PDE and Martingale Methods in Option Pricing / Edition 1

by Andrea Pascucci

Overview

This book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of shastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Lévy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform.

Product Details

ISBN-13:	9788847017801
Publisher:	Springer Milan
Publication date:	03/09/2011
Series:	Bocconi & Springer Series , #2
Edition description:	2011
Pages:	721
Product dimensions:	6.10(w) x 9.25(h) x 0.06(d)

About the Author

Andrea Pascucci is Professor of Mathematics at the University of Bologna where he is director of a master program in Quantitative Finance. His research interests include second order parabolic partial differential equations and shastic analysis with applications to finance (pricing of European, American and Asian options).

Table of Contents

Preface V

General notations XV

1 Derivatives and arbitrage pricing 1

1.1 Options 1

1.1.1 Main purposes 3

1.1.2 Main problems 4

1.1.3 Rules of compounding 4

1.1.4 Arbitrage opportunities and Put-Call parity formula 5

1.2 Risk-neutral price and arbitrage pricing 7

1.2.1 Risk-neutral price 7

1.2.2 Risk-neutral probability 8

1.2.3 Arbitrage price 8

1.2.4 A generalization of the Put-Call parity 10

1.2.5 Incomplete markets 11

2 Discrete market models 15

2.1 Discrete markets and arbitrage strategies 15

2.1.1 Self-financing and predictable strategies 16

2.1.2 Normalized market 19

2.1.3 Arbitrage opportunities and admissible strategies 20

2.1.4 Equivalent martingale measure 21

2.1.5 Change of numeraire 24

2.2 European derivatives 26

2.2.1 Pricing in an arbitrage-free market 27

2.2.2 Completeness 30

2.2.3 Fundamental theorems of asset pricing 31

2.2.4 Markov property 34

2.3 Binomial model 35

2.3.1 Martingale measure and arbitrage price 38

2.3.2 Hedging strategies 40

2.3.3 Binomial algorithm 45

2.3.4 Calibration 50

2.3.5 Binomial model and Black-Scholes formula 53

2.3.6 Black-Scholes differential equation 60

2.4 Trinomial model 62

2.4.1 Pricing and hedging in an incomplete market 66

2.5 American derivatives 72

2.5.1 Arbitrage price 74

2.5.2 Optimal exercise strategies 80

2.5.3 Pricing and hedging algorithms 83

2.5.4 Relations with European options 88

2.5.5 Free-boundary problem for American options 90

2.5.6 American and European options in the binomial model 93

3 Continuous-time stochastic processes 97

3.1 Stochastic processes and real Brownian motion 97

3.1.1 Markov property 100

3.1.2 Brownian motion and the heat equation 102

3.2 Uniqueness 103

3.2.1 Law of a continuous process 103

3.2.2 Equivalence of processes 105

3.2.3 Modifications and indistinguishable processes 107

3.2.4 Adapted and progressively measurable processes 110

3.3 Martingales 111

3.3.1 Doob's inequality 113

3.3.2 Martingale spaces: M2 and M2 c 114

3.3.3 The usual hypotheses 117

3.3.4 Stopping times and martingales 120

3.4 Riemann-Stieltjes integral 125

3.4.1 Bounded-variation functions 127

3.4.2 Riemann-Stieltjes integral and Itô formula 131

3.4.3 Regularity of the paths of a Brownian motion 134

4 Brownian integration 139

4.1 Stochastic integral of deterministic functions 140

4.2 Stochastic integral of simple processes 141

4.3 Integral of L2-processes 145

4.3.1 Itô and Riemann-Stieltjes integral 149

4.3.2 Itô integral and stopping times 151

4.3.3 Quadratic variation process 153

4.3.4 Martingales with bounded variation 156

4.3.5 Co-variation process 157

4.4 Integral of L2 loc-processes 159

4.4.1 Local martingales 161

4.4.2 Localization and quadratic variation 163

5 Itô calculus 167

5.1 Itô processes 168

5.1.1 Itô formula for Brownian motion 169

5.1.2 General formulation 174

5.1.3 Martingales+and parabolic equations 176

5.1.4 Geometric Brownian motion 176

5.2 Multi-dimensional Itô processes 179

5.2.1 Multi-dimensional Itô formula 183

5.2.2 Correlated Brownian motion+and martingales 188

5.3 Generalized Itô formulas 191

5.3.1 Itô formula and+weak derivatives 191

5.3.2 Tanaka formula+and local times 194

