Numerical Solution of Stochastic Differential Equations With Jumps in Finance

by Platen, Eckhard; Bruti-Liberati, Nicola

• ISBN13: 9783642120572
• ISBN10: 3642120571
• Format: Hardcover
• Copyright: 2010-10-15
• Publisher: Springer Nature

Summary

In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.

Table of Contents

Preface	p. V
Suggestions for the Reader	p. XV
Basic Notation	p. XIX
Motivation and Brief Survey	p. XXIII
Stochastic Differential Equations with Jumps	p. 1
Stochastic Processes	p. 1
Supermartingales and Martingales	p. 16
Quadratic Variation and Covariation	p. 23
Itô Integral	p. 26
Itô Formula	p. 34
Stochastic Differential Equations	p. 38
Linear SDEs	p. 45
SDEs with Jumps	p. 53
Existence and Uniqueness of Solutions of SDEs	p. 57
Exercises	p. 59
Exact Simulation of Solutions of SDEs	p. 61
Motivation of Exact Simulation	p. 61
Sampling from Transition Distributions	p. 63
Exact Solutions of Multi-dimensional SDEs	p. 78
Functions of Exact Solutions	p. 99
Almost Exact Solutions by Conditioning	p. 105
Almost Exact Simulation by Time Change	p. 113
Functionals of Solutions of SDEs	p. 123
Exercises	p. 136
Benchmark Approach to Finance and Insurance	p. 139
Market Model	p. 139
Best Performing Portfolio	p. 142
Supermartingale Property and Pricing	p. 145
Diversification	p. 149
Real World Pricing Under Some Models	p. 158
Real World Pricing Under the MMM	p. 168
Binomial Option Pricing	p. 176
Exercises	p. 185
Stochastic Expansions	p. 187
Introduction to Wagner-Platen Expansions	p. 187
Multiple Stochastic Integrals	p. 195
Coefficient Functions	p. 202
Wagner-Platen Expansions	p. 206
Moments of Multiple Stochastic Integrals	p. 211
Exercises	p. 230
Introduction to Scenario Simulation	p. 233
Approximating Solutions of ODEs	p. 233
Scenario Simulation	p. 245
Strong Taylor Schemes	p. 252
Derivative-Free Strong Schemes	p. 266
Exercises	p. 271
Regular Strong Taylor Approximations with Jumps	p. 273
Discrete-Time Approximation	p. 273
Strong Order 1.0 Taylor Scheme	p. 278
Commutativity Conditions	p. 286
Convergence Results	p. 289
Lemma on Multiple Itô Integrals	p. 292
Proof of the Convergence Theorem	p. 302
Exercises	p. 307
Regular Strong Itô Approximations	p. 309
Explicit Regular Strong Schemes	p. 309
Drift-Implicit Schemes	p. 316
Balanced Implicit Methods	p. 321
Predictor-Corrector Schemes	p. 326
Convergence Results	p. 331
Exercises	p. 346
Jump-Adapted Strong Approximations	p. 347
Introduction to Jump-Adapted Approximations	p. 347
Jump-Adapted Strong Taylor Schemes	p. 350
Jump-Adapted Derivative-Free Strong Schemes	p. 355
Jump-Adapted Drift-Implicit Schemes	p. 356
Predictor-Corrector Strong Schemes	p. 359
Jump-Adapted Exact Simulation	p. 361
Convergence Results	p. 362
Numerical Results on Strong Schemes	p. 368
Approximation of Pure Jump Processes	p. 375
Exercises	p. 388
Estimating Discretely Observed Diffusions	p. 389
Maximum Likelihood Estimation	p. 389
Discretization of Estimators	p. 393
Transform Functions for Diffusions	p. 397
Estimation of Affine Diffusions	p. 404
Asymptotics of Estimating Functions	p. 409
Estimating Jump Diffusions	p. 413
Exercises	p. 417
Filtering	p. 419
Kalman-Bucy Filter	p. 419
Hidden Markov Chain Filters	p. 424
Filtering a Mean Reverting Process	p. 433
Balanced Method in Filtering	p. 447
A Benchmark Approach to Filtering in Finance	p. 456
Exercises	p. 475
Monte Carlo Simulation of SDEs	p. 477
Introduction to Monte Carlo Simulation	p. 477
Weak Taylor Schemes	p. 481
Derivative-Free Weak Approximations	p. 491
Extrapolation Methods	p. 495
Implicit and Predictor-Corrector Methods	p. 497
Exercises	p. 504
Regular Weak Taylor Approximations	p. 507
Weak Taylor Schemes	p. 507
Commutativity Conditions	p. 514
Convergence Results	p. 517
Exercises	p. 522
Jump-Adapted Weak Approximations	p. 523
Jump-Adapted Weak Schemes	p. 523
Derivative-Free Schemes	p. 529
Predictor-Corrector Schemes	p. 530
Some Jump-Adapted Exact Weak Schemes	p. 533
Convergence of Jump-Adapted Weak Taylor Schemes	p. 534
Convergence of Jump-Adapted Weak Schemes	p. 543
Numerical Results on Weak Schemes	p. 548
Exercises	p. 569
Numerical Stability	p. 571
Asymptotic p-Stability	p. 571
Stability of Predictor-Corrector Methods	p. 576
Stability of Some Implicit Methods	p. 583
Stability of Simplified Schemes	p. 586
Exercises	p. 590
Martingale Representations and Hedge Ratios	p. 591
General Contingent Claim Pricing	p. 591
Hedge Ratios for One-dimensional Processes	p. 595
Explicit Hedge Ratios	p. 601
Martingale Representation for Non-Smooth Payoffs	p. 606
Absolutely Continuous Payoff Functions	p. 616
Maximum of Several Assets	p. 621
Hedge Ratios for Lookback Options	p. 627
Exercises	p. 635
Variance Reduction Techniques	p. 637
Various Variance Reduction Methods	p. 637
Measure Transformation Techniques	p. 645
Discrete-Time Variance Reduced Estimators	p. 658
Control Variates	p. 669
HP Variance Reduction	p. 677
Exercises	p. 694
Trees and Markov Chain Approximations	p. 697
Numerical Effects of Tree Methods	p. 697
Efficiency of Simplified Schemes	p. 712
Higher Order Markov Chain Approximations	p. 720
Finite Difference Methods	p. 734
Convergence Theorem for Markov Chains	p. 744
Exercises	p. 753
Solutions for Exercises	p. 755
Acknowledgements	p. 781
Bibliographical Notes	p. 783
References	p. 793
Author Index	p. 835
Index	p. 847
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