the Brownian increment W 3 (T ) 0 W 3 (t) and is thus independent of Ft under Q. By employing the property
of conditional 3ex2
Q e0(T 0t) 1 f (S(T )) F =
pectations
given
in
Proposition
88,
we
obtain
E
t
h
i
0 12 2 )(T 0t)+(W 3 (T )0W 3 (t))
Q
0(T
0t)
(
E e
1f e
1 S (t) F = F (t; S (t)), where
t
the function F is given by
h
i
1 2
3
3
F (t; x) = E Q e0(T 0t) 1 f (e(0 2 )(T 0t)+(W (T )0W (t)) 1 x) :
Indeed, using the notation of Proposition 88, we set s = t; X = S (t)
1 2
3
3
(i.e. X is an Ft 0measureable r.v.) and Y = e(0 2 )(T 0t)+(W (T )0W (t))
(i.e. Y is a r.v. independent of Ft under the measure Q). The function
g of Proposition 88 is in our case g(x; y) = e0(T 0t) 1 x 1 y (the discount
factor e0(T 0t) is a constant here). Then, Proposition 88 guarantees that
the conditional expected value of interest takes the following form
h
i
Q
0(T 0t)
E
e
1 f (S(T )) Ft = E Q [ g(X; Y )j Ft ] = G(X);
190
PRICING IN THE BLACK-SCHOLES MARKET
where
G(x)
:
h
i
= E Q [g(x; Y )] = E Q e0(T 0t) 1 x 1 Y =
= F (t; x):
We can thus conclude that
SX (t) = F (t; S (t))4
An important remark stems from the Proposition we have just proved.
We must stress that the time-t no-arbitrage price (discounted or not) of the
derivative X is a deterministic function of t and of the underlying security
price at time t. Once the two-variable deterministic function F has been
identi ed, we just need to know S(t) to obtain SX (t):
Remark 99 The previous results can be achieved by applying Proposition
79 of Chapter 11 about the Markov property of di¤usions.
We now derive the celebrated Black-Scholes formula for the price of a
call option.
Proposition 100 (Call Option Formula) Let us consider a European
call option with maturity T and strike price K: Its time-t no-arbitrage price
is then given by
c (t) = S (t) N (d1 ) 0 Ke0(T 0t) N (d2 ) ;
(14.1a)
where N (z) is the distribution function of a standard normal random variable, i.e.
Zy
z2
1
p e0 2 dz;
N (y) =
2
01
while
1
d1 = p
(T 0 t)
and
ln
S (t)
K
d 2 = d1 0 p
1 2
+ + (T 0 t)
2
(T 0 t):
Proof. The payo¤ of the call option at maturity is X(T ) = max(S(T ) 0
K; 0): With the notation of the previous proposition, we set
f (S(T )) = max(S(T ) 0 K; 0):
RISK-NEUTRAL VALUATION
191
All of the assumptions of the previous proposition are satis ed, since
h
i
h
i
E Q (max(S(T ) 0 K; 0))2 < E Q (S(T ))2 < 1
and hence we know that the discounted price of the call option at time t is
a function of t and S(t). As an exercise, we re-apply to this special case the
argument employed in the proof of the previous proposition. We know that
the payo¤ of the call at maturity has nite variance under Q; so the theorem
of market completeness ensures that the option is a redundant security. As
a result, we know that its time-t discounted no-arbitrage price must coincide
with that of the corresponding replicating strategy. The no-arbitrage value
of such strategy is given by
h
i
c(t) = E Q e0(T 0t) max(S(T ) 0 K; 0) Ft :
From (13:11) ; we can write
1 2
3
S(t) = et 1 S 3 (t) = S0 1 e(0 2 )t+W (t) ;
so that S(t) is a function of W 3 (t) and is Ft 0measurable. We can further
write
S(T ) =
1 2
3
3
S(T )
1 S (t) = e(0 2 )(T 0t)+(W (T )0W (t)) 1 S (t) :
S(t)
The rst factor,
1 2
3
3
e(0 2 )(T 0t)+(W (T )0W (t)) ;
is a function of the Brownian increment W 3 (T ) 0 W 3 (t) and is thus independent of Ft under Q. By the properties of conditional expectations, it
follows that
h
i
1 2
3
3
c(t) = E Q e0(T 0t) max e(0 2 )(T 0t)+(W (T )0W (t)) 1 S (t) 0 K; 0 Ft =
= F (t; S(t));
where
h
i
1 2
3
3
F (t; x) = E Q e0(T 0t) 1 max e(0 2 )(T 0t)+(W (T )0W (t)) 1 x 0 K; 0
(it must be noted that the result could have been achieved by applying the
previous proposition specialized to f (S(T )) = max(S(T ) 0 K; 0)).
