Mathematical Modeling in Economics and Finance

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Mathematical Modeling in Economics and Finance: Probability, Stochastic Processes, and Differential Equations (AMS/MAA Textbooks)Mathematical Modeling in Economics and Finance is designed as a textbook for an upper-division course on modeling in the economic sciences. The emphasis throughout is on the modeling process including post-modeling analysis and criticism. It is a textbook on modeling that happens to focus on financial instruments for the management of economic risk. The book combines a study of mathematical modeling with exposure to the tools of probability theory, difference and differential equations, numerical simulation, data analysis, and mathematical analysis. Students taking a course from Mathematical Modeling in Economics and Finance will come to understand some basic stochastic processes and the solutions to stochastic differential equations. They will understand how to use those tools to model the management of financial risk. They will gain a deep appreciation for the modeling process and learn methods of testing and evaluation driven by data. The reader of this book will be successfully positioned for an entry-level position in the financial services industry or for beginning graduate study in finance, economics, or actuarial science. The exposition in Mathematical Modeling in Economics and Finance is crystal clear and very student-friendly. The many exercises are extremely well designed. Steven Dunbar is Professor Emeritus of Mathematics at the University of Nebraska and he has won both university-wide and MAA prizes for extraordinary teaching. Dunbar served as Director of the MAA's American Mathematics Competitions from 2004 until 2015. His ability to communicate mathematics is on full display in this approachable, innovative text.


Contents
Preface iii
1 Background Ideas 1
1.1 Brief History of Mathematical Finance . . . . . . . . . . . . . 1
1.2 Options and Derivatives . . . . . . . . . . . . . . . . . . . . . 11
1.3 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . 19
1.4 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . 32
1.6 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.7 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 57
1.8 A Model of Collateralized Debt Obligations . . . . . . . . . . 66
2 Binomial Option Pricing Models 77
2.1 Single Period Binomial Models . . . . . . . . . . . . . . . . . . 77
2.2 Multiperiod Binomial Tree Models . . . . . . . . . . . . . . . 88
3 First Step Analysis for Stochastic Processes 101
3.1 A Coin Tossing Experiment . . . . . . . . . . . . . . . . . . . 101
3.2 Ruin Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3 Duration of the Gambler’s Ruin . . . . . . . . . . . . . . . . 133
3.4 A Stochastic Process Model of Cash Management . . . . . . . 148
4 Limit Theorems for Stochastic Processes 171
4.1 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . 171
4.2 Moment Generating Functions . . . . . . . . . . . . . . . . . . 179
4.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . 186
4.4 The Absolute Excess of Heads over Tails . . . . . . . . . . . . 200
xiii
xiv CONTENTS
5 Brownian Motion 213
5.1 Intuitive Introduction to Diffusions . . . . . . . . . . . . . . . 213
5.2 The Definition of Brownian Motion and the Wiener Process . 220
5.3 Approximation of Brownian Motion by Coin-Flipping Sums . . 235
5.4 Transformations of the Wiener Process . . . . . . . . . . . . . 244
5.5 Hitting Times and Ruin Probabilities . . . . . . . . . . . . . . 254
5.6 Path Properties of Brownian Motion . . . . . . . . . . . . . . 264
5.7 Quadratic Variation of the Wiener Process . . . . . . . . . . . 272
6 Stochastic Calculus 287
6.1 Stochastic Differential Equations and the Euler-Maruyama Method287
6.2 Itˆo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.3 Properties of Geometric Brownian Motion . . . . . . . . . . . 308
6.4 Models of Stock Market Prices . . . . . . . . . . . . . . . . . . 320
6.5 Monte Carlo Simulation of Option Prices . . . . . . . . . . . . 335
7 The Black-Scholes Model 359
7.1 Derivation of the Black-Scholes Equation . . . . . . . . . . . . 359
7.2 Solution of the Black-Scholes Equation . . . . . . . . . . . . . 366
7.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . 382
7.4 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 395
7.5 Sensitivity, Hedging and the “Greeks” . . . . . . . . . . . . . . 405
7.6 Limitations of the Black-Scholes Model . . . . . . . . . . . . . 419

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