Springer Undergraduate Texts in Mathematics
and Technology
Series Editors:
J.M. Borwein
H. Holden
V.H. Moll
More information about this series at http://www.springer.com/series/7438
Arlie O. Petters • Xiaoying Dong
An Introduction to Mathematical
Finance with Applications
Understanding and Building Financial Intuition
123
Arlie O. Petters
Department of Mathematics
Duke University
Durham, NC, USA
Xiaoying Dong
Department of Mathematics
Duke University
Durham, NC, USA
ISSN 1867-5506
ISSN 1867-5514 (electronic)
Springer Undergraduate Texts in Mathematics and Technology
ISBN 978-1-4939-3781-3
ISBN 978-1-4939-3783-7 (eBook)
DOI 10.1007/978-1-4939-3783-7
Library of Congress Control Number: 2016939449
Mathematics Subject Classification (2010): 91Gxx
© Arlie O. Petters and Xiaoying Dong 2016
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction
on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and
regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed
to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty,
express or implied, with respect to the material contained herein or for any errors or omissions that may have been
made.
Printed on acid-free paper
This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG Switzerland
To my loving wife, Elizabeth Petters, for being at my
side unconditionally and my child, Preston Petters,
who inspires me with his intense curiosity.
A.O. Petters
To my dear husband and best friend, Xin Zhou. I could
not imagine to complete my part of the contribution to
this book without his love and support.
X. Dong
Preface
Rationale and Aim
Given the increasing intricacies and interconnectedness of financial firms’ activities and the potential opportunities and risks to which they expose themselves and the world’s economy, the next generation of financial engineers
needs to master an extensive array of mathematical financial models. Indeed,
one of the current challenges in finance is that the complexity of modern securities and markets has forced modelers to employ increasingly sophisticated
mathematical tools to address financial issues, creating a widening gap between the qualitative and quantitative approaches to finance.
Our book seeks to address this gap by introducing the quantitative aspects
of finance to students with either a qualitative background or no background
in the subject. At a firm the traders, risk managers, etc. employ proprietary analytical and numerical models custom made to the needs of their firm. However, since open access to such models is prohibited, the book instead strives
to give students a fundamental understanding of key financial ideas and tools
that form the basis for building more realistic models, including those of a
proprietary nature.
Distinctive Features and Benefits
This book is distinct in how it emphasizes and pedagogically conveys in an
accessible manner the theoretical understanding and applications of the mathematical models forming key pillars of modern finance.
First, the book keeps a good balance between mathematical derivation and
description for the sake of providing an adequate level of rigor and depth in
mathematics and maintaining accessibility to the reader, which in turn adds
flexibility of material selection for the instructor (e.g., Chapter 7 may be taught
earlier). Specifically, this book addresses the gap between textbooks that ofvii
viii
Preface
fer a theoretical treatment without many applications and those that simply
present and apply formulas without appropriately deriving them. Indeed, theoretical understanding is incomplete without enough practice in applications,
and applications are risky without a rigorous theoretical understanding. To
accomplish this, the book contains numerous carefully chosen examples and
exercises that reinforce a student’s conceptual understanding and develop a
facility with applications. Indeed, the exercises are divided into conceptual,
application, and theoretical problems that probe the material deeper.
Second, beyond a few required undergraduate mathematics courses (see
Prerequisites below), this book is essentially self-contained. The large number of necessary financial terminologies and concepts can be overwhelming
to a student new to finance. For this reason, after introducing some central,
big-picture financial ideas in the first chapter, we present the financial minutia along the way as needed. We have tried to make the book self-contained
in this regard through thoughtfully chosen illustrative applications starting at
the ground level with simple interest. We then gradually increase the difficulty
as the book develops, ranging across compound interest, annuities, portfolio theory,
capital market theory, portfolio risk measures, the role of linear factor models in portfolio risk attribution, binomial tree models, stochastic calculus, derivatives, the martingale approach to derivative pricing, the Black-Scholes-Merton model, and the Merton
jump-diffusion model.
