Financial Mathematics: A Comprehensive Treatment in Discrete Time (Chapman and Hall/CRC Financial Mathematics Series)

590 Pages·2021·8.853 MB·English

by  Giuseppe Campolieti, Roman N. Makarov

Financial Mathematics
Chapman & Hall/CRC Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
appeal to both academics and practitioners. The inclusion of numerical code and concrete real-world
examples is highly encouraged.
Series Editors
M.A.H. Dempster
Centre for Financial Research
Department of Pure Mathematics and Statistics
University of Cambridge
Dilip B. Madan
Robert H. Smith School of Business
University of Maryland
Rama Cont
Department of Mathematics
Imperial College
Robert A. Jarrow
Lynch Professor of Investment Management
Johnson Graduate School of Management
Cornell University
Handbook of Financial Risk Management
Thierry Roncalli
Optional Processes
Stochastic Calculus and Applications
Mohamed Abdelghani, Alexander Melnikov
Machine Learning for Factor Investing
Guillaume Coqueret and Tony Guida
Malliavin Calculus in Finance
Theory and Practice
Elisa Alos, David Garcia Lorite
Risk Measures and Insurance Solvency Benchmarks
Fixed-Probability Levels in Renewal Risk Models
Vsevolod K. Malinovskii
Giuseppe Campolieti
Professor, Department of Mathematics,
Wilfrid Laurier University,
Waterloo, Ontario
Roman N. Makarov
Associate Professor, Department of Mathematics,
Wilfrid Laurier University,
Waterloo, Ontario
For more information about this series please visit: https://www.crcpress.com/Chapman-and-
HallCRC-Financial-Mathematics-Series/book-series/CHFINANCMTH
Financial Mathematics
A Comprehensive Treatment
in Discrete Time
Second Edition
by
Giuseppe Campolieti
Professor, Department of Mathematics,
Wilfrid Laurier University,
Waterloo, Ontario
Roman N. Makarov
Associate Professor, Department of Mathematics,
Wilfrid Laurier University,
Waterloo, Ontario
Second edition published 2021
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To our students