192
PRICING IN THE BLACK-SCHOLES MARKET
We now compute explicitly F (t; x) by exploiting the fact that W 3 (T ) 0
Q p
Q
W 3 (t) T 0 t 1 Z, with Z N (0; 1):
h
i
1 2
3
3
F (t; x) = e0(T 0t) 1 E Q max e(0 2 )(T 0t)+(W (T )0W (t)) 1 x 0 K; 0 =
Z
p
1 2
0(T 0t)
= e
1 max e(0 2 )(T 0t)+( T 0t1z) 1 x 0 K; 0 fZ (z) dz;
<
where fZ (z) is the density of a standard Normal random variable, i.e.
z2
1
fZ (z) = p e0 2 :
2
The integrand is always nonnegative. In particular, it is strictly positive if
and only if
p
1 2
e(0 2 )(T 0t)+( T 0t1z) 1 x 0 K > 0;
that is, if
1
z>z = p
T 0t
3
1 2
x
0 0 (T 0 t) :
0 ln
K
2
As a result, we have:
F (t; x) = e0(T 0t) 1
+1
Z
p
1 2
e(0 2 )(T 0t)+(
T 0t1z)
1x0K
fZ (z) dz =
z3
0 +1
1
+1
Z
Z
p
1 2
e(0 2 )(T 0t)+( T 0t1z) 1 x fZ (z) dz 0
K fZ (z) dz A
= e0(T 0t) 1 @
z3
z3
The second term in parenthesis can be expressed as
+1
Z
K fZ (z) dz = K 1 Q [Z > z 3 ] = K 1 Q [Z < 0z 3 ] ;
I2 =
z3
with the last equality following from the properties of the standard Normal
random varibales. As a result, we have
I2 =
+1
Z
K fZ (z) dz = K 1 Q [Z < 0z 3 ] =
z3
1
p
= K 1N
T 0t
1 2
x
+ 0 (T 0 t) ;
ln
K
2
RISK-NEUTRAL VALUATION
193
where we denote by N (1) the distribution function of a standard Normal
random variable. We now move on to the rst term within parenthesis in
the expression de ning F (t; x), i.e.
I1 =
+1
Z
p
1 2
e(0 2 )(T 0t)+( T 0t1z) 1 x fZ (z) dz =
z3
+1
Z
p
1 2
e(0 2 )(T 0t)+(
=
T 0t1z)
z3
z2
1
1 x p e0 2 dz:
2
By collecting terms in the exponent we get
(T 0t)
I1 = xe
+1
Z
1
z3
p
2
1
1
p e0 2 (z0 T 0t) dz:
2
Applying the change of variable
y=z0
we obtain
(T 0t)
I1 = xe
p
T 0 t;
+1
Z
1
y3
with
1 2
1
p e0 2 y dy;
2
p
y 3 = z 3 0 T 0 t =
1
1 2
x
p
0 + (T 0 t) :
0 ln
=
K
2
T 0t
The integral in dy on (y 3 ; +1) is that of the density of a standard normal
random variable and as such we have
(T 0t)
I1 = xe
+1
Z
1
y3
1 2
1
p e0 2 y dy = xe(T 0t) 1 (1 0 N [y 3 ]) ;
2
which reduces to (use the properties of the Normal distribution)
I1 = xe(T 0t) 1 N [0y 3 ] =
1 2
1
x
(T 0t)
p
= xe
+ + (T 0 t) :
1N
ln
K
2
T 0t
194
PRICING IN THE BLACK-SCHOLES MARKET
Summing up, we have:
F (t; x) = e0(T 0t) 1 (I1 0 I2 ) =
1
x
1
p
= x1N
ln
+ + 2 (T 0 t) 0
K
2
T 0t
1 2
1
x
0(T 0t)
p
0Ke
+ 0 (T 0 t) :
1N
ln
K
2
T 0t
Finally, by adopting the notations d1 and d2 de ned in the theorem statement, we obtain the result
c (t) = f (t; S(t)) = S (t) N (d1 ) 0 Ke0(T 0t) N (d2 ) 4
We now derive the analogous formula for the price of a European put
option written on the risky security S:
Proposition 101 (Put Option Price Formula) Let us consider a European
put option written on the underlying S; with maturity T and strike price K:
Then, its time-t no-arbitrage price is given by
p(t) = Ke0(T 0t) N (0d2 ) 0 S (t) N (0d1 ) ;
(14.2)
with N (z) denoting the distribution function of a standard normal random
variable,
Zy
z2
1
p e0 2 dz;
N (y) =
2
01
while
and
1
d1 = p
(T 0 t)
ln
S (t)
K
