数理金融初步 原书第3版 英文版PDF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 14

1.5 Conditional Expectation 16

1.6 Exercises 17

2 Normal Random Variables 22

2.1 Continuous Random Variables 22

2.2 Normal Random Variables 22

2.3 Properties of Normal Random Variables 26

2.4 The Central Limit Theorem 29

2.5 Exercises 31

3 Brownian Motion and Geometric Brownian Motion 34

3.1 Brownian Motion 34

3.2 Brownian Motion as a Limit of Simpler Models 35

3.3 Geometric Brownian Motion 38

3.3.1 Geometric Brownian Motion as a Limit of Simpler Models 40

3.4 The Maximum Variable 40

3.5 The Cameron-Martin Theorem 45

3.6 Exercises 46

4 Interest Rates and Present Value Analysis 48

4.1 Interest Rates 48

4.2 Present Value Analysis 52

4.3 Rate of Return 62

4.4 Continuously Varying Interest Rates 65

4.5 Exercises 67

5 Pricing Contracts via Arbitrage 73

5.1 An Example in Options Pricing 73

5.2 Other Examples of Pricing via Arbitrage 77

5.3 Exercises 86

6 The Arbitrage Theorem 92

6.1 The Arbitrage Theorem 92

6.2 The Mulfiperiod Binomial Model 96

6.3 Proof of the Arbitrage Theorem 98

6.4 Exercises 102

7 The Black-Scholes Formula 106

7.1 Introduction 106

7.2 The Black-Scholes Formula 106

7.3 Properties of the Black-Scholes Option Cost 110

7.4 The Delta Hedging Arbitrage Strategy 113

7.5 Some Derivations 118

7.5.1 The Black-Scholes Formula 119

7.5.2 The Partial Derivatives 121

7.6 European Put Options 126

7.7 Exercises 127

8 Additional Results on Options 131

8.1 Introduction 131

8.2 Call Options on Dividend-Paying Securities 131

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 132

8.2.2 For Each Share Owned, a Single Payment of fS(td) Is Made at Time td 133

8.2.3 For Each Share Owned, a Fixed Amount D Is to Be Paid at Time td 134

8.3 Pricing American Put Options 136

8.4 Adding Jumps to Geometric Brownian Motion 142

8.4.1 When the Jump Distribution Is Lognormal 144

8.4.2 When the Jump Distribution Is General 146

8.5 Estimating the Volatility Parameter 148

8.5.1 Estimating a Population Mean and Variance 149

8.5.2 The Standard Estimator of Volatility 150

8.5.3 Using Opening and Closing Data 152

8.5.4 Using Opening, Closing, and High-Low Data 153

8.6 Some Comments 155

8.6.1 When the Option Cost Differs from the Black-Scholes Formula 155

8.6.2 When the Interest Rate Changes 156

8.6.3 Final Comments 156

8.7 Appendix 158

8.8 Exercises 159

9 Valuing by Expected Utility 165

9.1 Limitations of Arbitrage Pricing 165

9.2 Valuing Investments by Expected Utility 166

9.3 The Portfolio Selection Problem 174

9.3.1 Estimating Covariances 184

9.4 Value at Risk and Conditional Value at Risk 184

9.5 The Capital Assets Pricing Model 187

9.6 Rates of Return: Single-Period and Geometric Brownian Motion 188

9.7 Exercises 190

10 Stochastic Order Relations 193

10.1 First-Order Stochastic Dominance 193

10.2 Using Coupling to Show Stochastic Dominance 196

10.3 Likelihood Ratio Ordering 198

10.4 A Single-Period Investment Problem 199

10.5 Second-Order Dominance 203

10.5.1 Normal Random Variables 204

10.5.2 More on Second-Order Dominance 207

10.6 Exercises 210

11 Optimization Models 212

11.1 Introduction 212

11.2 A Deterministic Optimization Model 212

11.2.1 A General Solution Technique Based on Dynamic Programming 213

11.2.2 A Solution Technique for Concave Return Functions 215

11.2.3 The Knapsack Problem 219

11.3 Probabilistic Optimization Problems 221

11.3.1 A Gambling Model with Unknown Win Probabilities 221

11.3.2 An Investment Allocation Model 222

11.4 Exercises 225

12 Stochastic Dynamic Programming 228

12.1 The Stochastic Dynamic Programming Problem 228

12.2 Infinite Time Models 234

12.3 Optimal Stopping Problems 239

12.4 Exercises 244

13 Exotic Options 247

13.1 Introduction 247

13.2 Barrier Options 247

13.3 Asian and Lookback Options 248

13.4 Monte Carlo Simulation 249

13.5 Pricing Exotic Options by Simulation 250

13.6 More Efficient Simulation Estimators 252

13.6.1 Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations 253

13.6.2 Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations 257

13.7 Options with Nonlinear Payoffs 258

13.8 Pricing Approximations via Multiperiod Binomial Models 259

13.9 Continuous Time Approximations of Barrier and Lookback Options 261

13.10 Exercises 262

14 Beyond Geometric Brownian Motion Models 265

14.1 Introduction 265

14.2 Crude Oil Data 266

14.3 Models for the Crude Oil Data 272

14.4 Final Comments 274

15 Autoregressive Models and Mean Reversion 285

15.1 The Autoregressive Model 285

15.2 Valuing Options by Their Expected Return 286

15.3 Mean Reversion 289

15.4 Exercises 291

Index 303