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64Edited by B. Rozovskiı

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Advisory Board M. HairerI. KaratzasF.P. KellyA. KyprianouY. Le JanB. ØksendalG. PapanicolaouE. PardouxE. Perkins

For other titles in this series, go tohttp://www.springer.com/series/602

Eckhard Platen � Nicola Bruti-Liberati

Numerical Solutionof StochasticDifferential Equationswith Jumps in Finance

Eckhard PlatenNicola Bruti-Liberati (1975–2007)School of Finance and EconomicsDepartment of Mathematical SciencesUniversity of Technology, SydneyPO Box 123Broadway NSW 2007Australiaeckhard.platen@uts.edu.au

Managing EditorsBoris RozovskiıDivision of Applied MathematicsBrown University182 George StProvidence, RI 02912USArozovsky@dam.brown.edu

Geoffrey GrimmettCentre for Mathematical SciencesUniversity of CambridgeWilberforce RoadCambridge CB3 0WBUKg.r.grimmett@statslab.cam.ac.uk

ISSN 0172-4568ISBN 978-3-642-12057-2 e-ISBN 978-3-642-13694-8DOI 10.1007/978-3-642-13694-8Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010931518

Mathematics Subject Classification (2010): 60H10, 65C05, 62P05

© Springer-Verlag Berlin Heidelberg 2010This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations areliable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.

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Preface

This research monograph concerns the design and analysis of discrete-timeapproximations for stochastic differential equations (SDEs) driven by Wienerprocesses and Poisson processes or Poisson jump measures. In financial andactuarial modeling and other areas of application, such jump diffusions areoften used to describe the dynamics of various state variables. In finance thesemay represent, for instance, asset prices, credit ratings, stock indices, interestrates, exchange rates or commodity prices. The jump component can captureevent-driven uncertainties, such as corporate defaults, operational failures orinsured events. The book focuses on efficient and numerically stable strongand weak discrete-time approximations of solutions of SDEs. Strong approx-imations provide efficient tools for simulation problems such as those arisingin filtering, scenario analysis and hedge simulation. Weak approximations, onthe other hand, are useful for handling problems via Monte Carlo simulationsuch as the evaluation of moments, derivative pricing, and the computation ofrisk measures and expected utilities. The discrete-time approximations con-sidered are divided into regular and jump-adapted schemes. Regular schemesemploy time discretizations that do not include the jump times of the Poissonjump measure. Jump-adapted time discretizations, on the other hand, includethese jump times.

The first part of the book provides a theoretical basis for working withSDEs and stochastic processes with jumps motivated by applications in fi-nance. This part also introduces stochastic expansions for jump diffusions.It further proves powerful results on moment estimates of multiple stochas-tic integrals. The second part presents strong discrete-time approximationsof SDEs with given strong order of convergence, including derivative-free andpredictor-corrector schemes. The strong convergence of higher order schemesfor pure jump SDEs is established under conditions weaker than those requiredfor jump diffusions. Estimation and filtering methods are discussed. The thirdpart of the book introduces a range of weak approximations with jumps.These weak approximations include derivative-free, predictor-corrector, and

VI Preface

simplified schemes. The final part of the research monograph raises questionson numerical stability and discusses powerful martingale representations andvariance reduction techniques in the context of derivative pricing.

The book does not claim to be a complete account of the state of theart of the subject. Rather it attempts to provide a systematic framework foran understanding of the basic concepts and tools needed when implement-ing simulation methods for the numerical solution of SDEs. In doing so thebook aims to follow up on the presentation of the topic in Kloeden & Platen(1999) where no jumps were considered and no particular field of applica-tion motivated the numerical methods. The book goes significantly beyondKloeden & Platen (1999). It is covering many new results for the approxi-mation of continuous solutions of SDEs. The discrete time approximation ofSDEs with jumps represents the focus of the monograph. The reader learnsabout powerful numerical methods for the solution of SDEs with jumps. Theseneed to be implemented with care. It is directed at readers from different fieldsand backgrounds.

The area of finance has been chosen to motivate the methods. It has beenalso a focus of research by the first author for many years that culminatedin the development of the benchmark approach, see Platen & Heath (2006),which provides a general framework for modeling risk in finance, insurance andother areas and may be new to most readers. The book is written at a levelthat is appropriate for a reader with an engineer’s or similar undergraduatetraining in mathematical methods. It is readily accessible to many who onlyrequire numerical recipes.