Third, the book is also useful for students preparing either for higher level
study in mathematical finance or for a career in actuarial science. For example,
the syllabi for the actuarial Financial Mathematics Exam (Exam 2/FM) and
Models of Financial Economics Exam (Exam 3F/MFE) include many topics
covered in the book.
Prerequisites
The required mathematics consists of introductory courses on multivariable
calculus, probability, and linear algebra. Along the way, we introduce additional mathematical tools as needed—e.g., some measure theory is presented
from scratch.
No background in finance is assumed. As noted above, the necessary financial
concepts and tools are introduced in the text, with the first chapter giving an
overview of several common finance terminologies associated with securities
and securities markets.
Our book does not require computer programming. In our experience, finance courses based on computer programming are best taken after students
have developed a fundamental understanding of the theoretical architecture
of financial models.
Preface
ix
Audience
The text is aimed at advanced undergraduates and master’s degree students who are
either new to finance or want a more rigorous treatment of the mathematical
models used in finance. The students typically are from economics, mathematics, engineering, physics, and computer science.
We also believe that a faculty member who is teaching finance for the first
time will find this introduction readily manageable. Professionals working in
finance who would like a refresher or even clarification on some of the theoretical and conceptual aspects of mathematical finance will benefit from the
text.
Scope and Guide
The chapters are organized naturally into four parts and range over the following topics:
- Part I (Chapters 1 and 2):
introduction to securities markets and the time value of money
- Part II (Chapters 3 and 4):
Markowitz portfolio theory, capital market theory, and portfolio risk measures
- Part III (Chapters 5 and 6):
modeling underlying securities using binomial trees and stochastic calculus
- Part IV (Chapters 7 and 8):
derivative securities, BSM model, and Merton jump-diffusion model
The material was tested in courses offered to upper-level undergraduates and
master’s degree students. Below are two examples of possible topics that may
serve as a guide for semester-long courses:
- Introduction to Mathematical Finance: securities markets (Chapter 1), the time
value of money (Chapter 2), Markowitz portfolio theory, capital market
theory, and portfolio risk measures (Chapters 3–4), binomial security pricing (Chapter 5, omit most derivations), Itô’s formula and geometric Brownian motion (Sections 6.8 and 6.9), forwards, futures, and options (Sections 7.2, 7.3, and 7.5), and call option pricing with applications (Sections 8.3,
8.2.2, 8.5, and 8.6.2).
- Introduction to Financial Derivatives: modeling underliers in discrete time
(Sections 5.1–5.3), stochastic calculus and modeling underliers in continuous
time (Section 5.4 and Chapter 6), general aspects of forwards, futures, swaps,
and options, including trading strategies (Chapters 7), the Black-ScholesMerton (BSM) model, BSM p.d.e. approach to pricing European-style options, risk-neutral approach to pricing European-style options, applications
to warrants, delta hedging, managing portfolio risk, and extension of the
BSM model to the Merton jump-diffusion model (Chapter 8).
x
Preface
A year-long course on introductory mathematical finance can be based on the
entire book. The book can also be used as a reference for students enrolled in a
mathematical finance independent study course.
