Numerical Solution of Stochastic Differential Equations with Jumps in Finance docx

This research monograph concerns the design and analysis of discrete-timeapproximations for stochastic differential equations SDEs driven by Wienerprocesses and Poisson processes or Poiss

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Eckhard Platen r Nicola Bruti-Liberati

Numerical Solution

of Stochastic

Differential Equations with Jumps in Finance

Eckhard Platen

Nicola Bruti-Liberati (1975–2007)

School of Finance and Economics

Department of Mathematical Sciences

University of Technology, Sydney

g.r.grimmett@statslab.cam.ac.uk

ISSN 0172-4568

ISBN 978-3-642-12057-2 e-ISBN 978-3-642-13694-8

DOI 10.1007/978-3-642-13694-8

Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010931518

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

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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 arising

in 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 tic integrals The second part presents strong discrete-time approximations

stochas-of SDEs with given strong order stochas-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 questions

on 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 for

an understanding of the basic concepts and tools needed when 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

implement-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 culminated

in 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 area

of 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 K¨uchler, 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, misprints

or suggested improvements to: eckhard.platen@uts.edu.au The interestedreader is likely to find updated information about the numerical solution

of stochastic differential equations on the webpage of the first author under

Preface V

Suggestions for the Reader XV

Basic Notation XIX Motivation and Brief Survey XXIII

1 Stochastic Differential Equations with Jumps 1

1.1 Stochastic Processes 1

1.2 Supermartingales and Martingales 16

1.3 Quadratic Variation and Covariation 23

1.4 Itˆo Integral 26

1.5 Itˆo Formula 34

1.6 Stochastic Differential Equations 38

1.7 Linear SDEs 45

1.8 SDEs with Jumps 53

1.9 Existence and Uniqueness of Solutions of SDEs 57

1.10 Exercises 59

2 Exact Simulation of Solutions of SDEs 61

2.1 Motivation of Exact Simulation 61

2.2 Sampling from Transition Distributions 63

2.3 Exact Solutions of Multi-dimensional SDEs 78

2.4 Functions of Exact Solutions 99

2.5 Almost Exact Solutions by Conditioning 105

2.6 Almost Exact Simulation by Time Change 113

2.7 Functionals of Solutions of SDEs 123

2.8 Exercises 136

X Contents

3 Benchmark Approach to Finance and Insurance 139

3.1 Market Model 139

3.2 Best Performing Portfolio 142

3.3 Supermartingale Property and Pricing 145

3.4 Diversification 149

3.5 Real World Pricing Under Some Models 158

3.6 Real World Pricing Under the MMM 168

3.7 Binomial Option Pricing 176

3.8 Exercises 185

4 Stochastic Expansions 187

4.1 Introduction to Wagner-Platen Expansions 187

4.2 Multiple Stochastic Integrals 195

4.3 Coefficient Functions 202

4.4 Wagner-Platen Expansions 206

4.5 Moments of Multiple Stochastic Integrals 211

4.6 Exercises 230

5 Introduction to Scenario Simulation 233

5.1 Approximating Solutions of ODEs 233

5.2 Scenario Simulation 245

5.3 Strong Taylor Schemes 252

5.4 Derivative-Free Strong Schemes 266

5.5 Exercises 271

6 Regular Strong Taylor Approximations with Jumps 273

6.1 Discrete-Time Approximation 273

6.2 Strong Order 1.0 Taylor Scheme 278

6.3 Commutativity Conditions 286

6.4 Convergence Results 289

6.5 Lemma on Multiple Itˆo Integrals 292

6.6 Proof of the Convergence Theorem 302

6.7 Exercises 307

7 Regular Strong Itˆ o Approximations 309

7.1 Explicit Regular Strong Schemes 309

7.2 Drift-Implicit Schemes 316

7.3 Balanced Implicit Methods 321

7.4 Predictor-Corrector Schemes 326

7.5 Convergence Results 331

7.6 Exercises 346

Contents XI

8 Jump-Adapted Strong Approximations 347

8.1 Introduction to Jump-Adapted Approximations 347

8.2 Jump-Adapted Strong Taylor Schemes 350

8.3 Jump-Adapted Derivative-Free Strong Schemes 355

8.4 Jump-Adapted Drift-Implicit Schemes 356

8.5 Predictor-Corrector Strong Schemes 359

8.6 Jump-Adapted Exact Simulation 361

8.7 Convergence Results 362

8.8 Numerical Results on Strong Schemes 368

8.9 Approximation of Pure Jump Processes 375

8.10 Exercises 388

9 Estimating Discretely Observed Diffusions 389

9.1 Maximum Likelihood Estimation 389

9.2 Discretization of Estimators 393

9.3 Transform Functions for Diffusions 397

9.4 Estimation of Affine Diffusions 404

9.5 Asymptotics of Estimating Functions 409

9.6 Estimating Jump Diffusions 413

9.7 Exercises 417

10 Filtering 419

10.1 Kalman-Bucy Filter 419

10.2 Hidden Markov Chain Filters 424

10.3 Filtering a Mean Reverting Process 433

10.4 Balanced Method in Filtering 447

10.5 A Benchmark Approach to Filtering in Finance 456

10.6 Exercises 475

11 Monte Carlo Simulation of SDEs 477

11.1 Introduction to Monte Carlo Simulation 477

11.2 Weak Taylor Schemes 481

11.3 Derivative-Free Weak Approximations 491

11.4 Extrapolation Methods 495

11.5 Implicit and Predictor-Corrector Methods 497

11.6 Exercises 504

12 Regular Weak Taylor Approximations 507

12.1 Weak Taylor Schemes 507