Together with Nicola Bruti-Liberati we had for several years planned abook to follow on the book with Peter Kloeden on the “Numerical Solution ofStochastic Differential Equations”, which first appeared in 1992 at SpringerVerlag and helped to develop the theory and practice of this field. Nicola’sPhD thesis was written to provide proofs for parts of such a book. It is verysad that Nicola died tragically in a traffic accident on 28 August 2007. Thiswas an enormous loss for his family and friends, his colleagues and the areaof quantitative methods in finance.

The writing of such a book was not yet started at the time of Nicola’stragic death. I wish to express my deep gratitude to Katrin Platen, whothen agreed to typeset an even more comprehensive book than was originallyenvisaged. She carefully and patiently wrote and revised several versions ofthe manuscript under difficult circumstances. The book now contains not onlyresults that we obtained with Nicola on the numerical solution of SDEs withjumps, but also presents methods for exact simulation, parameter estimation,filtering and efficient variance reduction, as well as the simulation of hedgeratios and the construction of martingale representations.

I would like to thank several colleagues for their collaboration in relatedresearch and valuable suggestions on the manuscript, including Kevin Bur-rage, Leunglung Chan, Kristoffer Glover, David Heath, Des Higham, HardyHulley, Constantinos Kardaras, Peter Kloeden, Uwe Kuchler, Herman Lukito,

Preface VII

Remigius Mikulevicius, Renata Rendek, Wolfgang Runggaldier, Lei Shi andAnthony Tooman. Particular thanks go to Rob Lynch, the former Dean ofthe Faculty of Business at the University of Technology Sydney, who madethe writing of the book possible through his direct support. Finally, I like tothank the Editor, Catriona Byrne, at Springer for her excellent work and herencouragement to write this book as a sequel of the previous book with PeterKloeden.

It is greatly appreciated if readers could forward any errors, misprintsor suggested improvements to: eckhard.platen@uts.edu.au. The interestedreader is likely to find updated information about the numerical solutionof stochastic differential equations on the webpage of the first author under“Numerical Methods”:

http://www.business.uts.edu.au/finance/staff/Eckhard/Numerical Methods.html

Sydney, January 2010 Eckhard Platen

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Suggestions for the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV

Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX

Motivation and Brief Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII

1 Stochastic Differential Equations with Jumps . . . . . . . . . . . . . . 11.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supermartingales and Martingales . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Quadratic Variation and Covariation . . . . . . . . . . . . . . . . . . . . . . . 231.4 Ito Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5 Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.6 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 381.7 Linear SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.8 SDEs with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.9 Existence and Uniqueness of Solutions of SDEs . . . . . . . . . . . . . . 571.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2 Exact Simulation of Solutions of SDEs . . . . . . . . . . . . . . . . . . . . . 612.1 Motivation of Exact Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2 Sampling from Transition Distributions . . . . . . . . . . . . . . . . . . . . 632.3 Exact Solutions of Multi-dimensional SDEs . . . . . . . . . . . . . . . . . 782.4 Functions of Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.5 Almost Exact Solutions by Conditioning. . . . . . . . . . . . . . . . . . . . 1052.6 Almost Exact Simulation by Time Change . . . . . . . . . . . . . . . . . . 1132.7 Functionals of Solutions of SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

X Contents

3 Benchmark Approach to Finance and Insurance . . . . . . . . . . . 1393.1 Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.2 Best Performing Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.3 Supermartingale Property and Pricing . . . . . . . . . . . . . . . . . . . . . 1453.4 Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.5 Real World Pricing Under Some Models . . . . . . . . . . . . . . . . . . . . 1583.6 Real World Pricing Under the MMM . . . . . . . . . . . . . . . . . . . . . . 1683.7 Binomial Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1763.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4 Stochastic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.1 Introduction to Wagner-Platen Expansions . . . . . . . . . . . . . . . . . 1874.2 Multiple Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.3 Coefficient Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024.4 Wagner-Platen Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.5 Moments of Multiple Stochastic Integrals . . . . . . . . . . . . . . . . . . . 2114.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5 Introduction to Scenario Simulation . . . . . . . . . . . . . . . . . . . . . . . 2335.1 Approximating Solutions of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 2335.2 Scenario Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2455.3 Strong Taylor Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.4 Derivative-Free Strong Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2665.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