Acknowledgments
Specials thanks to the following individuals for their feedback and assistance:
Daniel Aarhus
Amir Aazami
Stanley Absher
Vibhav Agarwal
Hengjie Ai
Mitesh Amarthaluru
Vlad Bouchouev
Michael Brandt
Esteban Chavez
Rui Chen
Kyuwon Choi
Qian Deng
Christian Drappi
Zachary Freeman
Tingran Gao
William Grisaitis
Xiaosheng Guo
Zhonglin Han
John Hyde
Huseyin Kortmaz
Baolei Li
Junchi Li
Li Li
Nan Li
Qiao Li
Li Liang
Lu Liu
Ruisi Ma
Tanya Mallavarapu
Xavier Mela
Vadim Mokhnatkin
Julia Ni
James Nolen
Vivek Oberoi
Feng Pan
Chloe Peng
Hal Press
Hui Qi
Zhaozhen Qian
Hayagreev Ramesh
Emma Rasiel
Tianhua Ren
Chelsea Richwine
Irving Salvatierra
Andrew Schretter
Yuhang Si
John Sias
Maxwell Stern
Lingran Sun
Alberto Teguia
Nicholas Tenev
Dominick Totino
Chi Trinh
Dan Turtel
Kari Vaughn
Robert Vanderbei
Kevin Wan
Chenyu Wang
David Williams
Chao Xu
Hangjun Xu
Lu Xu
Junkai Xue
Chao Yang
Jiahui Yang
Ashley Yeager
Jeong Yoo
Yanchi Yu
Yunliang Yu
Javier Zapata
Xiaodong Zhai
Biyuan Zhang
Yang Zhang
Bowen Zhao
Ruiyang Zhao
Xiaoyang Zhuang
Zilong Zou
We are also thankful to Elizabeth Loew of Springer for her support and guidance along the entire way and to Lisa Goldberg for her valuable comments and
constructive suggestions. AP is indebted to Duke University for providing
the financial support needed to hire many students who assisted with writing computer codes, checking calculations, etc. He is also extremely grateful to
his wife, Elizabeth Petters, for her patience, love, and steadfast encouragement
throughout the project. XD would like to express her gratitude to her husband
Xin Zhou who saw her through this book and offered great suggestions.
Durham, NC, USA
Durham, NC, USA
2016
A.O. Petters
X. Dong
Contents
1
Preliminaries on Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Primer on Banks and Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Banks and the Federal Funds Rate . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Short-Term and Long-Term Rates and Yield Curves . . . . . . . 4
1.2 A Primer on Securities Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Securities Markets Organization . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Professional Participants in Securities Markets . . . . . . . . . . . 8
1.2.3 Bid-Ask Spreads and Market Liquidity . . . . . . . . . . . . . . . . . . 8
1.2.4 Trading Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Economic Indicators That May Affect Financial Markets . . . . . . . . 10
2
The Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Interest Rate and Return Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Required Return Rate and the Risk-Free Rate . . . . . . . . . . . .
2.2.3 Total Return Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Compounding: Nonnegative Integer Number of Periods . .
2.4.2 Compounding: Nonnegative Real Number of Periods . . . .
2.4.3 Fractional Compounding Versus Simple Interest . . . . . . . . .
2.4.4 Continuous Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Generalized Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Varying Interest and Varying Compounding Periods . . . . .
2.5.2 APR Versus APY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Geometric Mean Return Versus Arithmetic Mean Return . .
2.6 The Net Present Value and Internal Rate of Return . . . . . . . . . . . . .
2.6.1 Present Value and NPV of a Sequence of Net Cash Flows .
2.6.2 The Internal Return Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
14
15
15
16
17
20
21
22
24
30
31
31
31
33
35
38
38
41
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2.6.3 NPV and IRR for General Net Cash Flows . . . . . . . . . . . . . . .
2.7 Annuity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Future and Present Values of Simple Ordinary Annuities . .
2.7.2 Amortization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Annuities with Varying Payments and Interest Rates . . . . .
2.8 Applications of Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 Saving, Borrowing, and Spending . . . . . . . . . . . . . . . . . . . . . . .
2.8.2 Equity in a House . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.3 Sinking Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Applications to Stock Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.1 The Dividend Discount Model . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.2 Present Value of Preferred and Common Stocks . . . . . . . . . .
2.10 Applications to Bond Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.1 Bond Terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.2 Bond Prices Versus Interest Rates and Yield to Maturity . . .