d 2 = d1 0 p
1 2
+ + (T 0 t)
2
(T 0 t):
Proof. We can follow the same approach employed to derive the call
option price, i.e. we solve explicitly the integral
h
i
p(t) = E Q e0(T 0t) max(K 0 S(T ); 0) Ft
Alternatively, given the call option price, we can apply the put-call parity
formula. Indeed, we know that the Black-Scholes market is arbitrage-free
RISK-NEUTRAL VALUATION
195
and hence that also violations of the law of one price are prevented. The
price of a call option (with same strike price and maturity as the put) being
available, we can replicate the put option by holding a portfolio invested long
in Ke0(T 0t) units of the risk-free asset2 B; short in one unit of S (short
sale of 1 stock S) and long in one call. The time-t value of the portfolio is
then given by
p(t) = c (t) 0 S (t) + Ke0(T 0t) :
By employing the call option price formula, we get
p(t) = c (t) 0 S (t) + Ke0(T 0t) =
= S (t) (N (d1 ) 0 1) 0 Ke0(T 0t) (N (d2 ) 0 1)
and, for the properties of the standard normal distribution,
p(t) = Ke0(T 0t) (1 0 N (d2 )) 0 S (t) (1 0 N (d1 )) =
= Ke0(T 0t) N (0d2 ) 0 S (t) N (0d1 ) 4
It is not always possible to obtain such an explicit formula formula for
derivatives prices. When the risk-neutral valuation formula
SX (t) = e0(T 0t) E Q [ X(T )j Ft ] ;
yielding the no-arbitrage price SX of a European derivative paying X(T )
at T , cannot be solved in closed form, one usually employs Monte Carlo
methods to carry out a numerical valuation. Monte Carlo methods are based
on the simulation of the realizations of X(T ), i.e. on the sampling of random
variables having the same distribution as X(T ), clearly under the measure Q
and conditional on the information Ft . The sample mean of the realizations
of X(T ) so generated provides a numerical extimate of E Q [ X(T )j Ft ] : There
are several techniques allowing to improve the numerical precision of the
estimate, to reduce the computational burden of the procedure or to adapt
the method to speci c features of the derivative being priced. Here, we limit
ourselves to underline the importance of risk-neutral valuation. Indeed, even
when explicit formulas are not available, it can be complemented by Monte
Carlo methods to provide numerical answers to the problem of derivative
pricing.
The alternative approach to the pricing of derivatives is solving numerically the partial di¤erential equation satis ed by the derivative security price.
2
As was already noted, it is possible to replicate a European put option with a buy and
hold strategy involving a call option (with same underlying, strike price and maturity as
the put), the underlying stock S and the risk-free asset B:
196
PRICING IN THE BLACK-SCHOLES MARKET
This can be done when the latter only depends on time and on the current
price of the underlying security S. We will obtain the so called Black-Scholes
partial di¤erential equation in the next section.