6 Regular Strong Taylor Approximations with Jumps . . . . . . . . 2736.1 Discrete-Time Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2736.2 Strong Order 1.0 Taylor Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2786.3 Commutativity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.4 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2896.5 Lemma on Multiple Ito Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 2926.6 Proof of the Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . 3026.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

7 Regular Strong Ito Approximations . . . . . . . . . . . . . . . . . . . . . . . . 3097.1 Explicit Regular Strong Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 3097.2 Drift-Implicit Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3167.3 Balanced Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3217.4 Predictor-Corrector Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3267.5 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3317.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Contents XI

8 Jump-Adapted Strong Approximations . . . . . . . . . . . . . . . . . . . . 3478.1 Introduction to Jump-Adapted Approximations . . . . . . . . . . . . . 3478.2 Jump-Adapted Strong Taylor Schemes . . . . . . . . . . . . . . . . . . . . . 3508.3 Jump-Adapted Derivative-Free Strong Schemes . . . . . . . . . . . . . . 3558.4 Jump-Adapted Drift-Implicit Schemes . . . . . . . . . . . . . . . . . . . . . . 3568.5 Predictor-Corrector Strong Schemes . . . . . . . . . . . . . . . . . . . . . . . 3598.6 Jump-Adapted Exact Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3618.7 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.8 Numerical Results on Strong Schemes . . . . . . . . . . . . . . . . . . . . . . 3688.9 Approximation of Pure Jump Processes . . . . . . . . . . . . . . . . . . . . 3758.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

9 Estimating Discretely Observed Diffusions . . . . . . . . . . . . . . . . . 3899.1 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 3899.2 Discretization of Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3939.3 Transform Functions for Diffusions . . . . . . . . . . . . . . . . . . . . . . . . 3979.4 Estimation of Affine Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4049.5 Asymptotics of Estimating Functions . . . . . . . . . . . . . . . . . . . . . . 4099.6 Estimating Jump Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4139.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

10 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41910.1 Kalman-Bucy Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41910.2 Hidden Markov Chain Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42410.3 Filtering a Mean Reverting Process . . . . . . . . . . . . . . . . . . . . . . . . 43310.4 Balanced Method in Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44710.5 A Benchmark Approach to Filtering in Finance . . . . . . . . . . . . . 45610.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

11 Monte Carlo Simulation of SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 47711.1 Introduction to Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . 47711.2 Weak Taylor Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48111.3 Derivative-Free Weak Approximations . . . . . . . . . . . . . . . . . . . . . . 49111.4 Extrapolation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49511.5 Implicit and Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . 49711.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

12 Regular Weak Taylor Approximations . . . . . . . . . . . . . . . . . . . . . 50712.1 Weak Taylor Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50712.2 Commutativity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51412.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51712.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

XII Contents

13 Jump-Adapted Weak Approximations . . . . . . . . . . . . . . . . . . . . . 52313.1 Jump-Adapted Weak Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52313.2 Derivative-Free Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52913.3 Predictor-Corrector Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53013.4 Some Jump-Adapted Exact Weak Schemes . . . . . . . . . . . . . . . . . 53313.5 Convergence of Jump-Adapted Weak Taylor Schemes . . . . . . . . 53413.6 Convergence of Jump-Adapted Weak Schemes . . . . . . . . . . . . . . . 54313.7 Numerical Results on Weak Schemes . . . . . . . . . . . . . . . . . . . . . . . 54813.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

14 Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57114.1 Asymptotic p-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57114.2 Stability of Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . 57614.3 Stability of Some Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . 58314.4 Stability of Simplified Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58614.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

15 Martingale Representations and Hedge Ratios . . . . . . . . . . . . . 59115.1 General Contingent Claim Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 59115.2 Hedge Ratios for One-dimensional Processes . . . . . . . . . . . . . . . . 59515.3 Explicit Hedge Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60115.4 Martingale Representation for Non-Smooth Payoffs . . . . . . . . . . 60615.5 Absolutely Continuous Payoff Functions . . . . . . . . . . . . . . . . . . . . 61615.6 Maximum of Several Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62115.7 Hedge Ratios for Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . 62715.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