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
42
46
47
53
56
59
59
61
62
63
64
65
66
66
69
72
72
73
78
81
Markowitz Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1 Markowitz Portfolio Model: The Setup . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 Security Return Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1.2 What About Multivariate Normality of Security Return
Rates? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.1.3 Investors and the Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . 87
3.1.4 The One-Period Assumption, Weights, and Short Selling . . 88
3.1.5 Expected Portfolio Return Rate . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.1.6 Portfolio Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1.7 Risks and Covariances of the Portfolio’s Securities . . . . . . . . 96
3.1.8 Expectation and Volatility of Portfolio Log Return . . . . . . . . 100
3.2 Two-Security Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.2 Efficient Frontier of a Two-Security Portfolio . . . . . . . . . . . . . 107
3.2.3 Reducing Risk Through Diversification . . . . . . . . . . . . . . . . . . 114
3.3 Efficient Frontier for N Securities with Short Selling . . . . . . . . . . . . 117
3.3.1 N-Security Portfolio Quantities in Matrix Notation . . . . . . . 118
3.3.2 Derivation of the N-Security Efficient Frontier . . . . . . . . . . . 120
3.4 N-Security Efficient Frontier Without Short Selling . . . . . . . . . . . . . 126
3.5 The Mutual Fund Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.5.1 The Global Minimum-Variance Portfolio . . . . . . . . . . . . . . . . . 128
3.5.2 The Diversified Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Contents
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3.5.3 The Mutual Fund Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.6 Investor Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.6.1 Utility Functions and Expected Utility Maximization . . . . . 131
3.6.2 Risk-Averse, Risk-Neutral, and Risk-Seeking Investors . . . . 133
3.7 Diversification and Randomly Selected Securities . . . . . . . . . . . . . . 138
3.7.1 Mean Portfolio Variance and the Uniform Dirichlet
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.7.2 Mean Portfolio Variance using the NASDAQ . . . . . . . . . . . . . 142
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.8.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.8.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3.8.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4
Capital Market Theory and Portfolio Risk Measures . . . . . . . . . . . . . . 151
4.1 The Capital Market Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.1.1 The Capital Market Line (CML) . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.1.2 Expected Return and Risk of the Market Portfolio . . . . . . . . 157
4.1.3 The Capital Asset Pricing Model (CAPM) . . . . . . . . . . . . . . . . 158
4.1.4 The Security Market Line (SML) . . . . . . . . . . . . . . . . . . . . . . . . 163
4.1.5 CAPM Security Risk Decomposition . . . . . . . . . . . . . . . . . . . . 164
4.2 Portfolio Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2.1 The Sharpe Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.2.2 The Sortino Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.2.3 The Maximum Drawdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.4 Quantile Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.2.5 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.2.6 Conditional Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.2.7 Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.3 Introduction to Linear Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.3.1 Definition and Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.3.2 Portfolio Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . 189
4.3.3 Factor Categorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.3.4 Alpha and Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.3.5 CAPM Beta Versus Linear Factor Beta . . . . . . . . . . . . . . . . . . . 195
4.3.6 Fama-French Three-Factor Model . . . . . . . . . . . . . . . . . . . . . . . 196
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.4.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.4.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.4.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
xiv
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5
Binomial Trees and Security Pricing Modeling . . . . . . . . . . . . . . . . . . . . 209
5.1 The General Binomial Tree Model of Security Prices . . . . . . . . . . . . 209
5.2 The Cox-Ross-Rubinstein Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.2.1 The Real-World CRR Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.2.2 The Risk-Neutral CRR Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.3 Continuous-Time Limit of the CRR Pricing Formula . . . . . . . . . . . . 237
