1 Probability 1
1.1 Probabilities and Events 1
1.2 Conditional Probability 5
1.3 Random Variables and Expected Values 9
1.4 Covariance and Correlation 13
1.5 Exercises 15
2 Normal Random Variables 20
2.1 Continuous Random Variables 20
2.2 Normal Random Variables 20
2.3 Properties of Normal Random Variables 24
2.4 The Central Limit Theorem 27
2.5 Exercises 29
3 Geometric Brownian Motion 32
3.1 Geometric Brownian Motion 32
3.2 Geometric Brownian Motion as a Limit of Simpler Models 33
3.3 Brownian Motion 35
3.4 Exercises 36
4 Interest Rates and Present Value Analysis 38
4.1 Interest Rates 38
4.2 Present Value Analysis 42
4.3 Rate of Return 52
4.4 Continuously Varying Interest Rates 55
4.5 Exercises 57
5 Pricing Contracts via Arbitrage 63
5.1 An Example in Options Pricing 63
5.2 Other Examples of Pricing via Arbitrage 67
5.3 Exercises 76
6 The Arbitrage Theorem 81
6.1 The Arbitrage Theorem 81
6.2 The Multiperiod Binomial Model 85
6.3 Proof of the Arbitrage Theorem 87
6.4 Exercises 91
7 The Black-Scholes Formula 95
7.1 Introduction 95
7.2 The Black-Scholes Formula 95
7.3 Properties of the Black-Scholes Option Cost 99
7.4 The Delta Hedging Arbitrage Strategy 102
7.5 Some Derivations 108
7.5.1 The Black-Scholes Formula 108
7.5.2 The Partial Derivatives 110
7.6 Exercises 115
8 Additional Results on Options 118
8.1 Introduction 118
8.2 Call Options on Dividend-Paying Securities 118
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 119
8.2.2 For Each Share Owned， a Single Payment of fS（td） Is Made at Time td 120
8.2.3 For Each Share Owned， a Fixed Amount D Is to Be Paid at Time td 121
8.3 Pricing American Put Options 123
8.4 Adding Jumps to Geometric Brownian Motion 129
8.4.1 When the Jump Distribution Is Lognormal 131
8.4.2 When the Jump Distribution Is General 133
8.5 Estimating the Volatility Parameter 135
8.5.1 Estimating a Population Mean and Variance 136
8.5.2 The Standard Estimator of Volatility 137
8.5.3 Using Opening and Closing Data 139
8.5.4 Using Opening， Closing， and High-Low Data 140
8.6 Some Comments 142
8.6.1 When the Option Cost Differs from the Black-Scholes Formula 142
8.6.2 When the Interest Rate Changes 143
8.6.3 Final Comments 143
8.7 Appendix 145
8.8 Exercises 146
9 Valuing by Expected Utility 152
9.1 Limitations of Arbitrage Pricing 152
9.2 Valuing Investments by Expected Utility 153
9.3 The Portfolio Selection Problem 160
9.3.1 Estimating Covariances 169
9.4 Value at Risk and Conditional Value at Risk 170
9.5 The Capital Assets Pricing Model 172
9.6 Mean Variance Analysis of Risk-Neutral-Priced Call Options 173
9.7 Rates of Return： Single-Period and Geometric Brownian Motion 176
9.8 Exercises 177
10 Optimization Models 181
10.1 Introduction 181
10.2 A Deterministic Optimization Model 181
10.2.1 A General Solution Technique Based on Dynamic Programming 182
10.2.2 A Solution Technique for Concave Return Functions 184
10.2.3 The Knapsack Problem 188
10.3 Probabilistic Optimization Problems 190
10.3.1 A Gambling Model with Unknown Win Probabilities 190
10.3.2 An Investment Allocation Model 191
10.4 Exercises 193
11 Exotic Options 196
11.1 Introduction 196
11.2 Barrier Options 196
11.3 Asian and Lookback Options 197
11.4 Monte Carlo Simulation 198
11.5 Pricing Exotic Options by Simulation 199
11.6 More Efficient Simulation Estimators 201
11.6.1 Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations 201
11.6.2 Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations 205
11.7 Options with Nonlinear Payoffs 207
11.8 Pricing Approximations via Multiperiod Binomial Models 208
11.9 Exercises 210
12 Beyond Geometric Brownian Motion Models 213
12.1 Introduction 213
12.2 Crude Oil Data 214
12.3 Models for the Crude Oil Data 220
12.4 Final Comments 222
13 Autogressive Models and Mean Reversion 233
13.1 The Autoregressive Model 233
13.2 Valuing Options by Their Expected Return 234
13.3 Mean Reversion 237
13.4 Exercises 239
Index 251