金融随机分析 第2卷PDF|Epub|txt|kindle电子书版本网盘下载

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1 General Probability Theory1

1.1 Infinite Probability Spaces1

1.2 Random Variables and Distributions7

1.3 Expectations13

1.4 Convergence of Integrals23

1.5 Computation of Expectations27

1.6 Change of Measure32

1.7 Summary39

1.8 Notes41

1.9 Exercises41

2 Information and Conditioning49

2.1 Information and σ-algebras49

2.2 Independence53

2.3 General Conditional Expectations66

2.4 Summary75

2.5 Notes77

2.6 Exercises77

3 Brownian Motion83

3.1 Introduction83

3.2 Scaled Random Walks83

3.2.1 Symmetric Random Walk83

3.2.2 Increments of the Symmetric Random Walk84

3.2.3 Martingale Property for the Symmetric Random Walk85

3.2.4 Quadratic Variation of the Symmetric Random Walk85

3.2.5 Scaled Symmetric Random Walk86

3.2.6 Limiting Distribution of the Scaled Random Walk88

3.2.7 Log-Normal Distribution as the Limit of the Binomial Model91

3.3 Brownian Motion93

3.3.1 Definition of Brownian Motion93

3.3.2 Distribution of Brownian Motion95

3.3.3 Filtration for Brownian Motion97

3.3.4 Martingale Property for Brownian Motion98

3.4 Quadratic Variation98

3.4.1 First-Order Variation99

3.4.2 Quadratic Variation101

3.4.3 Volatility of Geometric Brownian Motion106

3.5 Markov Property107

3.6 First Passage Time Distribution108

3.7 Reflection Principle111

3.7.1 Reflection Equality111

3.7.2 First Passage Time Distribution112

3.7.3 Distribution of Brownian Motion and Its Maximum113

3.8 Summary115

3.9 Notes116

3.10 Exercises117

4 Stochastic Calculus125

4.1 Introduction125

4.2 It？'s Integral for Simple Integrands125

4.2.1 Construction of the Integral126

4.2.2 Properties of the Integral128

4.3 It？'s Integral for General Integrands132

4.4 It？-Doeblin Formula137

4.4.1 Formula for Brownian Motion137

4.4.2 Formula for It？ Processes143

4.4.3 Examples147

4.5 Black-Scholes-Merton Equation153

4.5.1 Evolution of Portfolio Value154

4.5.2 Evolution of Option Value155

4.5.3 Equating the Evolutions156

4.5.4 Solution to the Black-Scholes-Merton Equation158

4.5.5 The Greeks159

4.5.6 Put-Call Parity162

4.6 Multivariable Stochastic Calculus164

4.6.1 Multiple Brownian Motions164

4.6.2 It？-Doeblin Formula for Multiple Processes165

4.6.3 Recognizing a Brownian Motion168

4.7 Brownian Bridge172

4.7.1 Gaussian Processes172

4.7.2 Brownian Bridge as a Gaussian Process175

4.7.3 Brownian Bridge as a Scaled Stochastic Integral176

4.7.4 Multidimensional Distribution of the Brownian Bridge178

4.7.5 Brownian Bridge as a Conditioned Brownian Motion182

4.8 Summary183

4.9 Notes187

4.10 Exercises189

5 Risk-Neutral Pricing209

5.1 Introduction209

5.2 Risk-Neutral Measure210

5.2.1 Girsanov's Theorem for a Single Brownian Motion210

5.2.2 Stock Under the Risk-Neutral Measure214

5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure217

5.2.4 Pricing Under the Risk-Neutral Measure218

5.2.5 Deriving the Black-Scholes-Merton Formula218

5.3 Martingale Representation Theorem221

5.3.1 Martingale Representation with One Brownian Motion221

5.3.2 Hedging with One Stock222

5.4 Fundamental Theorems of Asset Pricing224

5.4.1 Girsanov and Martingale Representation Theorems224

5.4.2 Multidimensional Market Model226

5.4.3 Existence of the Risk-Neutral Measure228

5.4.4 Uniqueness of the Risk-Neutral Measure231

5.5 Dividend-Paying Stocks234

5.5.1 Continuously Paying Dividend235

5.5.2 Continuously Paying Dividend with Constant Coefficients237

5.5.3 Lump Payments of Dividends238

5.5.4 Lump Payments of Dividends with239

Constant Coefficients239

5.6 Forwards and Futures240

5.6.1 Forward Contracts240

5.6.2 Futures Contracts241

5.6.3 Forward-Futures Spread247

5.7 Summary248

5.8 Notes250

5.9 Exercises251

6 Connections with Partial Differential Equations263

6.1 Introduction263

6.2 Stochastic Differential Equations263

6.3 The Markov Property266

6.4 Partial Differential Equations268

6.5 Interest Rate Models272

6.6 Multidimensional Feynman-Kac Theorems277