16 Variance Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63716.1 Various Variance Reduction Methods . . . . . . . . . . . . . . . . . . . . . . 63716.2 Measure Transformation Techniques . . . . . . . . . . . . . . . . . . . . . . . 64516.3 Discrete-Time Variance Reduced Estimators . . . . . . . . . . . . . . . . 65816.4 Control Variates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66916.5 HP Variance Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67716.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

17 Trees and Markov Chain Approximations . . . . . . . . . . . . . . . . . . 69717.1 Numerical Effects of Tree Methods . . . . . . . . . . . . . . . . . . . . . . . . 69717.2 Efficiency of Simplified Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 71217.3 Higher Order Markov Chain Approximations . . . . . . . . . . . . . . . . 72017.4 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73417.5 Convergence Theorem for Markov Chains . . . . . . . . . . . . . . . . . . . 74417.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

18 Solutions for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

Contents XIII

Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

Suggestions for the Reader

It has been mentioned in the Preface that the material of this book has beenarranged in a way that should make it accessible to as wide a readership aspossible. Prospective readers will have different backgrounds and objectives.The following four groups are suggestions to help use the book more efficiently.

(i) Let us begin with those readers who aim for a sufficient understanding,to be able to apply stochastic differential equations with jumps and ap-propriate simulation methods in their field of application, which may notbe finance. Deeper mathematical issues are avoided in the following sug-gested sequence of reading, which provides a guide to the book for thosewithout a strong mathematical background:

§1.1 → §1.2 → §1.3 → §1.4 → §1.5 → §1.6 → §1.7 → §1.8

↓§5.1 → §5.2 → §5.3 → §5.4

↓§6.1 → §6.2 → §6.3

↓§7.1 → §7.2 → §7.3 → §7.4

↓§8.1 → §8.2 → §8.3 → §8.4 → §8.5 → §8.6 → §8.8

↓§11.1 → §11.2 → §11.3 → §11.4 → §11.5

↓§12.1 → §12.2

↓§13.1 → §13.2 → §13.3 → §13.4 → §13.7

↓Chapter 14

↓§16.1

XVI Suggestions for the Reader

(ii) Engineers, quantitative analysts and others with a more technical back-ground in mathematical and quantitative methods who are interested inapplying stochastic differential equations with jumps, and in implement-ing efficient simulation methods or developing new schemes could use thebook according to the following suggested flowchart. Without too muchemphasis on proofs the selected material provides the underlying mathe-matics.

Chapter 1

↓Chapter 2

↓Chapter 4

↓Chapter 5

↓§6.1 → §6.2 → §6.3

↓§7.1 → §7.2 → §7.3 → §7.4

↓§8.1 → §8.2 → §8.3 → §8.4 → §8.5 → §8.6 → §8.8 → §8.9

↓Chapter 11

↓§12.1 → §12.2

↓§13.1 → §13.2 → §13.3 → §13.4 → §13.7

↓Chapter 14

↓Chapter 16

Suggestions for the Reader XVII

(iii) Readers with strong mathematical background and mathematicians mayomit the introductory Chap.1. The following flowchart focuses on the the-oretical aspects of the numerical approximation of solutions of stochasticdifferential equations with jumps while avoiding well-known or appliedtopics.

Chapter 2

↓Chapter 4

↓Chapter 5

↓Chapter 6

↓Chapter 7

↓Chapter 8

↓Chapter 11

↓Chapter 12

↓Chapter 13

↓Chapter 14

↓Chapter 16

XVIII Suggestions for the Reader

(iv) Financial engineers, quantitative analysts, risk managers, fund managers,insurance professionals and others who have no strong mathematical back-ground and are interested in finance, insurance and other areas of riskmanagement will find the following flowchart helpful. It suggests the read-ing for an introduction into quantitative methods in finance and relatedareas.

