金融市场用的数学方法 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Part Ⅰ Continuous Path Processes3

1 Continuous-Path Random Processes:Mathematical Prerequisites3

1.1 Some Definitions3

1.1.1 Measurability3

1.1.2 Monotone Class Theorem4

1.1.3 Probability Measures5

1.1.4 Filtration5

1.1.5 Law of a Random Variable,Expectation6

1.1.6 Independence6

1.1.7 Equivalent Probabilities and Radon-Nikod？m Densities7

1.1.8 Construction of Simple Probability Spaces8

1.1.9 Conditional Expectation9

1.1.10 Stochastic Processes10

1.1.11 Convergence12

1.1.12 Laplace Transform13

1.1.13 Gaussian Processes15

1.1.14 Markov Processes15

1.1.15 Uniform Integrability18

1.2 Martingales19

1.2.1 Definition and Main Properties19

1.2.2 Spaces of Martingales21

1.2.3 Stopping Times21

1.2.4 Local Martingales25

1.3 Continuous Semi-martingales27

1.3.1 Brackets of Continuous Local Martingales27

1.3.2 Brackets of Continuous Semi-martingales29

1.4 Brownian Motion30

1.4.1 One-dimensional Brownian Motion30

1.4.2 d-dimensional Brownian Motion34

1.4.3 Correlated Brownian Motions34

1.5 Stochastic Calculus35

1.5.1 Stochastic Integration36

1.5.2 Integration by Parts38

1.5.3 It？'s Formula:The Fundamental Formula of Stochastic Calculus38

1.5.4 Stochastic Differential Equations43

1.5.5 Stochastic Differential Equations:The Onedimensional Case47

1.5.6 Partial Differential Equations51

1.5.7 Doléans-Dade Exponential52

1.6 Predictable Representation Property55

1.6.1 Brownian Motion Case55

1.6.2 Towards a General Definition of the Predictable Representation Property57

1.6.3 Dudley's Theorem60

1.6.4 Backward Stochastic Differential Equations61

1.7 Change of Probability and Girsanov's Theorem66

1.7.1 Change of Probability66

1.7.2 Decomposition of P-Martingales as Q-semi-martingales68

1.7.3 Girsanov's Theorem:The One-dimensional Brownian Motion Case69

1.7.4 Multidimensional Case72

1.7.5 Absolute Continuity73

1.7.6 Condition for Martingale Property of Exponential Local Martingales74

1.7.7 Predictable Representation Property under a Change of Probability77

1.7.8 An Example of Invariance of BM under Change of Measure78

2 Basic Concepts and Examples in Finance79

2.1 A Semi-martingale Framework79

2.1.1 The Financial Market80

2.1.2 Arbitrage Opportunities83

2.1.3 Equivalent Martingale Measure85

2.1.4 Admissible Strategies85

2.1.5 Complete Market87

2.2 A Diffusion Model89

2.2.1 Absence of Arbitrage90

2.2.2 Completeness of the Market90

2.2.3 PDE Evaluation of Contingent Claims in a Complete Market92

2.3 The Black and Scholes Model93

2.3.1 The Model94

2.3.2 European Call and Put Options97

2.3.3 The Greeks101

2.3.4 General Case102

2.3.5 Dividend Paying Assets102

2.3.6 R？le of Information104

2.4 Change of Numéraire105

2.4.1 Change of Numéraire and Black-Scholes Formula106

2.4.2 Self-financing Strategy and Change of Numéraire107

2.4.3 Change of Numéraire and Change of Probability108

2.4.4 Forward Measure108

2.4.5 Self-financing Strategies:Constrained Strategies109

2.5 Feynman-Kac112

2.5.1 Feynman-Kac Formula112

2.5.2 Occupation Time for a Brownian Motion113

2.5.3 Occupation Time for a Drifted Brownian Motion114

2.5.4 Cumulative Options116

2.5.5 Quantiles118

2.6 Ornstein-Uhlenbeck Processes and Related Processes119

2.6.1 Definition and Properties119

2.6.2 Zero-coupon Bond123

