An Introduction to Financial Mathematics: Option Valuation (Chapman and Hall/CRC Financial Mathematics Series)

Introduction to Financial Mathematics: Option Valuation, Second Edition is a well-rounded primer to the mathematics and models used in the valuation of financial derivatives. The book consists of fteen chapters, the rst ten of which develop option valuation techniques in discrete time, the last ve describing the theory in continuous time. The first half of the textbook develops basic finance and probability. The author then treats the binomial model as the primary example of discrete-time option valuation. The final part of the textbook examines the Black-Scholes model. The book is written to provide a straightforward account of the principles of option pricing and examines these principles in detail using standard discrete and stochastic calculus models. Additionally, the second edition has new exercises and examples, and includes many tables and graphs generated by over 30 MS Excel VBA modules available on the author's webpage https://home.gwu.edu/~hdj/. Read more... Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Dedication 6 Table of Contents 8 Preface 12 1: Basic Finance 16 1.1 Interest 16 *1.2 Inflation 18 1.3 Annuities 19 1.4 Bonds 25 *1.5 Internal Rate of Return 26 1.6 Exercises 28 2: Probability Spaces 32 2.1 Sample Spaces and Events 32 2.2 Discrete Probability Spaces 33 2.3 General Probability Spaces 36 2.4 Conditional Probability 41 2.5 Independence 45 2.6 Exercises 46 3: Random Variables 50 3.1 Introduction 50 3.2 General Properties of Random Variables 52 3.3 Discrete Random Variables 53 3.4 Continuous Random Variables 57 3.5 Joint Distributions of Random Variables 59 3.6 Independent Random Variables 61 3.7 Identically Distributed Random Variables 63 3.8 Sums of Independent Random Variables 63 3.9 Exercises 66 4: Options and Arbitrage 70 4.1 The Price Process of an Asset 70 4.2 Arbitrage 71 4.3 Classification of Derivatives 73 4.4 Forwards 74 4.5 Currency Forwards 75 4.6 Futures 76 *4.7 Equality of Forward and Future Prices 78 4.8 Call and Put Options 79 4.9 Properties of Options 82 4.10 Dividend-Paying Stocks 84 4.11 Exotic Options 85 *4.12 Portfolios and Payoff Diagrams 88 4.13 Exercises 91 5: Discrete-Time Portfolio Processes 94 5.1 Discrete Time Stochastic Processes 94 5.2 Portfolio Processes and the Value Process 98 5.3 Self-Financing Trading Strategies 99 5.4 Equivalent Characterizations of Self-Financing 100 5.5 Option Valuation by Portfolios 102 5.6 Exercises 103 6: Expectation 106 6.1 Expectation of a Discrete Random Variable 106 6.2 Expectation of a Continuous Random Variable 108 6.3 Basic Properties of Expectation 110 6.4 Variance of a Random Variable 111 6.5 Moment Generating Functions 113 6.6 The Strong Law of Large Numbers 114 6.7 The Central Limit Theorem 115 6.8 Exercises 117 7: The Binomial Model 122 7.1 Construction of the Binomial Model 122 7.2 Completeness and Arbitrage in the Binomial Model 126 7.3 Path-Independent Claims 130 *7.4 Path-Dependent Claims 134 7.5 Exercises 136 8: Conditional Expectation 140 8.1 Definition of Conditional Expectation 140 8.2 Examples of Conditional Expectations 141 8.3 Properties of Conditional Expectation 143 8.4 Special Cases 145 *8.5 Existence of Conditional Expectation 147 8.6 Exercises 149 9: Martingales in Discrete Time Markets 150 9.1 Discrete Time Martingales 150 9.2 The Value Process as a Martingale 152 9.3 A Martingale View of the Binomial Model 153 9.4 The Fundamental Theorems of Asset Pricing 155 *9.5 Change of Probability 157 9.6 Exercises 159 10: American Claims in Discrete-Time Markets 162 10.1 Hedging an American Claim 162 10.2 Stopping Times 164 10.3 Submartingales and Supermartingales 166 10.4 Optimal Exercise of an American Claim 167 10.5 Hedging in the Binomial Model 169 10.6 Optimal Exercise in the Binomial Model 170 10.7 Exercises 171 11: Stochastic Calculus 174 11.1 Continuous-Time Stochastic Processes 174 11.2 Brownian Motion 175 11.3 Stochastic Integrals 179 11.4 The Ito-Doeblin Formula 185 11.5 Stochastic Differential Equations 191 11.6 Exercises 195 12: The Black-Scholes-Merton Model 198 12.1 The Stock Price SDE 198 12.2 Continuous-Time Portfolios 199 12.3 The Black-Scholes Formula 200 12.4 Properties of the Black-Scholes Call Function 203 *12.5 The BS Formula as a Limit of CRR Formulas 206 12.6 Exercises 209 13: Martingales in the Black-Scholes-Merton Model 212 13.1 Continuous-Time Martingales 212 13.2 Change of Probability and Girsanov's Theorem 216 13.3 Risk-Neutral Valuation of a Derivative 219 13.4 Proofs of the Valuation Formulas 220 *13.5 Valuation under P 223 *13.6 The Feynman-Kac Representation Theorem 224 13.7 Exercises 226 14: Path-Independent Options 228 14.1 Currency Options 228 14.2 Forward Start Options 231 14.3 Chooser Options 231 14.4 Compound Options 233 14.5 Quantos 234 14.6 Options on Dividend-Paying Stocks 236 14.7 American Claims 239 14.8 Exercises 241 15: Path-Dependent Options 244 15.1 Barrier Options 244 15.2 Lookback Options 249 15.3 Asian Options 255 15.4 Other Options 258 15.5 Exercises 259 A: Basic Combinatorics 264 B: Solution of the BSM PDE 270 C: Properties of the BSM Call Function 274 D: Solutions to Odd-Numbered Problems 280 Bibliography 312 Index 314 An,Introduction,to,Financial,Mathematics Introduction to Financial Mathematics: Option Valuation, Second Edition is a well-rounded primer to the mathematics and models used in the valuation of financial derivatives. The book consists of fifteen chapters, the first ten of which develop option valuation techniques in discrete time, the last five describing the theory in continuous time. The first half of the textbook develops basic finance and probability. The author then treats the binomial model as the primary example of discrete-time option valuation. The final part of the textbook examines the Black-Scholes model. The book is written to provide a straightforward account of the principles of option pricing and examines these principles in detail using standard discrete and stochastic calculus models. Additionally, the second edition has new exercises and examples, and includes many tables and graphs generated by over 30 MS Excel VBA modules available on the author's webpage https://home.gwu.edu/~hdj/. Designed for readers having a background in standard multivariable calculus, Introduction to Financial Mathematics: Option Valuation, Second Edition is a well-rounded primer to the mathematics and models used in the valuation of financial derivatives. New examples and exercises have been added in this second edition.