§1.1 → §1.2 → §1.3 → §1.4 → §1.5 → §1.6 → §1.7 → §1.8

↓Chapter 2

↓Chapter 3

↓Chapter 5

↓§6.1 → §6.2 → §6.3

↓§7.1 → §7.2 → §7.3 → §7.4

↓§8.1 → §8.2 → §8.3 → §8.4 → §8.5 → §8.6 → §8.8

↓Chapter 9

↓Chapter 10

↓Chapter 11

↓§12.1 → §12.2

↓§13.1 → §13.2 → §13.3 → §13.4 → §13.7

↓Chapter 14

↓Chapter 15

↓Chapter 16

↓Chapter 17

Basic Notation

μX mean of X

σ2X , Var(X) variance of X

Cov(X, Y ) covariance of X and Y

inf{·} greatest lower bound

sup{·} smallest upper bound

max(a, b) = a ∨ b maximum of a and b

min(a, b) = a ∧ b minimum of a and b

(a)+ = max(a, 0) maximum of a and 0

x� transpose of a vector or matrix x

x = (x1, x2, . . . , xd)� column vector x ∈ �d with ith component xi

|x| absolute value of x or Euclidean norm

A = [ai,j ]k,di,j=1 (k × d)-matrix A with ijth component ai,j

det (A) determinant of a matrix A

A−1 inverse of a matrix A

(x,y) inner product of vectors x and y

N = {1, 2, . . .} set of natural numbers

XX Basic Notation

∞ infinity

(a, b) open interval a < x < b in �

[a, b] closed interval a ≤ x ≤ b in �

� = (−∞,∞) set of real numbers

�+ = [0,∞) set of nonnegative real numbers

�d d-dimensional Euclidean space

Ω sample space

∅ empty set

A ∪ B the union of sets A and B

A ∩ B the intersection of sets A and B

A\B the set A without the elements of B

E = �\{0} � without origin

[X, Y ]t covariation of processes X and Y at time t

[X]t quadratic variation of process X at time t

n! = 1 · 2 · . . . · n factorial of n

[a] largest integer not exceeding a ∈ �

i.i.d. independent identically distributed

a.s. almost surely

f ′ first derivative of f : � → �

f ′′ second derivative of f : � → �

f : Q1 → Q2 function f from Q1 into Q2

∂u∂xi ith partial derivative of u : �d → �(

∂∂xi

)ku kth order partial derivative of u with respect to xi

∃ there exists

FX(·) distribution function of X

fX(·) probability density function of X

φX(·) characteristic function of X

1A indicator function for event A to be true

Basic Notation XXI

N(·) Gaussian distribution function

Γ (·) gamma function

Γ (·; ·) incomplete gamma function

(mod c) modulo c

A collection of events, sigma-algebra

A filtration

E(X) expectation of X

E(X | A) conditional expectation of X under A

P (A) probability of A

P (A |B) probability of A conditioned on B

∈ element of

∈ not element of

= not equal

≈ approximately equal

a � b a is significantly smaller than b

limN→∞ limit as N tends to infinity

lim infN→∞ lower limit as N tends to infinity

lim supN→∞ upper limit as N tends to infinity

ı square root of −1, imaginary unit

δ(·) Dirac delta function at zero

I unit matrix

sgn(x) sign of x ∈ �

L2T space of square integrable, progressively measurable

functions on [0, T ] × Ω

B(U) smallest sigma-algebra on U

ln(a) natural logarithm of a

MM Merton model

MMM minimal market model

XXII Basic Notation

GIG generalized inverse Gaussian

GH generalized hyperbolic

VG variance gamma

GOP growth optimal portfolio

EWI equi-value weighted index

ODE ordinary differential equation

SDE stochastic differential equation

PDE partial differential equation

PIDE partial integro differential equation

Iν(·) modified Bessel function of the first kind with indexν

Kλ(·) modified Bessel function of the third kind with indexλ

Δ time step size of a time discretization(il

)= i!

l!(i−l)! combinatorial coefficient

Ck(Rd,R) set of k times continuously differentiable functions

CkP (Rd,R) set of k times continuously differentiable functions

which, together with their partial derivatives of orderup to k, have at most polynomial growth

Letters such as K, K1, . . ., K, C, C1, . . ., C, . . . represent finite positive realconstants that can vary from line to line. All these constants are assumed tobe independent of the time step size Δ.