2.6.3 Absolute Continuity Relationship for Generalized Vasicek Processes124

2.6.4 Square of a Generalized Vasicek Process127

2.6.5 Powers of δ-Dimensional Radial OU Processes,Alias CIR Processes128

2.7 Valuation of European Options129

2.7.1 The Garman and Kohlhagen Model for Currency Options129

2.7.2 Evaluation of an Exchange Option130

2.7.3 Quanto Options132

3 Hitting Times:A Mix of Mathematics and Finance135

3.1 Hitting Times and the Law of the Maximum for Brownian Motion136

3.1.1 The Law of the Pair of Random Variables(Wt,Mt)136

3.1.2 Hitting Times Process138

3.1.3 Law of the Maximum of a Brownian Motion over[0,t]139

3.1.4 Laws of Hitting Times140

3.1.5 Law of the Infimum142

3.1.6 Laplace Transforms of Hitting Times143

3.2 Hitting Times for a Drifted Brownian Motion145

3.2.1 Joint Laws of (MX,X) and (mX,X) at Time t145

3.2.2 Laws of Maximum,Minimum,and Hitting Times147

3.2.3 Laplace Transforms148

3.2.4 Computation of W(v)(ll{Ty(X)<t}e-λTy(X))149

3.2.5 Normal Inverse Gaussian Law150

3.3 Hitting Times for Geometric Brownian Motion151

3.3.1 Laws of the Pairs(MSt,St)and(mSt,St)151

3.3.2 Laplace Transforms152

3.3.3 Computation of E(e-λTa(S)ll{Ta(S)<t})153

3.4 Hitting Times in Other Cases153

3.4.1 Ornstein-Uhlenbeck Processes153

3.4.2 Deterministic Volatility and Nonconstant Barrier154

3.5 Hitting Time of a Two-sided Barrier for BM and GBM156

3.5.1 Brownian Case156

3.5.2 Drifted Brownian Motion159

3.6 Barrier Options160

3.6.1 Put-Call Symmetry160

3.6.2 Binary Options and △'s163

3.6.3 Barrier Options:General Characteristics164

3.6.4 Valuation and Hedging of a Regular Down-and-In Call Option When the Underlying is a Martingale166

3.6.5 Mathematical Results Deduced from the Previous Approach169

3.6.6 Valuation and Hedging of Regular Down-and-In Call Options:The General Case172

3.6.7 Valuation and Hedging of Reverse Barrier Options175

3.6.8 The Emerging Calls Method177

3.6.9 Closed Form Expressions178

3.7 Lookback Options179

3.7.1 Using Binary Options179

3.7.2 Traditional Approach180

3.8 Double-barrier Options182

3.9 Other Options183

3.9.1 Options Involving a Hitting Time183

3.9.2 Boost Options184

3.9.3 Exponential Down Barrier Option186

3.10 A Structural Approach to Default Risk188

3.10.1 Merton's Model188

3.10.2 First Passage Time Models190

3.11 American Options191

3.11.1 American Stock Options192

3.11.2 American Currency Options193

3.11.3 Perpetual American Currency Options195

3.12 Real Options198

3.12.1 Optimal Entry with Stochastic Investment Costs198

3.12.2 Optimal Entry in the Presence of Competition201

3.12.3 Optimal Entry and Optimal Exit204

3.12.4 Optimal Exit and Optimal Entry in the Presence of Competition205

3.12.5 Optimal Entry and Exit Decisions206

4 Complements on Brownian Motion211

4.1 Local Time211

4.1.1 A Stochastic Fubini Theorem211

4.1.2 Occupation Time Formula211

4.1.3 An Approximation of Local Time213

4.1.4 Local Times for Semi-martingales214

4.1.5 Tanaka's Formula214

4.1.6 The Balayage Formula216

4.1.7 Skorokhod's Reflection Lemma217

4.1.8 Local Time of a Semi-martingale222

4.1.9 Generalized It？-Tanaka Formula226

4.2 Applications227

4.2.1 Dupire's Formula227

4.2.2 Stop-Loss Strategy229

4.2.3 Knock-out BOOST230

4.2.4 Passport Options232

4.3 Bridges,Excursions,and Meanders232

4.3.1 Brownian Motion Zeros232

4.3.2 Excursions232

4.3.3 Laws of Tx,dt and gt233