6.7 Summary280

6.8 Notes281

6.9 Exercises282

7 Exotic Options295

7.1 Introduction295

7.2 Maximum of Brownian Motion with Drift295

7.3 Knock-out Barrier Options299

7.3.1 Up-and-Out Call300

7.3.2 Black-Scholes-Merton Equation300

7.3.3 Computation of the Price of the Up-and-Out Call304

7.4 Lookback Options308

7.4.1 Floating Strike Lookback Option308

7.4.2 Black-Scholes-Merton Equation309

7.4.3 Reduction of Dimension312

7.4.4 Computation of the Price of the Lookback Option314

7.5 Asian Options320

7.5.1 Fixed-Strike Asian Call320

7.5.2 Augmentation of the State321

7.5.3 Change of Numéraire323

7.6 Summary331

7.7 Notes331

7.8 Exercises332

8 American Derivative Securities339

8.1 Introduction339

8.2 Stopping Times340

8.3 Perpetual American Put345

8.3.1 Price Under Arbitrary Exercise346

8.3.2 Price Under Optimal Exercise349

8.3.3 Analytical Characterization of the Put Price351

8.3.4 Probabilistic Characterization of the Put Price353

8.4 Finite-Expiration American Put356

8.4.1 Analytical Characterization of the Put Price357

8.4.2 Probabilistic Characterization of the Put Price359

8.5 American Call361

8.5.1 Underlying Asset Pays No Dividends361

8.5.2 Underlying Asset Pays Dividends363

8.6 Summary368

8.7 Notes369

8.8 Exercises370

9 Change of Numéraire375

9.1 Introduction375

9.2 Numéraire376

9.3 Foreign and Domestic Risk-Neutral Measures381

9.3.1 The Basic Processes381

9.3.2 Domestic Risk-Neutral Measure383

9.3.3 Foreign Risk-Neutral Measure385

9.3.4 Siegel's Exchange Rate Paradox387

9.3.5 Forward Exchange Rates388

9.3.6 Garman-Kohlhagen Formula390

9.3.7 Exchange Rate Put-Call Duality390

9.4 Forward Measures392

9.4.1 Forward Price392

9.4.2 Zero-Coupon Bond as Numéraire392

9.4.3 Option Pricing with a Random Interest Rate394

9.5 Summary397

9.6 Notes398

9.7 Exercises398

10 Term-Structure Models403

10.1 Introduction403

10.2 Affine-Yield Models405

10.2.1 Two-Factor Vasicek Model406

10.2.2 Two-Factor CIR Model420

10.2.3 Mixed Model422

10.3 Heath-Jarrow-Morton Model423

10.3.1 Forward Rates423

10.3.2 Dynamics of Forward Rates and Bond Prices425

10.3.3 No-Arbitrage Condition426

10.3.4 HJM Under Risk-Neutral Measure429

10.3.5 Relation to Affine-Yield Models430

10.3.6 Implementation of HJM432

10.4 Forward LIBOR Model435

10.4.1 The Problem with Forward Rates435

10.4.2 LIBOR and Forward LIBOR436

10.4.3 Pricing a Backset LIBOR Contract437

10.4.4 Black Caplet Formula438

10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities440

10.4.6 A Forward LIBOR Term-Structure Model442

10.5 Summary447

10.6 Notes450

10.7 Exercises451

11 Introduction to Jump Processes461

11.1 Introduction461

11.2 Poisson Process462

11.2.1 Exponential Random Variables462

11.2.2 Construction of a Poisson Process463

11.2.3 Distribution of Poisson Process Increments463

11.2.4 Mean and Variance of Poisson Increments466

11.2.5 Martingale Property467

11.3 Compound Poisson Process468

11.3.1 Construction of a Compound Poisson Process468

11.3.2 Moment-Generating Function470

11.4 Jump Processes and Their Integrals473

11.4.1 Jump Processes474

11.4.2 Quadratic Variation479

11.5 Stochastic Calculus for Jump Processes483

11.5.1 It？-Doeblin Formula for One Jump Process483

11.5.2 It？-Doeblin Formula for Multiple Jump Processes489

11.6 Change of Measure492

11.6.1 Change of Measure for a Poisson Process493

11.6.2 Change of Measure for a Compound Poisson Process495

11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion502

11.7 Pricing a European Call in a Jump Model505

11.7.1 Asset Driven by a Poisson Process505

11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process512

11.8 Summary523

11.9 Notes525

11.10 Exercises525

A Advanced Topics in Probability Theory527

A.1 Countable Additivity527

A.2 Generating σ-algebras530

A.3 Random Variable with Neither Density nor Probability Mass Function531

B Existence of Conditional Expectations533

C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing535

References537

Index545