An Elementary Introduction to Mathematical

This textbook on the basics of option pricing is accessible to readers with limited mathematical training. It is for both professional traders and undergraduates studying the basics of finance. Assuming no prior knowledge of probability, Sheldon M. Ross offers clear, simple explanations of arbitrage, the Black-Scholes option pricing formula, and other topics such as utility functions, optimal portfolio selections, and the capital assets pricing model. Among the many new features of this third edition are new chapters on Brownian motion and geometric Brownian motion, stochastic order relations, and stochastic dynamic programming, along with expanded sets of exercises and references for all the chapters.

Sheldon M. Ross is the Epstein Chair Professor in the Department of Industrial and Systems Engineering, University of Southern California. He received his Ph.D. in statistics from Stanford University in 1968 and was a Professor at the University of California, Berkeley, from 1976 until 2004. He has published more than 100 article and a variety of textbooks in the areas of statistics and applied probability, including Topics in Finite and Discrete Mathematics (2000), Introduction to Probability and Statistics for Engineers and Scientists, Fourth Edition (2009), A First Course in Probability, Eighth Edition (2009), and Introduction to Probability Models, Tenth Edition (2009). Dr. Ross serves as the editor for Probability in the Engineering and Informational Sciences.

Introduction and Preface	p. xi
Probability	p. 1
Probabilities and Events	p. 1
Conditional Probability	p. 5
Random Variables and Expected Values	p. 9
Covariance and Correlation	p. 14
Conditional Expectation	p. 16
Exercises	p. 17
Normal Random Variables	p. 22
Continuous Random Variables	p. 22
Normal Random Variables	p. 22
Properties of Normal Random Variables	p. 26
The Central Limit Theorem	p. 29
Exercises	p. 31
Brownian Motion and Geometric Brownian Motion	p. 34
Brownian Motion	p. 34
Brownian Motion as a Limit of Simpler Models	p. 35
Geometric Brownian Motion	p. 38
Geometric Brownian Motion as a Limit of Simpler Models	p. 40
*The Maximum Variable	p. 40
The Cameron-Martin Theorem	p. 45
Exercises	p. 46
Interest Rates and Present Value Analysis	p. 48
Interest Rates	p. 48
Present Value Analysis	p. 52
Rate of Return	p. 62
Continuously Varying Interest Rates	p. 65
Exercises	p. 67
Pricing Contracts via Arbitrage	p. 73
An Example in Options Pricing	p. 73
Other Examples of Pricing via Arbitrage	p. 77
Exercises	p. 86
The Arbitrage Theorem	p. 92
The Arbitrage Theorem	p. 92
The Multiperiod Binomial Model	p. 96
Proof of the Arbitrage Theorem	p. 98
Exercises	p. 102
The Black-Scholes Formula	p. 106
Introduction	p. 106
The Black-Scholes Formula	p. 106
Properties of the Black-Scholes Option Cost	p. 110
The Delta Hedging Arbitrage Strategy	p. 113
Some Derivations	p. 118
The Black-Scholes Formula	p. 119
The Partial Derivatives	p. 121
European Put Options	p. 126
Exercises	p. 127
Additional Results on Options	p. 131
Introduction	p. 131
Call Options on Dividend-Paying Securities	p. 131
The Dividend for Each Share of the Security Is Paid Continuously in Time at a Rate Equal to a Fixed Fraction f of the Price of the Security	p. 132
For Each Share Owned, a Single Payment of fS(t_d) Is Made at Time t_d	p. 133
For Each Share Owned, a Fixed Amount D Is to Be Paid at Time t_d	p. 134
Pricing American Put Options	p. 134
Adding Jumps to Geometric Brownian Motion	p. 142
When the Jump Distribution Is Lognormal	p. 144
When the Jump Distribution Is General	p. 146
Estimating the Volatility Parameter	p. 148
Estimating a Population Mean and Variance	p. 149
The Standard Estimator of Volatility	p. 150
Using Opening and Closing Data	p. 152
Using Opening, Closing, and High-Low Data	p. 153
Some Comments	p. 155
When the Option Cost Differs from the Black-Scholes Formula	p. 155
When the Interest Rate Changes	p. 156
Final Comments	p. 156
Appendix	p. 158
Exercises	p. 159
Valuing by Expected Utility	p. 165
Limitations of Arbitrage Pricing	p. 165
Valuing Investments by Expected Utility	p. 166
The Portfolio Selection Problem	p. 174
Estimating Covariances	p. 184
Value at Risk and Conditional Value at Risk	p. 184
The Capital Assets Pricing Model	p. 187
Rates of Return: Single-Period and Geometric Brownian Motion	p. 188
Exercises	p. 190
Stochastic Order Relations	p. 193
First-Order Stochastic Dominance	p. 193
Using Coupling to Show Stochastic Dominance	p. 196
Likelihood Ratio Ordering	p. 198
A Single-Period Investment Problem	p. 199
Second-Order Dominance	p. 203
Normal Random Variables	p. 204
More on Second-Order Dominance	p. 207
Exercises	p. 210
Optimization Models	p. 212
Introduction	p. 212
A Deterministic Optimization Model	p. 212
A General Solution Technique Based on Dynamic Programming	p. 213
A Solution Technique for Concave Return Functions	p. 215
The Knapsack Problem	p. 219
Probabilistic Optimization Problems	p. 221
A Gambling Model with Unknown Win Probabilities	p. 221
An Investment Allocation Model	p. 222
Exercises	p. 225
Stochastic Dynamic Programming	p. 228
The Stochastic Dynamic Programming Problem	p. 228
Infinite Time Models	p. 234
Optimal Stopping Problems	p. 239
Exercises	p. 244
Exotic Options	p. 247
Introduction	p. 247
Barrier Options	p. 247
Asian and Lookback Options	p. 248
Monte Carlo Simulation	p. 249
Pricing Exotic Options by Simulation	p. 250
More Efficient Simulation Estimators	p. 252
Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations	p. 253
Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations	p. 257
Options with Nonlinear Payoffs	p. 258
Pricing Approximations via Multiperiod Binomial Models	p. 259
Continuous Time Approximations of Barrier and Lookback Options	p. 261
Exercises	p. 262
Beyond Geometric Brownian Motion Models	p. 265
Introduction	p. 265
Crude Oil Data	p. 266
Models for the Crude Oil Data	p. 272
Final Comments	p. 274
Autoregressive Models and Mean Reversion	p. 285
The Autoregressive Model	p. 285
Valuing Options by Their Expected Return	p. 286
Mean Reversion	p. 289
Exercises	p. 291
Index	p. 303
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An Elementary Introduction to Mathematical Finance

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