4.3.4 Laws of(Bt,gt,dt)236

4.3.5 Brownian Bridge237

4.3.6 Slow Brownian Filtrations241

4.3.7 Meanders242

4.3.8 The Azéma Martingale243

4.3.9 Drifted Brownian Motion244

4.4 Parisian Options246

4.4.1 The Law of(G-,eD(W),WG-,eD)249

4.4.2 Valuation of a Down-and-In Parisian Option252

4.4.3 PDE Approach256

4.4.4 American Parisian Options257

5 Complements on Continuous Path Processes259

5.1 Time Changes259

5.1.1 Inverse of an Increasing Process259

5.1.2 Time Changes and Stopping Times260

5.1.3 Brownian Motion and Time Changes261

5.2 Dual Predictable Projections264

5.2.1 Definitions264

5.2.2 Examples266

5.3 Diffusions269

5.3.1 (Time-homogeneous)Diffusions270

5.3.2 Scale Function and Speed Measure270

5.3.3 Boundary Points273

5.3.4 Change of Time or Change of Space Variable275

5.3.5Recurrence277

5.3.6 Resolvent Kernel and Green Function277

5.3.7 Examples279

5.4 Non-homogeneous Diffusions281

5.4.1 Kolmogorov's Equations281

5.4.2 Application:Dupire's Formula284

5.4.3 Fokker-Planck Equation286

5.4.4 Valuation of Contingent Claims289

5.5 Local Times for a Diffusion290

5.5.1 Various Definitions of Local Times290

5.5.2 Some Diffusions Involving Local Time291

5.6 Last Passage Times294

5.6.1 Notation and Basic Results294

5.6.2 Last Passage Time of a Transient Diffusion294

5.6.3 Last Passage Time Before Hitting a Level297

5.6.4 Last Passage Time Before Maturity298

5.6.5 Absolutely Continuous Compensator301

5.6.6 Time When the Supremum is Reached302

5.6.7 Last Passage Times for Particular Martingales303

5.7 Pitman's Theorem about(2Mt-Wt)306

5.7.1 Time Reversal of Brownian Motion306

5.7.2 Pitman's Theorem307

5.8 Filtrations309

5.8.1 Strong and Weak Brownian Filtrations310

5.8.2 Some Examples312

5.9 Enlargements of Filtrations315

5.9.1 Immersion of Filtrations315

5.9.2 The Brownian Bridge as an Example of Initial Enlargement318

5.9.3 Initial Enlargement:General Results319

5.9.4 Progressive Enlargement323

5.10 Filtering the Information329

5.10.1 Independent Drift329

5.10.2 Other Examples of Canonical Decomposition330

5.10.3 Innovation Process331

6 A Special Family of Diffusions:Bessel Processes333

6.1 Definitions and First Properties333

6.1.1 The Euclidean Norm of the n-Dimensional Brownian Motion333

6.1.2 General Definitions334

6.1.3 Path Properties337

6.1.4 Infinitesimal Generator337

6.1.5 Absolute Continuity339

6.2 Properties342

6.2.1 Additivity of BESQ's342

6.2.2 Transition Densities343

6.2.3 Hitting Times for Bessel Processes345

6.2.4 Lamperti's Theorem347

6.2.5 Laplace Transforms349

6.2.6 BESQ Processes with Negative Dimensions353

6.2.7 Squared Radial Ornstein-Uhlenbeck356

6.3 Cox-Ingersoll-Ross Processes356

6.3.1 CIR Processes and BESQ357

6.3.2 Transition Probabilities for a CIR Process358

6.3.3 CIR Processes as Spot Rate Models359

6.3.4 Zero-coupon Bond361

6.3.5 Inhomogeneous CIR Process364

6.4 Constant Elasticity of Variance Process365

6.4.1 Particular Case μ=0366

6.4.2 CEV Processes and CIR Processes368

6.4.3 CEV Processes and BESQ Processes368

6.4.4 Properties370

6.4.5 Scale Functions for CEV Processes371

6.4.6 Option Pricing in a CEV Model372

6.5 Some Computations on Bessel Bridges373

6.5.1 Bessel Bridges373

6.5.2 Bessel Bridges and Ornstein-Uhlenbeck Processes374

6.5.3 European Bond Option376

6.5.4 American Bond Options and the CIR Model378