5.3.3 Tanaka+formula for Itô processes 197

5.3.4 Local+time and Black-Scholes formula 198

6 Parabolic PDEs with variable coefficients: uniqueness 203

6.1 Maximum principle and Cauchy-Dirichlet problem 206

6.2 Maximum principle and Cauchy problem 208

6.3 Non-negative solutions of the Cauchy problem 213

7 Black-Scholes model 219

7.1 Self-financing strategies 220

7.2 Markovian strategies and Black-Scholes equation 222

7.3 Pricing 225

7.3.1 Dividends and time-dependent parameters 228

7.3.2 Admissibility and absence of arbitrage 229

7.3.3 Black-Scholes analysis: heuristic approaches 231

7.3.4 Market price of risk 233

7.4 Hedging 236

7.4.1 The Greeks 236

7.4.2 Robustness of the model 245

7.4.3 Gamma and Vega-hedging 246

7.5 Implied volatility 248

7.6 Asian options 252

7.6.1 Arithmetic average 253

7.6.2 Geometric average 255

8 Parabolic PDEs with variable coefficients: existence 257

8.1 Cauchy problem and fundamental solution 258

8.1.1 Levi's parametrix method 260

8.1.2 Gaussian estimates and adjoint operator 261

8.2 Obstacle problem 263

8.2.1 Strong solutions 265

8.2.2 Penalization method 268

9 Stochastic differential equations 275

9.1 Strong solutions 276

9.1.1 Uniqueness 278

9.1.2 Existence 280

9.1.3 Properties of solutions 283

9.2 Weak solutions 286

9.2.1 Tanaka's example 286

9.2.2 Existence: the martingale problem 287

9.2.3 Uniqueness 290

9.3 Maximal estimates 292

9.3.1 Maximal estimates for martingales 293

9.3.2 Maximal estimates for diffusions 296

9.4 Feynman-Kac representation formulas 298

9.4.1 Exit time from a bounded domain 300

9.4.2 Elliptic-parabolic equations and Dirichlet-problem 302

9.4.3 Evolution equations and Cauchy-Dirichlet problem 307

9.4.4 Fundamental solution and transition density 308

9.4.5 Obstacle problem and optimal stopping 310

9.5 Linear equations 314

9.5.1 Kalman condition 318

9.5.2 Kolmogorov equations and Hörmander condition 323

9.5.3 Examples 326

10 Continuous market models 329

10.1 Change of measure 329

10.1.1 Exponential martingales 329

10.1.2 Girsanov's theorem 332

10.1.3 Representation of Brownian martingales 334

10.1.4 Change of drift 339

10.2 Arbitrage theory 340

10.2.1 Change of drift with correlation 343

10.2.2 Martingale measures and market prices of risk 345

10.2.3 Examples 348

10.2.4 Admissible strategies and arbitrage opportunities 352

10.2.5 Arbitrage pricing 355

10.2.6 Complete markets 357

10.2.7 Parity formulas 358

10.3 Markovian models: the PDE approach 359

10.3.1 Martingale models for the short rate 361

10.3.2 Pricing and hedging in a complete model 364

10.4 Change of numeraire 366

10.4.1 LIBOR market model 370

10.4.2 Change of numeraire for Itô processes 372

10.4.3 Pricing with stochastic interest rate 374

10.5 Diffusion-based volatility models 376

10.5.1 Local and path-dependent volatility 377

10.5.2 CEV model 379

10.5.3 Stochastic volatility and the SABR model 386

11 American options 389

11.1 Pricing and hedging in the Black-Scholes model 389

11.2 American Call and Put options in the Black-Scholes model 395

11.3 Pricing and hedging in a complete market 398

12 Numerical methods 403

12.1 Euler method for ordinary equations 403

12.1.1 Higher order schemes 407

12.2 Euler method for stochastic differential equations 408

12.2.1 Milstein scheme 411

12.3 Finite-difference methods for parabolic equations 412

12.3.1 Localization 413

12.3.2 θ-schemes for the Cauchy-Dirichlet problem 414

12.3.3 Free-boundary problem 419

12.4 Monte Carlo methods 420

12.4.1 Simulation 423

12.4.2 Computation of the Greeks 425

12.4.3 Error analysis 427

13 Introduction to Lévy processes 429

13.1 Beyond Brownian motion 429

13.2 Poisson process 432

13.3 Lévy processes 437

13.3.1 Infinite divisibility and characteristic function 439

13.3.2 Jump measures of compound Poisson processes 444

13.3.3 Lévy-Itô decomposition 450

13.3.4 Lévy-Khintchine representation 457

13.3.5 Cumulants and Lévy martingales 460

13.4 Examples of Lévy processes 463

13.4.1 Jump-diffusion processes 464