Motivation and Brief Survey

Key features of advanced models in many areas of application with uncer-tainties are often event-driven. In finance and insurance one has to dealwith events such as corporate defaults, operational failures or insured acci-dents. By analyzing time series of historical data, such as prices and otherfinancial quantities, many authors have argued in the area of finance forthe presence of jumps, see Jorion (1988) and Ait-Sahalia (2004) for foreignexchange and stock markets, and Johannes (2004) for short-term interestrates. Jumps are also used to generate the short-term smile effect observedin implied volatilities of option prices, see Cont & Tankov (2004). Further-more, jumps are needed to properly model credit events like defaults andcredit rating changes, see for instance Jarrow, Lando & Turnbull (1997). Theshort rate, typically set by a central bank, jumps up or down, usually bysome quarters of a percent, see Babbs & Webber (1995). Models for the dy-namics of financial quantities specified by stochastic differential equations(SDEs) with jumps have become increasingly popular. Models of this kindcan be found, for instance, in Merton (1976), Bjork, Kabanov & Runggaldier(1997), Duffie, Pan & Singleton (2000), Kou (2002), Schonbucher (2003),Glasserman & Kou (2003), Cont & Tankov (2004) and Geman & Roncoroni(2006). The areas of application of SDEs with jumps go far beyond fi-nance. Other areas of application include economics, insurance, popula-tion dynamics, epidemiology, structural mechanics, physics, chemistry andbiotechnology. In chemistry, for instance, the reactions of single moleculesor coupled reactions yield stochastic models with jumps, see, for instance,Turner, Schnell & Burrage (2004), to indicate just one such application.

Since only a small class of jump diffusion SDEs admits explicit solutions,it is important to construct discrete-time approximations. The focus of thismonograph is the numerical solution of SDEs with jumps via simulation. Weconsider pathwise scenario simulation, for which strong schemes are used, andMonte Carlo simulation, for which weak schemes are employed. Of course,there exist various alternative methods to Monte Carlo simulation that we onlyconsider peripherally in this book when it is related to the idea of discrete-time

XXIV Motivation and Brief Survey

numerical approximations. These methods include Markov chain approxima-tions, tree-based, and finite difference methods. The class of SDEs consideredhere are those driven by Wiener processes and Poisson random measures.Some authors consider the smaller class of SDEs driven by Wiener processesand homogeneous Poisson processes, while other authors analyze the largerclass of SDEs driven by fairly general semimartingales. The class of SDEsdriven by Wiener processes and Poisson jump measures with finite intensityappears to be large enough for realistic modeling of the dynamics of quantitiesin finance. Here continuous trading noise and a few single events model thetypical sources of uncertainty. Furthermore, stochastic jump sizes and stochas-tic intensities, can be conveniently covered by using a Poisson jump measure.As we will explain, there are some numerical and theoretical advantages whenmodeling jumps with predescribed size. The simulation of some Levy processdriven dynamics will also be discussed. The development of a rich theory onsimulation methods for SDEs with jumps, similar to that established for purediffusion SDEs in Kloeden & Platen (1992), is still under way. This book aimsto contribute to this theory motivated by applications in the area of finance.However, challenging problems in insurance, biology, chemistry, physics andmany other areas can readily apply the presented numerical methods.

We consider discrete-time approximations of solutions of SDEs constructedon time discretizations (t)Δ, with maximum step size Δ ∈ (0, Δ0), withΔ0 ∈ (0, 1). We call a time discretization regular if the jump times, gen-erated by the Poisson measure, are not discretization times. On the otherhand, if the jump times are included in the time discretization, then a jump-adapted time discretization is obtained. Accordingly, discrete-time approxima-tions constructed on regular time discretizations are called regular schemes,while approximations constructed on jump-adapted time discretizations arecalled jump-adapted schemes.

Discrete-time approximations can be divided into two major classes: strongapproximations and weak approximations, see Kloeden & Platen (1999). Wesay that a discrete-time approximation YΔ, constructed on a time discretiza-tion (t)Δ, with maximum step size Δ > 0, converges with strong order γ attime T to the solution X of a given SDE, if there exists a positive constantC, independent of Δ, and a finite number Δ0 ∈ (0, 1), such that

E(|XT − YΔT |) ≤ CΔγ , (0.0.1)

for all Δ ∈ (0, Δ0). From the definition of the strong error on the left hand sideof (0.0.1) one notices that strong schemes provide pathwise approximations ofthe original solution X of the given SDE. These methods are therefore suitablefor problems such as filtering, scenario simulation and hedge simulation, aswell as the testing of statistical and other quantitative methods. In insurance,the area of dynamic financial analysis is well suited for applications of strongapproximations.

