An Elementary Intoduction to Mathematical Finance:Options and Other TopicsPDF电子书下载

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