13.4.2 Stable processes 466

13.4.3 Tempered stable processes 469

13.4.4 Subordination 471

13.4.5 Hyperbolic processes 478

13.5 Option pricing under exponential Lévy processes 480

13.5.1 Martingale modeling in Lévy markets 480

13.5.2 Incompleteness and choice of an EMM 485

13.5.3 Esscher transform 486

13.5.4 Exotic option pricing 491

13.5.5 Beyond Lévy processes 494

14 Stochastic calculus for jump processes 497

14.1 Stochastic integrals 497

14.1.1 Predictable processes 500

14.1.2 Semimartingales 504

14.1.3 Integrals with respect to jump measures 507

14.1.4 Lévy-type stochastic integrals 511

14.2 Stochastic differentials 514

14.2.1 Itô formula for discontinuous functions 514

14.2.2 Quadratic variation 515

14.2.3 Itô formula for semimartingales 518

14.2.4 Itô formula for Lévy processes 520

14.2.5 SDEs with jumps and Itô formula 525

14.2.6 PIDEs and Feynman-Kac representation 529

14.2.7 Linear SDEs with jumps 532

14.3 Lévy models with stochastic volatility 534

14.3.1 Lévy-driven models and pricing PIDEs 534

14.3.2 Bates model 537

14.3.3 Barndorff-Nielsen and Shephard model 539

15 Fourier methods 541

15.1 Characteristic functions and branch cut 542

15.2 Integral pricing formulas 545

15.2.1 Damping method 546

15.2.2 Pricing formulas 547

15.2.3 Implementation 551

15.2.4 Choice of the damping parameter 553

15.3 Fourier-cosine series expansions 562

15.3.1 Implementation 567

16 Elements of Malliavin calculus 577

16.1 Stochastic derivative 578

16.1.1 Examples 580

16.1.2 Chain rule 582

16.2 Duality 586

16.2.1 Clark-Ocone formula 588

16.2.2 Integration by parts and computation of the Greeks 590

16.2.3 Examples 594

Appendix: a primer in probability and parabolic PDEs 599

A.1 Probability spaces 599

A.1.1 Dynkin's theorems 601

A.1.2 Distributions 605

A.1.3 Random variables 608

A.1.4 Integration 610

A.1.5 Mean and variance 612

A.1.6 σ-algebras and information 618

A.1.7 Independence 619

A.1.8 Product measure and joint distribution 622

A.1.9 Markov inequality 625

A.2 Fourier transform 626

A.3 Parabolic equations with constant coefficients 630

A.3.1 A special case 631

A.3.2 General case 636

A.3.3 Locally integrable initial datum 637

A.3.4 Non-homogeneous Cauchy problem 638

A.3.5 Adjoint operator 639

A.4 Characteristic function and normal distribution 641

A.4.1 Multi-normal distribution 643

A.5 Conditional expectation 646

A.5.1 Radon-Nikodym theorem 646

A.5.2 Conditional expectation 648

A.5.3 Conditional expectation and discrete random variables 650

A.5.4 Properties of the conditional expectation 652

A.5.5 Conditional expectation in L2 655

A.5.6 Change of measure 656

A.6 Stochastic processes in discrete time 657

A.6.1 Doob's decomposition 659

A.6.2 Stopping times 661

A.6.3 Doob's maximal inequality 665

A.7 Convergence of random variables 669

A.7.1 Characteristic function and convergence of variables 670

A.7.2 Uniform integrability 674

A.8 Topologies and σ-algebras 676

A.9 Generalized derivatives 678

A.9.1 Weak derivatives in R 678

A.9.2 Sobolev spaces and embedding theorems 681

A.9.3 Distributions 682

A.9.4 Mollifiers 687

A.10 Separation of convex sets 690

References 691

Index 713

Editorial Reviews

From the reviews:

“The author provides an excellent overview of methods from the theory of partial differential equations and shastic processes used in mathematical finance. … The book is well written, the mathematical level is quite sophisticated and a broad range of material is covered. At the same time, there is a clear focus on applications. The book is therefore warmly recommended for graduate students as well as for professionals in the financial industry.” (Johan Tysk, Mathematical Reviews, Issue 2012 i)

“The book is written for graduate and advanced undergraduate students and gives an introduction to the modern theory of option pricing. … this book covers a wide range of topics with good motivations on a rigorous mathematical level.” (Sören Christensen, Zentralblatt MATH, Vol. 1214, 2011)

From the Publisher