Contents
List of Figures xiii
List of Tables xvii
Preface xix
I Introduction to Pricing and Management of Financial Secu-
rities 1
1 Mathematics of Compounding 3
1.1 Interest and Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Amount Function and Return . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Periodic Compound Interest . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Continuous Compound Interest . . . . . . . . . . . . . . . . . . . . . 10
1.1.5 Equivalent Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.6 Continuously Varying Interest Rates . . . . . . . . . . . . . . . . . . 14
1.2 Time Value of Money and Cash Flows . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 Equations of Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2 Deterministic Cash Flows and Their Net Present Values . . . . . . . 18
1.3 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Simple Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.2 Determining the Term of an Annuity . . . . . . . . . . . . . . . . . . 26
1.3.3 General Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.4 Perpetuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.5 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.1 Introduction and Terminology. . . . . . . . . . . . . . . . . . . . . . 31
1.4.2 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.3 Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.4 Serial Bonds, Strip Bonds, and Callable Bonds . . . . . . . . . . . . 34
1.5 Yield Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.1 Internal Rate of Return and Evaluation Criteria . . . . . . . . . . . 36
1.5.2 Determining Yield Rates for Bonds . . . . . . . . . . . . . . . . . . . 38
1.5.3 Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5.4 The Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.6 Yield Risk and Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.6.1 Immunization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vii
viii Contents
2 Primer on Pricing Risky Securities 65
2.1 Stocks and Stock Price Models . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.1 Underlying Assets and Derivative Securities . . . . . . . . . . . . . . 65
2.1.2 Basic Assumptions for Asset Price Models . . . . . . . . . . . . . . . 66
2.2 Basic Price Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.1 A Single-Period Binomial Model . . . . . . . . . . . . . . . . . . . . 67
2.2.2 A Discrete-Time Model with a Finite Number of States . . . . . . . 74
2.2.3 Introducing the Binomial Tree Model . . . . . . . . . . . . . . . . . 77
2.2.4 Recursive Construction of a Binomial Tree . . . . . . . . . . . . . . 82
2.2.5 Self-Financing Investment Strategies in the Binomial Model . . . . . 83
2.2.6 Log-Normal Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 86
2.3 Arbitrage and Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . 91
2.3.1 The Law of One Price . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.3.2 A First Look at Arbitrage in the Single-Period Binomial Model . . . 93
2.3.3 Arbitrage in the Binomial Tree Model . . . . . . . . . . . . . . . . . 95
2.3.4 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . . 96
2.3.5 Martingale Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.3.6 Risk-Neutral Log-Normal Model . . . . . . . . . . . . . . . . . . . . 99
2.4 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.5 Dividend Paying Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3 Portfolio Management 113
3.1 Expected Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.1.1 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.1.2 Mean-Variance Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.2 Portfolio Optimization for Two Assets . . . . . . . . . . . . . . . . . . . . . 122
3.2.1 Portfolio of Two Risky Assets . . . . . . . . . . . . . . . . . . . . . . 122
3.2.2 Portfolio Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.2.3 The Minimum Variance Portfolio . . . . . . . . . . . . . . . . . . . . 129
3.2.4 Selection of Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . 131
3.3 Portfolio Optimization for N Assets . . . . . . . . . . . . . . . . . . . . . . 137
3.3.1 Portfolios of Several Assets . . . . . . . . . . . . . . . . . . . . . . . 137
3.3.2 The Minimum Variance Portfolio . . . . . . . . . . . . . . . . . . . . 140
3.3.3 Minimum Variance Portfolio Line . . . . . . . . . . . . . . . . . . . . 141
3.3.4 Case without Short Selling . . . . . . . . . . . . . . . . . . . . . . . 144
3.3.5 Maximum Expected Utility Portfolio . . . . . . . . . . . . . . . . . . 145
3.3.6 Efficient Frontier and Capital Market Line. . . . . . . . . . . . . . . 147
3.4 The Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . 152
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4 Primer on Derivative Securities 163
4.1 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.1.1 No-Arbitrage Evaluation of Forward Contracts . . . . . . . . . . . . 165
4.1.2 Value of a Forward Contract . . . . . . . . . . . . . . . . . . . . . . 170
4.2 Basic Options Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.2.1 Payoffs of Standard Options . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.2 Put-Call Parities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.2.3 Properties of European Options. . . . . . . . . . . . . . . . . . . . . 176
4.2.4 Early Exercise and American Options . . . . . . . . . . . . . . . . . 178
4.2.5 Nonstandard European-Style Options . . . . . . . . . . . . . . . . . 180
Contents ix
4.3 Fundamentals of Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . 183
4.3.1 Pricing of European-Style Derivatives in the Binomial Tree Model . 183
4.3.2 Pricing of American Options in the Binomial Tree Model . . . . . . 191
4.3.3 OptionPricingintheLog-NormalModel:TheBlack–Scholes–Merton
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.3.4 Greeks and Hedging of Options . . . . . . . . . . . . . . . . . . . . . 196
4.3.5 Black–Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
II Discrete-Time Modelling 217
5 Single-Period Arrow–Debreu Models 219
5.1 Specification of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.1.1 Finite-State Economy. Vector Space of Payoffs. Securities . . . . . . 219
5.1.2 Initial Price Vector and Payoff Matrix . . . . . . . . . . . . . . . . . 222
5.1.3 Portfolios of Base Securities . . . . . . . . . . . . . . . . . . . . . . . 223
5.2 Analysis of the Arrow–Debreu Model . . . . . . . . . . . . . . . . . . . . . 224
5.2.1 Redundant Assets and Attainable Securities . . . . . . . . . . . . . . 224
5.2.2 Completeness of the Model . . . . . . . . . . . . . . . . . . . . . . . 227
5.3 No-Arbitrage Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
5.3.1 The Law of One Price . . . . . . . . . . . . . . . . . . . . . . . . . . 229
5.3.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.3.3 The First Fundamental Theorem of Asset Pricing. . . . . . . . . . . 232
5.3.4 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . . 236
5.3.5 The Second Fundamental Theorem of Asset Pricing . . . . . . . . . 239
5.3.6 Investment Portfolio Optimization . . . . . . . . . . . . . . . . . . . 240
5.4 Pricing in an Incomplete Market . . . . . . . . . . . . . . . . . . . . . . . . 243
5.4.1 A Trinomial Model of an Incomplete Market . . . . . . . . . . . . . 243
5.4.2 Pricing Unattainable Payoffs: The Bid-Ask Spread . . . . . . . . . . 246
5.5 Change of Num´eraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
5.5.1 The Concept of a Num´eraire Asset . . . . . . . . . . . . . . . . . . . 252
5.5.2 Change of Num´eraire in a Binomial Model. . . . . . . . . . . . . . . 253
5.5.3 Change of Num´eraire in a General Single Period Model . . . . . . . 254
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
6 Introduction to Discrete-Time Stochastic Calculus 271
6.1 A Multi-Period Binomial Probability Model . . . . . . . . . . . . . . . . . 271
6.1.1 The Binomial Probability Space . . . . . . . . . . . . . . . . . . . . 271
6.1.2 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
6.2 Information Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6.2.1 Partitions and Their Refinements . . . . . . . . . . . . . . . . . . . . 280
6.2.2 Sigma-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
6.2.3 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
6.2.4 Filtered Probability Space . . . . . . . . . . . . . . . . . . . . . . . . 292
6.3 Conditional Expectation and Martingales . . . . . . . . . . . . . . . . . . . 293
6.3.1 Measurability of Random Variables and Processes . . . . . . . . . . 293
6.3.2 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . 295
6.3.3 Properties of Conditional Expectations. . . . . . . . . . . . . . . . . 301
6.3.4 Conditioning in the Binomial Model . . . . . . . . . . . . . . . . . . 305
6.3.5 Binomial Model with Interdependent Market Moves . . . . . . . . . 308
6.3.6 Sub-, Super-, and True Martingales. . . . . . . . . . . . . . . . . . . 312