14.2
Pricing of European Derivatives via the BlackScholes PDE
In this section, we derive the Black-Scholes partial di¤erential equation
(PDE). A PDE is an equation expressing the unknown function in terms
of its partial derivatives. We shall consider functions of two variables: the
time variable t and the space variable x; the latter representing the price of
the risky security S at time t: The unknown function is expressed in terms
of its partial derivatives with respect to t and with respect to x and for this
reason we call the equation partial di¤erential equation: in ordinary di¤erential equations instead, the unknown function depends on a single variable
(in the examples examined so far, the variable was the time t). In the
Black-Scholes PDE, the unknown is the function F (t; x) of Proposition 98,
expressing the price of a European derivative written on the risky stock S as
a function of time and of the current price of the underlying. The equation
is deterministic, because the Black-Scholes model is Markovian (see Proposition 98), i.e. the time-t price of a European derivative is a deterministic
function of t and S(t):
We provide two di¤erent derivations. The rst one is a straightforward
result of the no-arbitrage analysis of the previous section. The second is the
one originally adopted by Black and Scholes and based on the construction
of a locally risk-free portfolio invested in the underlying stock S and in
the derivative itself. No-arbitrage arguments imply that such portfolio has
local rate of return equal to the risk-free rate and in this way we obtain the
partial di¤erential equation. From the Black-Scholes PDE we then go back
to risk-neutral valuation via the Feynman-Kac Formula.
14.2.1
The Black-Scholes PDE
We derive the PDE satis ed by a European derivative security.
Proposition 102 (Black-Scholes PDE) Let us consider a European derivative security whose payo¤ at maturity T is a function of S(T ), i.e.
V (T )(!) = f (S(T )(!))
PRICING VIA THE BLACK-SCHOLES PDE
197
h
i
for each ! 2 ; with f : <+ ! <. Let assume that E Q (X(T ))2 < 1
denote by F , as in Proposition 98, the function giving the time-t no-arbitrage
price of the derivative as a function of t and S(t), i.e.
SX (t) = F (t; S(t)):
Suppose that F is continuously di¤erentiable once with respect to time, t,
and twice with respect to space, S. Then, F solves the following PDE for
all t 2 [0; T [ and S 2 <+ with boundary condition at time T :
@
@t F (t; S)
@
+ @S
F (t; S) 1 S +
F (T; S) = f (S)
1 @2
2 @S 2 F (t; S)
1 2 S 2 = F (t; S)
(14.3)
Proof. The discounted no-arbitrage price of the derivative security is a
martingale under Q: We thus set
3
Y (t) = SX
(t) = e0t 1 F (t; S(t))
and compute its di¤erential by means of Ito’s Lemma. With the notation
used in (12:11), recalling the SDE solved by S(t) under Q, we have
X(t) = S(t)
a (t; S(t)) = 1 S (t)
b (t; S(t)) = 1 S (t)
' (t; S(t)) = e0t 1 F (t; S(t))
In order to apply Ito’s Lemma, we compute the derivatives of ' with respect
to t and to S :
@
@
'(t; S) = 0e0t 1 F (t; S) + e0t 1 F (t; S)
@t
@t
@
@
' = e0t 1
F (t; S)
@S
@S
@2
@2
0t
'
=
e
1
F (t; S):
@S 2
@S 2
As a result, we have
1
20
@
+ e0t 1 @t
F (t; S)
+
0e0t 1 F (t; S)
i
1
0 0t @
1
@2
0t
1 @S F (t; S) 1 1 S + 2 e
1 @S 2 F (t; S) ( 1 S)2 dt
+ e
0
1
@
+ e0t 1 @S
F (t; S) ( 1 S) dW 3 (t) ;
dY =
198
PRICING IN THE BLACK-SCHOLES MARKET
with S = S(t): For Y (t) to be a martingale under Q; the di¤erential dY (t)
must be merely stochastic (see Proposition 76). The coe¢cient multiplying
dt must not show up, i.e.
1
20
@
F (t; S) +
0e0t 1 F (t; S) + e0t 1 @t
i
1
0
2
@
@2
=0
F (t; S) 1 1 S (t) + 12 e0t 1 @S
F
(t;
S)
(
1
S)
+ e0t 1 @S
2
After simple passages we get to the Black-Scholes PDE. The boundary condition at t = T simply requires the option price at maturity to coincide with
its payo¤.
Remark 103 In the case of a call option, the boundary condition in (14:3)
is
F (T; S) = max (S 0 K; 0)
for all S 2 <+ . It can be veri ed that the no-arbitrage call price c(t), given
by formula (14:1a) as a function of t and S, satis es the PDE (14:3) :
Similarly, in the case of a put option we have
F (T; S) = max (K 0 S; 0)
and one can verify that the no-arbitrage price put p(t), given by formula
(14:2) as a function of t and S, satis es the PDE (14:3).