6.6 Asian Options381

6.6.1 Parity and Symmetry Formulae382

6.6.2 Laws of A(v)θ and A(v)t383

6.6.3 The Moments of At388

6.6.4 Laplace Transform Approach389

6.6.5 PDE Approach391

6.7 Stochastic Volatility392

6.7.1 Black and Scholes Implied Volatility392

6.7.2 A General Stochastic Volatility Model392

6.7.3 Option Pricing in Presence of Non-normality of Returns:The Martingale Approach393

6.7.4 Hull and White Model396

6.7.5 Closed-form Solutions in Some Correlated Cases398

6.7.6 PDE Approach401

6.7.7 Heston's Model401

6.7.8 Mellin Transform403

Part Ⅱ Jump Processes407

7 Default Risk:An Enlargement of Filtration Approach407

7.1 A Toy Model407

7.1.1 Defaultable Zero-coupon with Payment at Maturity408

7.1.2 Defaultable Zero-coupon with Payment at Hit410

7.2 Toy Model and Martingales412

7.2.1 Key Lemma412

7.2.2 The Fundamental Martingale412

7.2.3 Hazard Function413

7.2.4 Incompleteness of the Toy Model,non Arbitrage Prices415

7.2.5 Predictable Representation Theorem415

7.2.6 Risk-neutral Probability Measures416

7.2.7 Partial Information:Duffie and Lando's Model418

7.3 Default Times with a Given Stochastic Intensity418

7.3.1 Construction of Default Time with a Given Stochastic Intensity418

7.3.2 Conditional Expectation with Respect to Ft419

7.3.3 Enlargements of Filtrations420

7.3.4 Conditional Expectations with Respect to ？t420

7.3.5 Conditional Expectations of F∞-Measurable Random Variables422

7.3.6 Correlated Defaults:Copula Approach423

7.3.7 Correlated Defaults:Jarrow and Yu's Model425

7.4 Conditional Survival Probability Approach426

7.4.1 Conditional Expectations427

7.5 Conditional Survival Probability Approach and Immersion428

7.5.1 (H)-Hypothesis and Arbitrages429

7.5.2 Pricing Contingent Claims430

7.5.3 Correlated Defaults:Kusuoka's Example431

7.5.4 Stochastic Barrier432

7.5.5 Predictable Representation Theorems432

7.5.6 Hedging Contingent Claims with DZC434

7.6 General Case:Without the(H)-Hypothesis437

7.6.1 An Example of Partial Observation437

7.6.2 Two Defaults,Trivial Reference Filtration440

7.6.3 Initial Times442

7.6.4 Explosive Defaults444

7.7 Intensity Approach445

7.7.1 Definition445

7.7.2 Valuation Formula446

7.8 Credit Default Swaps446

7.8.1 Dynamics of the CDS's Price in a single name setting447

7.8.2 Dynamics of the CDS's Price in a multi-name setting448

7.9 PDE Approach for Hedging Defaultable Claims449

7.9.1 Defaultable Asset with Total Default449

7.9.2 PDE for Valuation450

7.9.3 General Case454

8 Poisson Processes and Ruin Theory457

8.1 Counting Processes and Stochastic Integrals457

8.2 Standard Poisson Process459

8.2.1 Definition and First Properties459

8.2.2 Martingale Properties461

8.2.3 Infinitesimal Generator464

8.2.4 Change of Probability Measure:An Example465

8.2.5 Hitting Times466

8.3 Inhomogeneous Poisson Processes467

8.3.1 Definition467

8.3.2 Martingale Properties467

8.3.3 Watanabe's Characterization of Inhomogeneous Poisson Processes468

8.3.4 Stochastic Calculus469

8.3.5 Predictable Representation Property473

8.3.6 Multidimensional Poisson Processes474

8.4 Stochastic Intensity Processes475

8.4.1 Doubly Stochastic Poisson Processes475

8.4.2 Inhomogeneous Poisson Processes with Stochastic Intensity476

8.4.3 It？'s Formula476

8.4.4 Exponential Martingales477

8.4.5 Change of Probability Measure478