Motivation and Brief Survey XXV

On the other hand, we say that a discrete-time approximation YΔ con-verges weakly with order β to X at time T , if for each g ∈ C2(β+1)

P (�d,�)there exists a positive constant C, independent of Δ, and a finite number,Δ0 ∈ (0, 1), such that

|E(g(XT )) − E(g(YΔT ))| ≤ CΔβ , (0.0.2)

for each Δ ∈ (0, Δ0). Here C2(β+1)P (�d,�) denotes the set of 2(β + 1) con-

tinuously differentiable functions which, together with their partial deriva-tives of order up to 2(β + 1), have polynomial growth. This means that forg ∈ C2(β+1)

P (�d,�) there exist constants K > 0 and r ∈ N , possibly dependingon g, such that

|∂jyg(y)| ≤ K(1 + |y|2r), (0.0.3)

for all y ∈ �d and any partial derivative ∂jyig(y) of order j ≤ 2(β + 1). Weak

schemes provide approximations of the probability measure generated by thesolution of a given SDE. These schemes are appropriate for problems suchas derivative pricing, the evaluation of moments and the computation of riskmeasures and expected utilities.

Let us briefly discuss some relationships between strong and weak approx-imations. Let YΔ be a discrete-time approximation, constructed on a timediscretization (t)Δ, with strong order of convergence γ, see (0.0.1). Considera function g : �d → � satisfying the Lipschitz condition

|g(x) − g(y)| ≤ K|x − y|, (0.0.4)

for every x,y ∈ �d, where K is a positive constant. Then there exists apositive constant C, independent of Δ, and a finite number, Δ0 ∈ (0, 1), suchthat by the Lipschitz condition (0.0.4) and the strong order γ we have

∣∣∣E (g(XT )) − E

(g

(YΔ

T

))∣∣∣ ≤ K E

(∣∣∣XT − YΔ

T

∣∣∣)≤ Δγ (0.0.5)

for each Δ ∈ (0, Δ0).Since the set of Lipschitz continuous functions includes the set C2(β+1)

P (�d,

�), the above result implies that if a discrete-time approximation YΔ achievesan order γ of strong convergence, then it also achieves at least an order β = γof weak convergence. We emphasize that the weak order obtained above isusually not sharp and, thus, the order of weak convergence could actually behigher than that of strong convergence. For instance, it is well-known andwill later be shown that the Euler scheme typically achieves only strong orderγ = 0.5 but weak order β = 1.0.

In the light of the estimate (0.0.5), one could think that the design ofstrong approximations is sufficient for any type of application, since these ap-proximations can be also applied to weak problems. This is in principle true,but the resulting schemes might remain far from being optimal in terms of

XXVI Motivation and Brief Survey

computational efficiency. Let us consider as an example the strong Milsteinscheme for pure diffusion SDEs, see Milstein (1974). By adding the doubleWiener integrals to the Euler scheme one obtains the Milstein scheme, thusenhancing the order of strong convergence from γ = 0.5 to γ = 1.0. Nonethe-less, the order of weak convergence of the Milstein scheme equals β = 1.0,which is not an improvement over the order of weak convergence of the Eu-ler scheme. Therefore, to price a European call option, for example, the Eulerscheme is often computationally more efficient than the Milstein scheme, sinceit has fewer terms and the same order of weak convergence. Furthermore, thenumerical stability of the Milstein scheme can be worse than that of the Eu-ler scheme. This simple example indicates that to construct efficient higherorder weak approximations, one should not take the naive approach of justusing higher order strong approximations. Furthermore, as will be discussed,when designing weak schemes one has the freedom of using simple multi-pointdistributed random variables to approximate the underlying multiple stochas-tic integrals. These multi-point distributed random variables lead to highlyefficient implementations of weak schemes.

For the approximation of the expected value of a function g of the solu-tion XT at a final time T , there exist alternative numerical methods. Undersuitable conditions, the pricing function u(x, t) = E(g(XT )|Xt = x) canbe expressed as a solution of a partial integro differential equation (PIDE).Therefore, an approximation of the pricing function u(x, t) can be obtained bysolving the corresponding PIDE via finite difference or finite element methods,see, for instance, D’Halluin, Forsyth & Vetzal (2005) and Cont & Voltchkova(2005). These methods are computationally efficient when we have a low di-mensional underlying factor process X. Moreover, it is easy to incorporateearly exercise features, as those arising in the pricing of Bermudan and Amer-ican options. However, when the underlying stochastic process X has dimen-sion higher than two or three, finite difference and finite element methodsbecome difficult to be implemented and turn out to be computationally, pro-hibitively expensive.

