金融随机分析 第2卷PDF电子书下载

1 General Probability Theory 1

1.1 Infinite Probability Spaces 1

1.2 Random Variables and Distributions 7

1.3 Expectations 13

1.4 Convergence of Integrals 23

1.5 Computation of Expectations 27

1.6 Change of Measure 32

1.7 Summary 39

1.8 Notes 41

1.9 Exercises 41

2 Information and Conditioning 49

2.1 Information and σ-algebras 49

2.2 Independence 53

2.3 General Conditional Expectations 66

2.4 Summary 75

2.5 Notes 77

2.6 Exercises 77

3 Brownian Motion 83

3.1 Introduction 83

3.2 Scaled Random Walks 83

3.2.1 Symmetric Random Walk 83

3.2.2 Increments of the Symmetric Random Walk 84

3.2.3 Martingale Property for the Symmetric Random Walk 85

3.2.4 Quadratic Variation of the Symmetric Random Walk 85

3.2.5 Scaled Symmetric Random Walk 86

3.2.6 Limiting Distribution of the Scaled Random Walk 88

3.2.7 Log-Normal Distribution as the Limit of the Binomial Model 91

3.3 Brownian Motion 93

3.3.1 Definition of Brownian Motion 93

3.3.2 Distribution of Brownian Motion 95

3.3.3 Filtration for Brownian Motion 97

3.3.4 Martingale Property for Brownian Motion 98

3.4 Quadratic Variation 98

3.4.1 First-Order Variation 99

3.4.2 Quadratic Variation 101

3.4.3 Volatility of Geometric Brownian Motion 106

3.5 Markov Property 107

3.6 First Passage Time Distribution 108

3.7 Reflection Principle 111

3.7.1 Reflection Equality 111

3.7.2 First Passage Time Distribution 112

3.7.3 Distribution of Brownian Motion and Its Maximum 113

3.8 Summary 115

3.9 Notes 116

3.10 Exercises 117

4 Stochastic Calculus 125

4.1 Introduction 125

4.2 It？'s Integral for Simple Integrands 125

4.2.1 Construction of the Integral 126

4.2.2 Properties of the Integral 128

4.3 It？'s Integral for General Integrands 132

4.4 It？-Doeblin Formula 137

4.4.1 Formula for Brownian Motion 137

4.4.2 Formula for It？ Processes 143

4.4.3 Examples 147

4.5 Black-Scholes-Merton Equation 153

4.5.1 Evolution of Portfolio Value 154

4.5.2 Evolution of Option Value 155

4.5.3 Equating the Evolutions 156

4.5.4 Solution to the Black-Scholes-Merton Equation 158

4.5.5 The Greeks 159

4.5.6 Put-Call Parity 162

4.6 Multivariable Stochastic Calculus 164

4.6.1 Multiple Brownian Motions 164

4.6.2 It？-Doeblin Formula for Multiple Processes 165

4.6.3 Recognizing a Brownian Motion 168

4.7 Brownian Bridge 172

4.7.1 Gaussian Processes 172

4.7.2 Brownian Bridge as a Gaussian Process 175

4.7.3 Brownian Bridge as a Scaled Stochastic Integral 176

4.7.4 Multidimensional Distribution of the Brownian Bridge 178

4.7.5 Brownian Bridge as a Conditioned Brownian Motion 182

4.8 Summary 183

4.9 Notes 187

4.10 Exercises 189

5 Risk-Neutral Pricing 209

5.1 Introduction 209

5.2 Risk-Neutral Measure 210

5.2.1 Girsanov's Theorem for a Single Brownian Motion 210

5.2.2 Stock Under the Risk-Neutral Measure 214

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure 217

5.2.4 Pricing Under the Risk-Neutral Measure 218

5.2.5 Deriving the Black-Scholes-Merton Formula 218

5.3 Martingale Representation Theorem 221

5.3.1 Martingale Representation with One Brownian Motion 221

5.3.2 Hedging with One Stock 222

5.4 Fundamental Theorems of Asset Pricing 224

5.4.1 Girsanov and Martingale Representation Theorems 224

5.4.2 Multidimensional Market Model 226

5.4.3 Existence of the Risk-Neutral Measure 228

5.4.4 Uniqueness of the Risk-Neutral Measure 231

5.5 Dividend-Paying Stocks 234

5.5.1 Continuously Paying Dividend 235

5.5.2 Continuously Paying Dividend with Constant Coefficients 237

5.5.3 Lump Payments of Dividends 238

5.5.4 Lump Payments of Dividends with 239

Constant Coefficients 239

5.6 Forwards and Futures 240

5.6.1 Forward Contracts 240

5.6.2 Futures Contracts 241

5.6.3 Forward-Futures Spread 247

5.7 Summary 248

5.8 Notes 250

5.9 Exercises 251

6 Connections with Partial Differential Equations 263

6.1 Introduction 263

6.2 Stochastic Differential Equations 263

6.3 The Markov Property 266

6.4 Partial Differential Equations 268

6.5 Interest Rate Models 272

6.6 Multidimensional Feynman-Kac Theorems 277

6.7 Summary 280

6.8 Notes 281

