PDE and Martingale Methods in Option Pricing

By: Andrea Pascucci

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Hardcover | 28 December 2010 | Edition Number 2

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740 Pages

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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 stochastic 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 Levy 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.

Industry Reviews

From the reviews:

"The author provides an excellent overview of methods from the theory of partial differential equations and stochastic 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." (Soeren Christensen, Zentralblatt MATH, Vol. 1214, 2011)

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Stochastics Economics Econometrics Accounting & Finance Finance Investment & Securities Calculus & Mathematical Analysis Differential Calculus & Equations Banking

Probability & Statistics Optimisation Game Theory

Preface	p. V
General notations	p. XV
Derivatives and arbitrage pricing	p. 1
Options	p. 1
Main purposes	p. 3
Main problems	p. 4
Rules of compounding	p. 4
Arbitrage opportunities and Put-Call parity formula	p. 5
Risk-neutral price and arbitrage pricing	p. 7
Risk-neutral price	p. 7
Risk-neutral probability	p. 8
Arbitrage price	p. 8
A generalization of the Put-Call parity	p. 10
Incomplete markets	p. 11
Discrete market models	p. 15
Discrete markets and arbitrage strategies	p. 15
Self-financing and predictable strategies	p. 16
Normalized market	p. 19
Arbitrage opportunities and admissible strategies	p. 20
Equivalent martingale measure	p. 21
Change of numeraire	p. 24
European derivatives	p. 26
Pricing in an arbitrage-free market	p. 27
Completeness	p. 30
Fundamental theorems of asset pricing	p. 31
Markov property	p. 34
Binomial model	p. 35
Martingale measure and arbitrage price	p. 38
Hedging strategies	p. 40
Binomial algorithm	p. 45
Calibration	p. 50
Binomial model and Black-Scholes formula	p. 53
Black-Scholes differential equation	p. 60
Trinomial model	p. 62
Pricing and hedging in an incomplete market	p. 66
American derivatives	p. 72
Arbitrage price	p. 74
Optimal exercise strategies	p. 80
Pricing and hedging algorithms	p. 83
Relations with European options	p. 88
Free-boundary problem for American options	p. 90
American and European options in the binomial model	p. 93
Continuous-time stochastic processes	p. 97
Stochastic processes and real Brownian motion	p. 97
Markov property	p. 100
Brownian motion and the heat equation	p. 102
Uniqueness	p. 103
Law of a continuous process	p. 103
Equivalence of processes	p. 105
Modifications and indistinguishable processes	p. 107
Adapted and progressively measurable processes	p. 110
Martingales	p. 111
Doob's inequality	p. 113
Martingale spaces: M2 and M2 c	p. 114
The usual hypotheses	p. 117
Stopping times and martingales	p. 120
Riemann-Stieltjes integral	p. 125
Bounded-variation functions	p. 127
Riemann-Stieltjes integral and Itô formula	p. 131
Regularity of the paths of a Brownian motion	p. 134
Brownian integration	p. 139
Stochastic integral of deterministic functions	p. 140
Stochastic integral of simple processes	p. 141
Integral of L2-processes	p. 145
Itô and Riemann-Stieltjes integral	p. 149
Itô integral and stopping times	p. 151
Quadratic variation process	p. 153
Martingales with bounded variation	p. 156
Co-variation process	p. 157
Integral of L2 loc-processes	p. 159
Local martingales	p. 161
Localization and quadratic variation	p. 163
Itô calculus	p. 167
Itô processes	p. 168
Itô formula for Brownian motion	p. 169
General formulation	p. 174
Martingales+and parabolic equations	p. 176
Geometric Brownian motion	p. 176
Multi-dimensional Itô processes	p. 179
Multi-dimensional Itô formula	p. 183
Correlated Brownian motion+and martingales	p. 188
Generalized Itô formulas	p. 191
Itô formula and+weak derivatives	p. 191
Tanaka formula+and local times	p. 194
Tanaka+formula for Itô processes	p. 197
Local+time and Black-Scholes formula	p. 198
Parabolic PDEs with variable coefficients: uniqueness	p. 203
Maximum principle and Cauchy-Dirichlet problem	p. 206
Maximum principle and Cauchy problem	p. 208
Non-negative solutions of the Cauchy problem	p. 213
Black-Scholes model	p. 219
Self-financing strategies	p. 220
Markovian strategies and Black-Scholes equation	p. 222
Pricing	p. 225
Dividends and time-dependent parameters	p. 228
Admissibility and absence of arbitrage	p. 229
Black-Scholes analysis: heuristic approaches	p. 231
Market price of risk	p. 233
Hedging	p. 236
The Greeks	p. 236
Robustness of the model	p. 245
Gamma and Vega-hedging	p. 246
Implied volatility	p. 248
Asian options	p. 252
Arithmetic average	p. 253
Geometric average	p. 255
Parabolic PDEs with variable coefficients: existence	p. 257
Cauchy problem and fundamental solution	p. 258
Levi's parametrix method	p. 260
Gaussian estimates and adjoint operator	p. 261
Obstacle problem	p. 263
Strong solutions	p. 265
Penalization method	p. 268
Stochastic differential equations	p. 275
Strong solutions	p. 276
Uniqueness	p. 278
Existence	p. 280
Properties of solutions	p. 283
Weak solutions	p. 286
Tanaka's example	p. 286
Existence: the martingale problem	p. 287
Uniqueness	p. 290
Maximal estimates	p. 292
Maximal estimates for martingales	p. 293
Maximal estimates for diffusions	p. 296
Feynman-Kac representation formulas	p. 298