5.3.1 The Lindeberg Central Limit Theorem . . . . . . . . . . . . . . . . . . . 237
5.3.2 The Continuous-Time Security Price Formula . . . . . . . . . . . . 241
5.4 Basic Properties of Continuous-Time Security Prices . . . . . . . . . . . . 246
5.4.1 Some Statistical Formulas for Continuous-Time Security
Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.4.2 Some Probability Formulas for Continuous-Time
Security Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.5.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.5.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.5.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
6
Stochastic Calculus and Geometric Brownian Motion Model . . . . . . 253
6.1 Stochastic Processes: The Evolution of Randomness . . . . . . . . . . . . 253
6.1.1 Notation for Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.1.2 Basic Concepts of Random Variables . . . . . . . . . . . . . . . . . . . . 257
6.1.3 Basic Concepts of Stochastic Processes . . . . . . . . . . . . . . . . . . 260
6.1.4 Convergence of Random Variables . . . . . . . . . . . . . . . . . . . . . . 265
6.1.5 Skewness and Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.2 Filtrations and Adapted Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.2.1 Filtrations: The Evolution of Information . . . . . . . . . . . . . . . . 268
6.2.2 Conditional Expectations: Properties and Intuition . . . . . . . 270
6.2.3 Adapted Processes: Definition and Intuition . . . . . . . . . . . . . 273
6.3 Martingales: A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.3.2 Martingale as a Necessary Condition of an Efficient Market 277
6.4 Modeling Security Price Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.4.1 From Deterministic Model to Stochastic Model . . . . . . . . . . . 278
6.4.2 Innovation Processes: An Intuition . . . . . . . . . . . . . . . . . . . . . . 279
6.4.3 Securities Paying a Continuous Cash Dividend . . . . . . . . . . . 281
6.5 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.5.1 Definition of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.5.2 Some Properties of Brownian Motion Paths . . . . . . . . . . . . . . 284
6.5.3 Visualization of Brownian Paths . . . . . . . . . . . . . . . . . . . . . . . . 285
6.5.4 Markov Property for Brownian Motion . . . . . . . . . . . . . . . . . . 288
6.6 Quadratic Variation and Covariation . . . . . . . . . . . . . . . . . . . . . . . . . . 289
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6.6.1 Motivation, Definition, and Notation . . . . . . . . . . . . . . . . . . . . 289
6.6.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6.6.3 Quadratic Variation and Covariation Properties of BM . . . . 293
6.6.4 Significance of Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . 297
6.7 Itô Integral: A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
6.7.1 Importance of Itô Integral with Respect to BM . . . . . . . . . . . . 299
6.7.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
6.7.3 A Famous Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.8 Itô’s Formula for Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.8.1 Itô Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.8.2 Itô’s Lemma for Brownian Motion . . . . . . . . . . . . . . . . . . . . . . 304
6.8.3 Risk-Neutral Probability Measure . . . . . . . . . . . . . . . . . . . . . . . 309
6.8.4 Girsanov Theorem for a Single Brownian Motion . . . . . . . . . 311
6.9 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
6.9.1 GBM: Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
6.9.2 GBM: Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
6.9.3 Relation Between Binomial Tree Model and GBM Model . . 317
6.10 BM as a Limit of Simple Symmetric RW . . . . . . . . . . . . . . . . . . . . . . . 320
6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
6.11.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
6.11.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
6.11.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
7
Derivatives: Forwards, Futures, Swaps, and Options . . . . . . . . . . . . . . 329
7.1 Derivative Securities: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.1.2 Basic Functions of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 331
7.1.3 Characteristics of Derivative Valuation . . . . . . . . . . . . . . . . . . 332
7.1.4 No-Arbitrage Principle and Law of One Price . . . . . . . . . . . . 334
7.2 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7.2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7.2.2 Forwards on Assets Paying a Continuous Cash Dividend . 340
7.2.3 Forward Price Formula and the Spot-Forward Parity . . . . . 341
7.2.4 Forward Value Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
7.3 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
7.3.1 Evolution from Forwards to Futures . . . . . . . . . . . . . . . . . . . . . 345