6.9 Exercises 282

7 Exotic Options 295

7.1 Introduction 295

7.2 Maximum of Brownian Motion with Drift 295

7.3 Knock-out Barrier Options 299

7.3.1 Up-and-Out Call 300

7.3.2 Black-Scholes-Merton Equation 300

7.3.3 Computation of the Price of the Up-and-Out Call 304

7.4 Lookback Options 308

7.4.1 Floating Strike Lookback Option 308

7.4.2 Black-Scholes-Merton Equation 309

7.4.3 Reduction of Dimension 312

7.4.4 Computation of the Price of the Lookback Option 314

7.5 Asian Options 320

7.5.1 Fixed-Strike Asian Call 320

7.5.2 Augmentation of the State 321

7.5.3 Change of Numéraire 323

7.6 Summary 331

7.7 Notes 331

7.8 Exercises 332

8 American Derivative Securities 339

8.1 Introduction 339

8.2 Stopping Times 340

8.3 Perpetual American Put 345

8.3.1 Price Under Arbitrary Exercise 346

8.3.2 Price Under Optimal Exercise 349

8.3.3 Analytical Characterization of the Put Price 351

8.3.4 Probabilistic Characterization of the Put Price 353

8.4 Finite-Expiration American Put 356

8.4.1 Analytical Characterization of the Put Price 357

8.4.2 Probabilistic Characterization of the Put Price 359

8.5 American Call 361

8.5.1 Underlying Asset Pays No Dividends 361

8.5.2 Underlying Asset Pays Dividends 363

8.6 Summary 368

8.7 Notes 369

8.8 Exercises 370

9 Change of Numéraire 375

9.1 Introduction 375

9.2 Numéraire 376

9.3 Foreign and Domestic Risk-Neutral Measures 381

9.3.1 The Basic Processes 381

9.3.2 Domestic Risk-Neutral Measure 383

9.3.3 Foreign Risk-Neutral Measure 385

9.3.4 Siegel's Exchange Rate Paradox 387

9.3.5 Forward Exchange Rates 388

9.3.6 Garman-Kohlhagen Formula 390

9.3.7 Exchange Rate Put-Call Duality 390

9.4 Forward Measures 392

9.4.1 Forward Price 392

9.4.2 Zero-Coupon Bond as Numéraire 392

9.4.3 Option Pricing with a Random Interest Rate 394

9.5 Summary 397

9.6 Notes 398

9.7 Exercises 398

10 Term-Structure Models 403

10.1 Introduction 403

10.2 Affine-Yield Models 405

10.2.1 Two-Factor Vasicek Model 406

10.2.2 Two-Factor CIR Model 420

10.2.3 Mixed Model 422

10.3 Heath-Jarrow-Morton Model 423

10.3.1 Forward Rates 423

10.3.2 Dynamics of Forward Rates and Bond Prices 425

10.3.3 No-Arbitrage Condition 426

10.3.4 HJM Under Risk-Neutral Measure 429

10.3.5 Relation to Affine-Yield Models 430

10.3.6 Implementation of HJM 432

10.4 Forward LIBOR Model 435

10.4.1 The Problem with Forward Rates 435

10.4.2 LIBOR and Forward LIBOR 436

10.4.3 Pricing a Backset LIBOR Contract 437

10.4.4 Black Caplet Formula 438

10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities 440

10.4.6 A Forward LIBOR Term-Structure Model 442

10.5 Summary 447

10.6 Notes 450

10.7 Exercises 451

11 Introduction to Jump Processes 461

11.1 Introduction 461

11.2 Poisson Process 462

11.2.1 Exponential Random Variables 462

11.2.2 Construction of a Poisson Process 463

11.2.3 Distribution of Poisson Process Increments 463

11.2.4 Mean and Variance of Poisson Increments 466

11.2.5 Martingale Property 467

11.3 Compound Poisson Process 468

11.3.1 Construction of a Compound Poisson Process 468

11.3.2 Moment-Generating Function 470

11.4 Jump Processes and Their Integrals 473

11.4.1 Jump Processes 474

11.4.2 Quadratic Variation 479

11.5 Stochastic Calculus for Jump Processes 483

11.5.1 It？-Doeblin Formula for One Jump Process 483

11.5.2 It？-Doeblin Formula for Multiple Jump Processes 489

11.6 Change of Measure 492

11.6.1 Change of Measure for a Poisson Process 493

11.6.2 Change of Measure for a Compound Poisson Process 495

11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion 502

11.7 Pricing a European Call in a Jump Model 505

11.7.1 Asset Driven by a Poisson Process 505

11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process 512

11.8 Summary 523

11.9 Notes 525

11.10 Exercises 525

A Advanced Topics in Probability Theory 527

A.1 Countable Additivity 527

A.2 Generating σ-algebras 530

A.3 Random Variable with Neither Density nor Probability Mass Function 531

B Existence of Conditional Expectations 533

C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing 535

References 537

Index 545