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CONTENTS

Introduction and Preface = xi

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 = 51

 4.4 Continuously Varying Interest Rates = 54

 4.5 Exercises = 56

5 Pricing Contracts via Arbitrage = 61

 5.1 An Example in Options Pricing = 61

 5.2 Other Examples of Pricing via Arbitrage = 64

 5.3 Exercises = 70

6 The Arbitrage Theorem = 72

 6.1 The Arbitrage Theorem = 72

 6.2 The Multiperiod Binomial Model = 76

 6.3 Proof of the Arbitrage Theorem = 78

 6.4 Exercises = 81

7 The Black-Scholes Formula = 85

 7.1 The Black-Scholes Formula = 85

 7.2 Properties of Black-Scholes Option Cost = 90

 7.3 Estimating σ = 91

 7.4 Pricing American Put Options = 92

 7.5 Comments = 97

  7.5.1 When the Option Cost Differs from the Black-Scholes Formula = 97

  7.5.2 When the Interest Rate Changes = 98

  7.5.3 Final Comments = 99

 7.6 Exercises = 100

8 Valuing by Expected Utility = 103

 8.1 Limitations of Arbitrage Pricing = 103

 8.2 Valuing Investments by Expected Utility = 104

 8.3 The Portfolio Selection Problem = 110

  8.3.1 Estimating Covariances = 119

 8.4 The Capital Assets Pricing Model = 120

 8.5 Mean Variance Analysis of Risk-Neutral-Priced Call Options = 121

 8.6 Rates of Return : Single-Period and Geometric Brownian Motion = 124

 8.7 Exercises = 125

9 Exotic Options = 129

 9.1 Introduction = 129

 9.2 Barrier Options = 129

 9.3 Asian and Lookback Options = 130

 9.4 Monte Carlo Simulation = 131

 9.5 Pricing Exotic Options by Simulation = 132

 9.6 More Efficient Simulation Estimators = 134

  9.6.1 Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations = 134

  9.6.2 Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations = 138

 9.7 Options with Nonlinear Payoffs = 140

 9.8 Pricing Approximations via Multiperiod Binomial Models = 141

 9.9 Exercises = 143

10 Beyond Geometric Brownian Motion Models = 146

 10.1 Introduction = 146

 10.2 Crude Oil Data = 147

 10.3 Models for the Crude Oil Data = 153

 10.4 Final Comments = 155

11 Autogressive Models and Mean Reversion = 166

 11.1 The Autoregressive Model = 166

 11.2 Valuing Options by Their Expected Return = 167

 11.3 Mean Reversion = 170

 11.4 Exercises = 172

Index = 183