7.3.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
7.3.3 Impact of Daily Settlement: A Brief Discussion . . . . . . . . . . . 347
7.4 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
7.4.1 A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
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7.5 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
7.5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
7.5.2 How Options Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
7.5.3 Terminal Payoff and Profit Diagrams . . . . . . . . . . . . . . . . . . . . 359
7.5.4 Market Sentiment Terminologies and Option Moneyness . 363
7.5.5 Option Strategies: Straddle, Strangle, and Spread . . . . . . . . . 365
7.5.6 Put-Call Parity for European Options Revisited . . . . . . . . . . 368
7.5.7 Relation Among Put, Call, and Forward . . . . . . . . . . . . . . . . . 369
7.5.8 Intrinsic Value and Time Value . . . . . . . . . . . . . . . . . . . . . . . . . . 370
7.5.9 Some General Relations of Options . . . . . . . . . . . . . . . . . . . . . . 372
7.5.10 Put-Call Parity for American Options . . . . . . . . . . . . . . . . . . . 373
7.5.11 Boundary Conditions for European Options . . . . . . . . . . . . . 376
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7.6.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7.6.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
7.6.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8
The BSM Model and European Option Pricing . . . . . . . . . . . . . . . . . . . . 383
8.1 The BSM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
8.1.1 Marketplace Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
8.1.2 Money Market Account and the Underlier Model . . . . . . . . 386
8.1.3 Self-Financing, Replicating Portfolio . . . . . . . . . . . . . . . . . . . . . 388
8.1.4 Derivation of the BSM p.d.e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
8.2 Applications of BSM Pricing to European Calls and Puts . . . . . . . . 392
8.2.1 Solving the BSM p.d.e. for European Calls . . . . . . . . . . . . . . . 392
8.2.2 BSM Pricing Formula for European Puts . . . . . . . . . . . . . . . . . 395
8.2.3 Delta and the Partial Derivative Relative to Strike Price . . . 396
8.2.4 European Call and Put Deltas at Expiration . . . . . . . . . . . . . . 398
8.3 Application to Pricing Warrants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
8.4 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
8.4.1 Review of Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 400
8.4.2 From BSM Pricing to Risk-Neutral Pricing . . . . . . . . . . . . . . . 404
8.4.3 The Fundamental Theorems of Asset Pricing . . . . . . . . . . . . . 408
8.4.4 Risk-Neutral Pricing of European Calls and Puts . . . . . . . . . 410
8.5 Binomial Approach to Pricing European Options . . . . . . . . . . . . . . . 411
8.5.1 One-Period Binomial Pricing by Self-Financing Replication 412
8.5.2 One-Period Binomial Pricing by Risk Neutrality . . . . . . . . . . 415
8.5.3 Two-Period Binomial Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
8.5.4 n-Period Binomial Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
8.6 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
8.6.1 Theoretical Delta Hedging for European Calls . . . . . . . . . . . . 422
8.6.2 Application of Delta Hedging to Selling European Calls . . 427
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8.7 Option Greeks and Managing Portfolio Risk . . . . . . . . . . . . . . . . . . . 433
8.7.1 Option Greeks for Portfolios and the BSM p.d.e. . . . . . . . . . . 433
8.7.2 Delta-Neutral Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
8.7.3 Delta-Gamma-Neutral Portfolios . . . . . . . . . . . . . . . . . . . . . . . . 438
8.8 The BSM Model Versus Market Data . . . . . . . . . . . . . . . . . . . . . . . . . . 440
8.8.1 Jumps in Security Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
8.8.2 Skewness and Kurtosis in Security Log Returns . . . . . . . . . . 441
8.8.3 Volatility Skews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
8.9 A Step Beyond the BSM Model: Merton Jump Diffusion . . . . . . . . 448
8.9.1 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
8.9.2 The MJD Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
8.9.3 Illustration of MJD Jump, Skewness, and Kurtosis . . . . . . . . 456
8.9.4 No-Arbitrage Condition and Market Incompleteness . . . . . 458
8.9.5 Pricing European Calls with an MJD Underlier . . . . . . . . . . . 461
8.9.6 MJD Volatility Smile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
8.10 A Glimpse Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
8.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
8.11.1 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
8.11.2 Application Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
8.11.3 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477