14.2.2
The Classical Derivation of the Black-Scholes Equation
In the previous section, we derived the Black-Scholes PDE (14:3) by employing the key principle underlying no-arbitrage valuations: the discounted
prices of nancial securities are martingales under the risk-neutral measure
Q: Historically, the Black-Scholes PDE has a di¤erent derivation, based on
the no-arbitrage and market completeness assumptions. We now examine it
in detail.
We make the same assumptions as in the Proposition of the previous
section on the European derivative we want to price. We suppose that the
time-t price of the derivative is a function F (t; S(t)), di¤erentiable once with
respect to t and twice with respect to S: Regularity conditions apart, we have
seen that the Markov property allows us to express in this way the price of
any derivative whose random behavior at maturity only depends on S(T )
(see Proposition 98).
We construct a self- nancing portfolio involving a short position on
the derivative F and h units of the underlying security S:
PRICING VIA THE BLACK-SCHOLES PDE
199
The value of at time t is given by
= h 1 S 0 F;
where we have omitted the dependence on t in S, F and also in h, as we
will see.
Our aim is to make portfolio locally risk-free. The self- nancing condition is such that the variation of value between time t and time t + dt
is
d = hdS (t) 0 dF (t; S(t)):
To compute the di¤erential of F , we apply Ito’s Formula, recalling the decomposition of dS under P. With the notation of Ito’s Lemma, we can
write
X(t) = S(t)
a (t; S(t)) = 1 S (t)
b (t; S(t)) = 1 S (t)
' (t; S(t)) = F (t; S(t))
and
@F
@F
1 @2F 2 2
@F
+
S +
SdW (t) ;
S dt +
dF (t) =
2
@t
@S
2 @S
@S
so that the di¤erential of is equal to
2 @F
@F
@F
1
@ F
2 2
d = 0
+ h0
S +
0 2 S dt+ h 0
SdW (t) :
@t
@S
2
@S
@S
(14.4)
We want to be locally risk-free. In order to obtain this, we need the
di¤usion coe¢cient to be zero. We thus set
h0
@F
= 0;
@S
from which, for all t, we obtain
h = h (t) =
@F
(t; S(t)):
@S
The number h is called hedging ratio. The random variation of S between t
and t + dt makes random the value of the derivative F; but h is such that the
variation of the portfolio is instead locally deterministic. There is another
price variation which is deterministic, the one involved by the risk-free asset
200
PRICING IN THE BLACK-SCHOLES MARKET
B: Since NA holds, the risk-free asset B and portfolio must have the same
instantaneous return between t and t + dt, for otherwise there would be
(local) arbitrage. For this reason, the following holds
d = 1 dt = (h 1 S 0 F ) 1 dt:
(14.5)
By equating (14:4) to (14:5), we get
2 @F
@F
1
@ F
+ h0
S +
0 2 2 S 2 = (h 1 S 0 F ) 1 ;
0
@t
@S
2
@S
@F
(t; S(t)), we have
@S
2 @F
@F
1
@ F
2 2
0
+
0 2 S =
1S0F 1
@t
2
@S
@S
from which, recalling that h = h (t) =
and thus the Black-Scholes PDE results.4
We note that the hedging ratio h has the e¤ect of killing the di¤usion
coe¢cient of d; but also that of killing the parameter characterizing the
dynamics of the stock S under the measure P: Why have we considered
the dynamics of S under the measure P in the classical derivation of the
Black-Scholes PDE? The classical approach is aimed at setting up a locally
risk-free portfolio. Risk is accounted for by the Brownian motion W , which
is such under the measure P, i.e. the measure under which the instantaneous
expected return on the stock is equal to : The measure P is the so called
physical or historical measure, and is typically taken into account when
dealing with hedging issues. We focused on killing the di¤usion coe¢cient
of dW rather than that of dW 3 , since we actually wanted to eliminate the
random component (represented by W ) from the local variation of under
the measure P. We could have obtained the same result by exploiting the
decomposition of S under Q and killing the di¤usion coe¢cient of dW 3 ; but
we would then have missed the meaning of the reasoning developed so far.