8.4.6 An Elementary Model of Prices Involving Jumps479

8.5 Poisson Bridges480

8.5.1 Definition of the Poisson Bridge480

8.5.2 Harness Property481

8.6 Compound Poisson Processes483

8.6.1 Definition and Properties483

8.6.2 Integration Formula484

8.6.3 Martingales485

8.6.4 It？'s Formula492

8.6.5 Hitting Times492

8.6.6 Change of Probability Measure494

8.6.7 Price Process495

8.6.8 Martingale Representation Theorem496

8.6.9 Option Pricing497

8.7 Ruin Process497

8.7.1 Ruin Probability497

8.7.2 Integral Equation498

8.7.3 An Example498

8.8 Marked Point Processes501

8.8.1 Random Measure501

8.8.2 Definition501

8.8.3 An Integration Formula503

8.8.4 Marked Point Processes with Intensity and Associated Martingales503

8.8.5 Girsanov's Theorem504

8.8.6 Predictable Representation Theorem504

8.9 Poisson Point Processes505

8.9.1 Poisson Measures505

8.9.2 Point Processes506

8.9.3 Poisson Point Processes506

8.9.4 The It？ Measure of Brownian Excursions507

9 General Processes:Mathematical Facts509

9.1 Some Basic Facts about càdlàg Processes509

9.1.1 An Illustrative Lemma509

9.1.2 Finite Variation Processes,Pure Jump Processes510

9.1.3 Some σ-algebras512

9.2 Stochastic Integration for Square Integrable Martingales513

9.2.1 Square Integrable Martingales513

9.2.2 Stochastic Integral516

9.3 Stochastic Integration for Semi-martingales517

9.3.1 Local Martingales517

9.3.2 Quadratic Covariation and Predictable Bracket of Two Local Martingales519

9.3.3 Orthogonality521

9.3.4 Semi-martingales522

9.3.5 Stochastic Integration for Semi-martingales524

9.3.6 Quadratic Covariation of Two Semi-martingales525

9.3.7 Particular Cases525

9.3.8 Predictable Bracket of Two Semi-martingales527

9.4 It？'s Formula and Girsanov's Theorem528

9.4.1 It？'s Formula:Optional and Predictable Forms528

9.4.2 Semi-martingale Local Times531

9.4.3 Exponential Semi-martingales532

9.4.4 Change of Probability, Girsanov's Theorem534

9.5 Existence and Uniqueness of the e.m.m537

9.5.1 Predictable Representation Property537

9.5.2 Necessary Conditions for Existence538

9.5.3 Uniqueness Property542

9.5.4 Examples543

9.6 Self-financing Strategies and Integration by Parts544

9.6.1 The Model545

9.6.2 Self-financing Strategies and Change of Numéraire545

9.7 Valuation in an Incomplete Market547

9.7.1 Replication Criteria548

9.7.2 Choice of an Equivalent Martingale Measure549

9.7.3 Indifference Prices550

10 Mixed Processes551

10.1 Definition551

10.2 It？'s Formula552

10.2.1 Integration by Parts552

10.2.2 It？'s Formula:One-dimensional Case553

10.2.3 Multidimensional Case555

10.2.4 Stochastic Differential Equations556

10.2.5 Feynman-Kac Formula557

10.2.6 Predictable Representation Theorem558

10.3 Change of Probability559

10.3.1 Exponential Local Martingales559

10.3.2 Girsanov's Theorem560

10.4 Mixed Processes in Finance561

10.4.1 Computation of the Moments561

10.4.2 Symmetry562

10.4.3 Hitting Times563

10.4.4 Affine Jump-Diffusion Model565

10.4.5 General Jump-Diffusion Processes569

10.5 Incompleteness569

10.5.1 The Set of Risk-neutral Probability Measures570

10.5.2 The Range of Prices for European Call Options572

10.5.3 General Contingent Claims575

10.6 Complete Markets with Jumps578

10.6.1 A Three Assets Model578

10.6.2 Structure Equations579

10.7 Valuation of Options582

10.7.1 The Valuation of European Options584

10.7.2 American Option586