Exit time from a bounded domain	p. 300
Elliptic-parabolic equations and Dirichlet-problem	p. 302
Evolution equations and Cauchy-Dirichlet problem	p. 307
Fundamental solution and transition density	p. 308
Obstacle problem and optimal stopping	p. 310
Linear equations	p. 314
Kalman condition	p. 318
Kolmogorov equations and Hörmander condition	p. 323
Examples	p. 326
Continuous market models	p. 329
Change of measure	p. 329
Exponential martingales	p. 329
Girsanov's theorem	p. 332
Representation of Brownian martingales	p. 334
Change of drift	p. 339
Arbitrage theory	p. 340
Change of drift with correlation	p. 343
Martingale measures and market prices of risk	p. 345
Examples	p. 348
Admissible strategies and arbitrage opportunities	p. 352
Arbitrage pricing	p. 355
Complete markets	p. 357
Parity formulas	p. 358
Markovian models: the PDE approach	p. 359
Martingale models for the short rate	p. 361
Pricing and hedging in a complete model	p. 364
Change of numeraire	p. 366
LIBOR market model	p. 370
Change of numeraire for Itô processes	p. 372
Pricing with stochastic interest rate	p. 374
Diffusion-based volatility models	p. 376
Local and path-dependent volatility	p. 377
CEV model	p. 379
Stochastic volatility and the SABR model	p. 386
American options	p. 389
Pricing and hedging in the Black-Scholes model	p. 389
American Call and Put options in the Black-Scholes model	p. 395
Pricing and hedging in a complete market	p. 398
Numerical methods	p. 403
Euler method for ordinary equations	p. 403
Higher order schemes	p. 407
Euler method for stochastic differential equations	p. 408
Milstein scheme	p. 411
Finite-difference methods for parabolic equations	p. 412
Localization	p. 413
-schemes for the Cauchy-Dirichlet problem	p. 414
Free-boundary problem	p. 419
Monte Carlo methods	p. 420
Simulation	p. 423
Computation of the Greeks	p. 425
Error analysis	p. 427
Introduction to Lévy processes	p. 429
Beyond Brownian motion	p. 429
Poisson process	p. 432
Lévy processes	p. 437
Infinite divisibility and characteristic function	p. 439
Jump measures of compound Poisson processes	p. 444
Lévy-Itô decomposition	p. 450
Lévy-Khintchine representation	p. 457
Cumulants and Lévy martingales	p. 460
Examples of Lévy processes	p. 463
Jump-diffusion processes	p. 464
Stable processes	p. 466
Tempered stable processes	p. 469
Subordination	p. 471
Hyperbolic processes	p. 478
Option pricing under exponential Lévy processes	p. 480
Martingale modeling in Lévy markets	p. 480
Incompleteness and choice of an EMM	p. 485
Esscher transform	p. 486
Exotic option pricing	p. 491
Beyond Lévy processes	p. 494
Stochastic calculus for jump processes	p. 497
Stochastic integrals	p. 497
Predictable processes	p. 500
Semimartingales	p. 504
Integrals with respect to jump measures	p. 507
Lévy-type stochastic integrals	p. 511
Stochastic differentials	p. 514
Itô formula for discontinuous functions	p. 514
Quadratic variation	p. 515
Itô formula for semimartingales	p. 518
Itô formula for Lévy processes	p. 520
SDEs with jumps and Itô formula	p. 525
PIDEs and Feynman-Kac representation	p. 529
Linear SDEs with jumps	p. 532
Lévy models with stochastic volatility	p. 534
Lévy-driven models and pricing PIDEs	p. 534
Bates model	p. 537
Barndorff-Nielsen and Shephard model	p. 539
Fourier methods	p. 541
Characteristic functions and branch cut	p. 542
Integral pricing formulas	p. 545
Damping method	p. 546
Pricing formulas	p. 547
Implementation	p. 551
Choice of the damping parameter	p. 553
Fourier-cosine series expansions	p. 562
Implementation	p. 567
Elements of Malliavin calculus	p. 577
Stochastic derivative	p. 578
Examples	p. 580
Chain rule	p. 582
Duality	p. 586
Clark-Ocone formula	p. 588
Integration by parts and computation of the Greeks	p. 590
Examples	p. 594
Appendix: a primer in probability and parabolic PDEs	p. 599
Probability spaces	p. 599
Dynkin's theorems	p. 601
Distributions	p. 605
Random variables	p. 608
Integration	p. 610
Mean and variance	p. 612
-algebras and information	p. 618
Independence	p. 619
Product measure and joint distribution	p. 622
Markov inequality	p. 625
Fourier transform	p. 626
Parabolic equations with constant coefficients	p. 630
A special case	p. 631
General case	p. 636
Locally integrable initial datum	p. 637
Non-homogeneous Cauchy problem	p. 638
Adjoint operator	p. 639
Characteristic function and normal distribution	p. 641
Multi-normal distribution	p. 643
Conditional expectation	p. 646
Radon-Nikodym theorem	p. 646
Conditional expectation	p. 648
Conditional expectation and discrete random variables	p. 650
Properties of the conditional expectation	p. 652
Conditional expectation in L2	p. 655
Change of measure	p. 656
Stochastic processes in discrete time	p. 657
Doob's decomposition	p. 659
Stopping times	p. 661
Doob's maximal inequality	p. 665
Convergence of random variables	p. 669
Characteristic function and convergence of variables	p. 670
Uniform integrability	p. 674
Topologies and -algebras	p. 676
Generalized derivatives	p. 678
Weak derivatives in R	p. 678
Sobolev spaces and embedding theorems	p. 681
Distributions	p. 682
Mollifiers	p. 687
Separation of convex sets	p. 690
References	p. 691
Index	p. 713
Table of Contents provided by Ingram. All Rights Reserved.

PDE and Martingale Methods in Option Pricing

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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