We nally note that although we have initially decided to work under
P; the coe¢cient characterizing the dynamics of S under P disappears by
the NA condition, since the locally risk-free portfolio must have the same
instantaneous return as the risk-free asset.
The classical derivation of the Black-Scholes PDE has been obtained
under the NA assumption, without mentioning the risk-neutral or equivalent
martingale measure. Nevertheless, we can get to the risk-neutral valuation
formula by applying the Feynman-Kac Formula, which we state below. The
PRICING VIA THE BLACK-SCHOLES PDE
201
picture is now complete: the risk-neutral valuation formula has allowed us
to derive the Black-Scholes PDE in Proposition 102. But the Black-Scholes
PDE can be derived more directly by employing replication and no-arbitrage
arguments without mentioning the equivalent martingale measure Q; by
following the classical derivation of the Black-Scholes formula we have just
seen. At this point, the Feyman-Kac Formula links the PDE to a probability
measure under which the stock S is lognormally distributed with drift and
volatility : Such measure is thus the equivalent martingale measure, so that
we are back to the risk-neutral valuation.
Theorem 104 (Corollary from the Feyman-Kac Formula) Suppose
that F solves the following PDE with boundary condition:
@
@
1 2 2 @2
@t F (t; x) + x @x F (t; x) + 2 x @x2 F (t; x) = F (t; x)
(14.6)
F (T; x) = f (x)
Under suitable regularity conditions,3 F admits the following representation
F (t; x) = e0(T 0t) E Q [ f (S(T ))j Ft ]
where the process S satis es the following SDE
dS(s) = S(s) ds + S(s) dW 3 (s) per s t
S(t) = x;
with W 3 standard Brownian motion under Q.
We can then say that the derivative price F (t; S(t)) solving the PDE with
the boundary condition in (14:3) is the expected value under the risk-neutral
measure Q of the discounted payo¤ at maturity f (S(T )). The Black-Scholes
PDE is then equivalent to the pricing approach based on the computation
of expected values under the risk-neutral probability measure Q.
In some lucky cases (for example for call and put options) it is possible
to give an explicit solution to (14:3), i.e. to get to an analytical formula
for the computation of the expected discounted value of the payo¤. In
the other cases, numerical techniques are adopted. The PDE (14:3) can
3
For example, we could assume the function F to be such the solution S(t) to the SDE
starting from x satis es the condition
2
Z T @2
E 1 S(t) 2 F (t; S(t)) dt < 1:
@x
0
202
PRICING IN THE BLACK-SCHOLES MARKET
be solved through so called nite-di¤erence methods. These are based on
the approximation of the derivatives of the function F by suitable rates of
variation computed along a grid of points on the plane (t; x). Equation
(14:3) can then be transformed into a system of equations in which the
unknown is the function F computed at the di¤erent points of the grid:
a nite-di¤erence equation. Depending on the discretization made on the
derivatives of the function F , we can obtain di¤erent systems that can be
solved numerically by employing suitable algorithms.
14.3
Market Price of Risk
In this section we de ne the market price of risk. In the Black-Scholes model,
risk is represented by a single Brownian motion and there is a unique market
price associated with it. When the market is subject to several sources of
risk and the tradeable securities are not enough to complete the market, the
market price of risk is not unique anymore.
Let us consider a market where traded are the risk-free asset B, yielding
the risk-free rate , and two risky securities S1 and S2 , with dynamics driven
by a single source of risk represented by a standard Brownian motion W
under the physical P :
dS1 (t) = S1 (t) (1 (t; S1 (t)) dt + 1 (t; S1 (t)) dW )
dS2 (t) = S2 (t) (2 (t; S2 (t)) dt + 2 (t; S2 (t)) dW )
The market price attached to the risk involved by S1 is given by
1 (t; S1 (t)) 0 ;
1 (t; S1 (t))
and thus represents the excess return over the risk-free rate divided by the
volatility of security S1 (the volatility measures the riskiness of security S1 ):
Analogously, the market price risk attached to S2 is given by:
2 (t; S2 (t)) 0 :
2 (t; S2 (t))
Conclusion 105 Under NA the following equality holds
1 (t; S1 (t)) 0 (t; S2 (t)) 0 = 2
1 (t; S1 (t))
2 (t; S2 (t))
14.3. MARKET PRICE OF RISK
203
so that it is possible to speak of market price of the risk W , the unique risk
source in the market. We denote their price by:
=
1 (t; S1 (t)) 0 (t; S2 (t)) 0 = 2
:
1 (t; S1 (t))
2 (t; S2 (t))
Proof. Since the two securities S1 and S2 depend on the same source
of risk, they can be used to construct a dynamic portfolio that is locally
risk-free. Let us denote by h1 the number of units of stock S1 and by h2 the
number of units of stock S2 at time t. To have a self- nancing portfolio, we
require its variation between t and t + dt to be given by
d (t) = h1 dS1 (t) + h2 dS2 (t) :
We denote by w1 = w1 (t) the percentage invested at time t in the security
S1 , i.e.