Monte Carlo simulation is well suited to tackle high dimensional problems.It has the great advantage that its computational complexity increases, inprinciple, polynomially with the dimension of the problem. Consequently, thecurse of dimensionality applies in a milder fashion to Monte Carlo simulationthan it does to most other numerical methods. Additionally, Monte Carlosimulation is well suited to parallel hardware devices and seems to providesolutions where no alternative is known.

The focus of this book is on the numerical solution of stochastic differentialequations (SDEs) with jumps via simulation methods, motivated by problemsin finance. The monograph is divided into three parts. The first part, cover-ing Chaps. 1 up to 4, introduces SDEs with jumps, presents exact simulationmethods, describes the benchmark approach as a general financial modelingframework and introduces Wagner-Platen expansions. The second part, com-prising Chaps. 5 up to 10, considers strong approximations of jump diffusion

Motivation and Brief Survey XXVII

and pure jump SDEs. It also includes some discussions on parameter estima-tion and filtering as well as their relation to strong approximation methods.Finally, the third part, which is composed of Chaps. 11 up to 17, introducesweak approximations for Monte Carlo simulation and discusses efficient im-plementations of weak schemes and numerical stability. Here the simulationof hedge ratios, efficient variance reduction techniques, Markov chain approx-imations and finite difference methods are discussed in the context of weakapproximation.

The monograph Kloeden & Platen (1992) and its printings in (1995) and(1999) aimed to give a reasonable overview on the literature on the numericalsolution of SDEs via simulation methods. Over the last two decades the fieldhas grown so rapidly that it is no longer possible to provide a reasonably fairpresentation of the area. This book is, therefore, simply presenting results thatthe authors were in some form involved with. There are several other lines ofresearch that may be of considerable value to those who have an interest inthis field. We apologize to those who would have expected other interestingtopics to be covered by the book. Unfortunately, this was not possible due tolimitations of space.

For further reading also in areas that are related but could not be cov-ered we may refer the reader to various well-written books, including Ikeda& Watanabe (1989), Niederreiter (1992), Elliott, Aggoun & Moore (1995),Milstein (1995a), Embrechts, Kluppelberg & Mikosch (1997), Bjork (1998),Karatzas & Shreve (1998), Mikosch (1998), Kloeden & Platen (1999), Shiryaev(1999), Bielecki & Rutkowski (2002), Borodin & Salminen (2002), Jackel(2002), Joshi (2003), Schonbucher (2003), Shreve (2003a, 2003b), Cont &Tankov (2004), Glasserman (2004), Higham (2004), Milstein & Tretjakov(2004), Achdou & Pironneau (2005), Brigo & Mercurio (2005), Elliott & Kopp(2005), Klebaner (2005), McLeish (2005), McNeil, Frey & Embrechts (2005),Musiela & Rutkowski (2005), Øksendal & Sulem (2005), Protter (2005), Chan& Wong (2006), Delbaen & Schachermayer (2006), Elliott & van der Hoek(2006), Malliavin & Thalmaier (2006), Platen & Heath (2006), Seydel (2006),Asmussen & Glynn (2007), Lamberton & Lapeyre (2007) and Jeanblanc, Yor& Chesney (2009).

The book has been used as reference for the Masters in Quantitative Fi-nance and the PhD program at the University of Technology in Sydney, aswell as for courses and workshops that the first author has presented in variousplaces.

The formulas in the book are numbered according to the chapter andsection where they appear. Assumptions, theorems, lemmas, definitions andcorollaries are numbered sequentially in each section. The most common no-tations are listed at the beginning and an Index of Keywords is given at theend of the book. Some readers may find the Author Index at the end of thebook useful. Each chapter finishes with some Exercises with Solutions givenin Chap. 18. These are aimed to support the study of the material.

XXVIII Motivation and Brief Survey

We conclude this brief survey with the remark that the practical applica-tion and theoretical understanding of numerical methods for stochastic differ-ential equations with jumps are still under development. This book shall stim-ulate interest and further work on such methods. The Bibliographical Notesat the end of this research monograph may be of assistance.