An elementary introduction to mathematical finance

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Introduction and Preface......Page 13
New to This Edition......Page 16
1.1 Probabilities and Events......Page 19
1.2 Conditional Probability......Page 23
1.3 Random Variables and Expected Values......Page 27
1.4 Covariance and Correlation......Page 32
1.5 Conditional Expectation......Page 34
1.6 Exercises......Page 35
REFERENCE......Page 39
2.2 Normal Random Variables......Page 40
2.3 Properties of Normal Random Variables......Page 44
2.4 The Central Limit Theorem......Page 47
2.5 Exercises......Page 49
REFERENCE......Page 51
3.1 Brownian Motion......Page 52
3.2 Brownian Motion as a Limit of Simpler Models......Page 53
3.3 Geometric Brownian Motion......Page 56
3.4 The Maximum Variable......Page 58
3.5 The Cameron-Martin Theorem......Page 63
3.6 Exercises......Page 64
4.1 Interest Rates......Page 66
4.2 Present Value Analysis......Page 70
4.3 Rate of Return......Page 80
4.4 Continuously Varying Interest Rates......Page 83
4.5 Exercises......Page 85
5.1 An Example in Options Pricing......Page 91
5.2 Other Examples of Pricing via Arbitrage......Page 95
5.3 Exercises......Page 104
REFERENCES......Page 109
6.1 The Arbitrage Theorem......Page 110
6.2 The Multiperiod Binomial Model......Page 114
6.3 Proof of the Arbitrage Theorem......Page 116
6.4 Exercises......Page 120
REFERENCES......Page 123
7.2 The Black–Scholes Formula......Page 124
7.3 Properties of the Black–Scholes Option Cost......Page 128
7.4 The Delta Hedging Arbitrage Strategy......Page 131
7.5 Some Derivations......Page 136
7.5.1 The Black–Scholes Formula......Page 137
7.5.2 The Partial Derivatives......Page 139
7.6 European Put Options......Page 144
7.7 Exercises......Page 145
REFERENCES......Page 148
8.2 Call Options on Dividend-Paying Securities......Page 149
8.2.1 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......Page 150
8.2.2 For Each Share Owned, a Single Payment of f S(td) Is Made at Time td......Page 151
8.2.3 For Each Share Owned, a Fixed Amount D Is to Be Paid at Time td......Page 152
8.3 Pricing American Put Options......Page 154
8.4 Adding Jumps to Geometric Brownian Motion......Page 160
8.4.1 When the Jump Distribution Is Lognormal......Page 162
8.4.2 When the Jump Distribution Is General......Page 164
8.5 Estimating the Volatility Parameter......Page 166
8.5.1 Estimating a Population Mean and Variance......Page 167
8.5.2 The Standard Estimator of Volatility......Page 168
8.5.3 Using Opening and Closing Data......Page 170
8.5.4 Using Opening, Closing, and High–Low Data......Page 171
8.6.1 When the Option Cost Differs from the Black–Scholes Formula......Page 173
8.6.3 Final Comments......Page 174
8.7 Appendix......Page 176
8.8 Exercises......Page 177
REFERENCES......Page 180
9.1 Limitations of Arbitrage Pricing......Page 183
9.2 Valuing Investments by Expected Utility......Page 184
9.3 The Portfolio Selection Problem......Page 192
9.4 Value at Risk and Conditional Value at Risk......Page 202
9.5 The Capital Assets Pricing Model......Page 205
9.6 Rates of Return: Single-Period and Geometric Brownian Motion......Page 206
9.7 Exercises......Page 208
REFERENCES......Page 210
10.1 First-Order Stochastic Dominance......Page 211
10.2 Using Coupling to Show Stochastic Dominance......Page 214
10.3 Likelihood Ratio Ordering......Page 216
10.4 A Single-Period Investment Problem......Page 217
10.5 Second-Order Dominance......Page 221
10.5.1 Normal Random Variables......Page 222
10.5.2 More on Second-Order Dominance......Page 225
10.6 Exercises......Page 228
REFERENCES......Page 229
11.2 A Deterministic Optimization Model......Page 230
11.2.1 A General Solution Technique Based on Dynamic Programming......Page 231
11.2.2 A Solution Technique for Concave Return Functions......Page 233
11.2.3 The Knapsack Problem......Page 237
11.3.1 A Gambling Model with Unknown Win Probabilities......Page 239
11.3.2 An Investment Allocation Model......Page 240
11.4 Exercises......Page 243
12.1 The Stochastic Dynamic Programming Problem......Page 246
12.2 Infinite Time Models......Page 252
12.3 Optimal Stopping Problems......Page 257
12.4 Exercises......Page 262
REFERENCE......Page 264
13.2 Barrier Options......Page 265
13.3 Asian and Lookback Options......Page 266
13.4 Monte Carlo Simulation......Page 267
13.5 Pricing Exotic Options by Simulation......Page 268
13.6 More Efficient Simulation Estimators......Page 270
13.6.1 Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations......Page 271
13.6.2 Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations......Page 275
13.7 Options with Nonlinear Payoffs......Page 276
13.8 Pricing Approximations via Multiperiod Binomial Models......Page 277
13.9 Continuous Time Approximations of Barrier and Lookback Options......Page 279
13.10 Exercises......Page 280
REFERENCES......Page 282
14.1 Introduction......Page 283
14.2 Crude Oil Data......Page 284
14.3 Models for the Crude Oil Data......Page 290
14.4 Final Comments......Page 292
REFERENCES......Page 293
15.1 The Autoregressive Model......Page 303
15.2 Valuing Options by Their Expected Return......Page 304
15.3 Mean Reversion......Page 307
15.4 Exercises......Page 309
Index......Page 321