h1 S 1
w1 =
;
and by w2 = w2 (t) = 1 0 w1 (t) the percentage invested in the security S2 ,
i.e.
h2 S 2
:
w2 =
The dynamics of the portfolio set up in this way are described by the SDE
d (t) = (t) ((w1 1 + (1 0 w1 )2 ) dt + (w1 1 + (1 0 w1 ) 2 ) dW )
To make locally risk-free, i.e. to kill the di¤usion coe¢cient, we need to
have
2
w1 =
2 0 1
1
and hence 1 0 w1 = 0 20
:
1
Under NA, the portfolio must yield the same instantaneous return as
the risk-free asset, i.e.
w1 1 + (1 0 w1 ) 2 = By substituting the expression obtained for w1 into (14.7), we get
2
1
1
2 = + 0
2 0 1
2 0 1
(1 0 ) 2 = (2 0 ) 1
(14.7)
204
PRICING IN THE BLACK-SCHOLES MARKET
from which the following results:
(t; S2 (t)) 0 1 (t; S1 (t)) 0 = 2
=: 1 (t; S1 (t))
2 (t; S2 (t))
The ratio is by de nition the market price of the risk W . The previous
equality shows that it is possible to de ne the market price of risk associated
with the risk source W because it is univocally determined in the market
(possibly as a function of time), since it is the same for two securities S1
and S2 depending on the very same source of randomness. 4
The existence of a unique market price of the risk associated with W is
due to the interaction between NA and market completeness, i.e. by the possibility of replicating any derivative dependent on the source of randomness
W . Moreover, the market considered has the following features:
Remark 106 It is possible to synthesize the risk-free asset with the two
securities S1 and S2 .
Proof. The portfolio that we have constructed in the previous proof is
exactly the one replicating the risk-free asset. In order to have
(0) = 1;
the percentages w1 and w2 invested in S1 and S2 must be linked to the corresponding number of units h1 and h2 through the following simple relation
wi (0) = hi (0)Si (0);
for i = 1; 2, from which
hi (0) =
wi (0)
:
Si (0)
The di¤erential equation satis ed by ; given by the choice of percentages
w1 and w2 , is the following
d (t) = (t) 1 dt
(0) = 1
and admits a unique solution equal to
(t) = 1 1 et
14.3. MARKET PRICE OF RISK
205
Hence, the number of units of stocks in are given for each t by
h1 (t) = w1
and
h2 (t) = w2
(t)
et
2
=
S1 (t)
2 0 1 S1 (t)
(t)
et
1
=0
:
S2 (t)
2 0 1 S2 (t)
Remark 107 If the market price of risk is unique, there exists a martingale
measure Q under which the discounted price processes of both securities S1
and S2 are martingales.
Proof.