11 Lévy Processes591

11.1 Infinitely Divisible Random Variables592

11.1.1 Definition592

11.1.2 Self-decomposable Random Variables596

11.1.3 Stable Random Variables598

11.2 Lévy Processes599

11.2.1 Definition and Main Properties599

11.2.2 Poisson Point Processes,Lévy Measures601

11.2.3 Lévy-Khintchine Formula for a Lévy Process606

11.2.4 It？'s Formulae for a One-dimensional Lévy Process612

11.2.5 It？'s Formula for Lévy-It？ Processes613

11.2.6 Martingales615

11.2.7 Harness Property620

11.2.8 Representation Theorem of Martingales in a Lévy Setting621

11.3 Absolutely Continuous Changes of Measures623

11.3.1 Esscher Transform623

11.3.2 Preserving the Lévy Property with Absolute Continuity625

11.3.3 General Case627

11.4 Fluctuation Theory628

11.4.1 Maximum and Minimum628

11.4.2 Pecherskii-Rogozin Identity631

11.5 Spectrally Negative Lévy Processes632

11.5.1 Two-sided Exit Times632

11.5.2 Laplace Exponent of the Ladder Process633

11.5.3 D.Kendall's Identity633

11.6 Subordinators634

11.6.1 Definition and Examples634

11.6.2 Lévy Characteristics of a Subordinated Process636

11.7 Exponential Lévy Processes as Stock Price Processes636

11.7.1 Option Pricing with Esscher Transform636

11.7.2 A Differential Equation for Option Pricing637

11.7.3 Put-call Symmetry638

11.7.4 Arbitrage and Completeness639

11.8 Variance-Gamma Model639

11.9 Valuation of Contingent Claims641

11.9.1 Perpetual American Options641

A List of Special Features,Probability Laws,and Functions647

A.1 Main Formulae647

A.1.1 Absolute Continuity Relationships647

A.1.2 Bessel Processes648

A.1.3 Brownian Motion649

A.1.4 Diffusions650

A.1.5 Finance650

A.1.6 Girsanov's Theorem651

A.1.7 Hitting Times651

A.1.8 It？'s Formulae651

A.1.9 Lévy Processes653

A.1.10 Semi-martingales654

A.2 Processes655

A.3 Some Main Models655

A.4 Some Important Probability Distributions656

A.4.1 Laws with Density656

A.4.2 Some Algebraic Properties for Special r.v.'s656

A.4.3 Poisson Law657

A.4.4 Gamma and Inverse Gaussian Law658

A.4.5 Generalized Inverse Gaussian and Normal Inverse Gaussian659

A.4.6 Variance Gamma VG(σ,ν,θ)661

A.4.7 Tempered Stable TS(Y±,C±,M±)661

A.5 Special Functions662

A.5.1 Gamma and Beta Functions662

A.5.2 Bessel Functions662

A.5.3 Hermite Functions663

A.5.4 Parabolic Cylinder Functions663

A.5.5 Airy Function663

A.5.6 Kummer Functions664

A.5.7 Whittaker Functions664

A.5.8 Some Laplace Transforms665

References667

B Some Papers and Books on Specific Subjects709

B.1 Theory of Continuous Processes709

B.1.1 Books709

B.1.2 Stochastic Differential Equations709

B.1.3 Backward SDE709

B.1.4 Martingale Representation Theorems710

B.1.5 Enlargement of Filtrations710

B.1.6 Exponential Functionals710

B.1.7 Uniform Integrability of Martingales710

B.2 Particular Processes710

B.2.1 Ornstein-Uhlenbeck Processes710

B.2.2 CIR Processes710

B.2.3 CEV Processes711

B.2.4 Bessel Processes711

B.3 Processes with Discontinuous Paths711

B.3.1 Some Books711

B.3.2 Survey Papers711

B.4 Hitting Times711

B.5 Lévy Processes712

B.5.1 Books712

B.5.2 Some Papers712

B.6 Some Books on Finance712

B.6.1 Discrete Time712

B.6.2 Continuous Time712

B.6.3 Collective Books712

B.6.4 History713

B.7 Arbitrage713

B.8 Exotic Options713

B.8.1 Books713

B.8.2 Articles713

Index of Authors715

Index of Symbols723

Subject Index725