We prove the result in the particular case of constant di¤usion and drift coe¢cients. The processes S13 (t) = e0t S1 (t) and S23 (t) =
e0t S2 (t) obey (under the physical measure P) the SDEs
dS13 (t) = S13 (t) ((1 0 ) dt + 1 dW )
dS23 (t) = S23 (t) ((2 0 ) dt + 2 dW ) :
Collecting 1 in the righthand side of the rst equation and 2 in the second
equation, we obtain
dS13 (t) = S13 (t) 1 (dt + dW )
dS23 (t) = S23 (t) 2 (dt + dW ) :
Guided by Girsanov’s Theorem (Theorem 87), we set
=
0
1 0 = 2
1
2
and recall that the probability measure Q, under which W 3 (t) = t + W (t)
is a standard Brownian motion, has density with respect to P given by
1
2
L = exp 0W (T ) 0 () T :
2
Hence
dS13 (t) = S13 (t) 1 dW 3 (t)
dS23 (t) = S23 (t) 2 dW 3 (t);
from which follows that S13 and S23 are martingales under Q:4
206
PRICING IN THE BLACK-SCHOLES MARKET
The existence of a measure Q making discounted asset prices martingales
is essentially equivalent to the absence of arbitrage (First Fundamental Theorem of Asset Pricing). As we have already seen, the market made of the
risk-free asset and of the two risky securities S1 and S2 is arbitrage-free if
1 0 0
= 2
;
1
2
because then the measure Q, de ned through
1
2
L = exp 0W (T ) 0 () T
2
in Girsanov’s Theorem, makes martingales both S13 and S23 (the absence of
arbitrage is hence guaranteed by the First Fundamental Theorem of Asset
Pricing). If we introduce in the market a European derivative security,
its no-arbitrage price is given by the expected value under the measure Q
of the discounted payo¤ at maturity. Speci cally, the market is complete,
the measure Q is unique and the price thus obtained is the only one not
generating arbitrages in the market. After this reminder, we are ready to
price an option on the maximum between two security prices.
Example 108 Option on the maximum of two securities.
Let us consider a market with two securities, S1 and S2 , having lognormal
dynamics:
dS1 = 1 S1 dt + 1 S1 dW
dS2 = 2 S2 dt + 2 S2 dW;
with 10
= 20
, and the risk-free asset B yielding the risk-free rate .
1
2
Compute the time-0 no-arbitrage price of the European derivative with the
following payo¤ at maturity T :
f (S1 (T ) ; S2 (T )) = max [S1 (T ) ; S2 (T )]
Solution. The no-arbitrage price of the derivative security is given by
h
i
V (0) = EQ e0T max (S1 (T ) ; S2 (T )) :
From the dynamics of S13 and S23 under Q, we deduce that S1 and S2 can
be expressed as
(
1 2
3
S1 (t) = S1 (0) e(0 2 1 )t+1 W (t)
1 2
3
S2 (t) = S2 (0) e(0 2 2 )t+2 W (t) ;
14.3. MARKET PRICE OF RISK
207
where W 3 is a standard Brownian motion under Q:
The payo¤ of the derivative security is given by
max (S1 (T ) ; S2 (T )) = S1 (T ) se S1 (T ) S2 (T )
max (S1 (T ) ; S2 (T )) = S2 (T ) se S1 (T ) < S2 (T )
Hence, we have
V (0) = I1 + I2
with
I1 = e0T
and
I2 = e
0T
Z
S1 (T )S2 (T )
S1 (T ) dQ
Z
S1 (T )<S2 (T )
S2 (T ) dQ
Recalling the distribution of S1 and S2 under Q, we have that S1 (T ) S2 (T ) if and only if
S1 (0) e(0 2 1 )T +1 W
1
that is
2
3 (T )
S2 (0) e(0 2 2 )T +2 W
1
2
3 (T )
;
1
S2 (0) 1 0 2
2
+
+ 2 T
ln
S1 (0) 2 1
Q p
Q
when 1 > 2 : Recalling now that W 3 (T ) T 1 Z, with Z N (0; 1); we
set
1
1
S2 (0) 1 0 2
p
ln
z=
+
1 + 22 T
S1 (0) 2
( 1 0 2 ) T
and have that I1 is equal to
Z +1
p
1 2
I1 = e0T
S1 (0) e(0 2 1 )T +1 T 1z fZ (z) dz
z
Z +1
p 2
1
1
=
S1 (0) p e0 2 (z01 T ) dz =
2
z
p = S1 (0) 1 0 N z 0 1 T
;
1
W (T ) 1 0 2
3
while I2 is equal to
I2 = e
0T
Z
z
Z
z
01
S2 (0) e(0 2 2 )T +2
1
2
p
T 1z
fZ (z) dz
p 2
1
1
S2 (0) p e0 2 (z02 T ) dz =
2
01
p = S2 (0) N z 0 2 T 4
=