12.2 Commutativity Conditions 514

12.3 Convergence Results 517

12.4 Exercises 522

XII Contents

13 Jump-Adapted Weak Approximations 523

13.1 Jump-Adapted Weak Schemes 523

13.2 Derivative-Free Schemes 529

13.3 Predictor-Corrector Schemes 530

13.4 Some Jump-Adapted Exact Weak Schemes 533

13.5 Convergence of Jump-Adapted Weak Taylor Schemes 534

13.6 Convergence of Jump-Adapted Weak Schemes 543

13.7 Numerical Results on Weak Schemes 548

13.8 Exercises 569

14 Numerical Stability 571

14.1 Asymptotic p-Stability 571

14.2 Stability of Predictor-Corrector Methods 576

14.3 Stability of Some Implicit Methods 583

14.4 Stability of Simplified Schemes 586

14.5 Exercises 590

15 Martingale Representations and Hedge Ratios 591

15.1 General Contingent Claim Pricing 591

15.2 Hedge Ratios for One-dimensional Processes 595

15.3 Explicit Hedge Ratios 601

15.4 Martingale Representation for Non-Smooth Payoffs 606

15.5 Absolutely Continuous Payoff Functions 616

15.6 Maximum of Several Assets 621

15.7 Hedge Ratios for Lookback Options 627

15.8 Exercises 635

16 Variance Reduction Techniques 637

16.1 Various Variance Reduction Methods 637

16.2 Measure Transformation Techniques 645

16.3 Discrete-Time Variance Reduced Estimators 658

16.4 Control Variates 669

16.5 HP Variance Reduction 677

16.6 Exercises 694

17 Trees and Markov Chain Approximations 697

17.1 Numerical Effects of Tree Methods 697

17.2 Efficiency of Simplified Schemes 712

17.3 Higher Order Markov Chain Approximations 720

17.4 Finite Difference Methods 734

17.5 Convergence Theorem for Markov Chains 744

17.6 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 propriate simulation methods in their field of application, which may not

ap-be finance Deeper mathematical issues are avoided in the following gested sequence of reading, which provides a guide to the book for thosewithout a strong mathematical background:

XVI Suggestions for the Reader

(ii) Engineers, quantitative analysts and others with a more technical 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

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

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

Basic Notation

σ2

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 = (x1, x2, , x d) column vector x ∈  d with ith component x i

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

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

XX Basic Notation

(a, b) open interval a < x < b in 

 = (−∞, ∞) set of real numbers

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

E = \{0}  without origin

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

n! = 1 · 2 · · n factorial of n

Basic Notation XXI

a  b a is significantly smaller than b

lim infN →∞ lower limit as N tends to infinity

lim supN →∞ upper limit as N tends to infinity

XXII Basic Notation

I ν( ·) modified Bessel function of the first kind with index

l!(i −l)! combinatorial coefficient

C k(R d , R) set of k times continuously differentiable functions

C k

P(R d , R) set of k times continuously differentiable functions

which, together with their partial derivatives of order

up to k, have at most polynomial growth Letters such as K, K1, , ˜ K, C, C1, , ˜ C, represent finite positive real

constants that can vary from line to line All these constants are assumed to

be independent of the time step size Δ.

Motivation and Brief Survey

Key features of advanced models in many areas of application with 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 observed

uncer-in implied volatilities of option prices, see Cont & Tankov (2004) 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), Bj¨ork, Kabanov & Runggaldier(1997), Duffie, Pan & Singleton (2000), Kou (2002), Sch¨onbucher (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 molecules

Further-or coupled reactions yield stochastic models with jumps, see, fFurther-or 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 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 quantities

approxima-in finance Here contapproxima-inuous tradapproxima-ing noise and a few sapproxima-ingle 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 L´evy 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 aims

to 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 constructed

on time discretizations (t)Δ , with maximum step size Δ ∈ (0, Δ0), with

Δ0 ∈ (0, 1) We call a time discretization regular if the jump times,

gen-erated by the Poisson measure, are not discretization times On the other

hand, 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 discretizaapproxima-tions are called regular schemes,

while approximations constructed on jump-adapted time discretizations are

called jump-adapted schemes.

Discrete-time approximations can be divided into two major classes: strong approximations and weak approximations, see Kloeden & Platen (1999) We

say that a discrete-time approximation Y Δ, constructed on a time

discretiza-tion (t)Δ, with maximum step size Δ > 0, converges with strong order γ at

time T to the solution X of a given SDE, if there exists a positive constant

C, independent of Δ, and a finite number Δ0∈ (0, 1), such that

E( |X T − Y Δ

for all Δ ∈ (0, Δ0) From the definition of the strong error on the left hand side

of (0.0.1) one notices that strong schemes provide pathwise approximations of

the original solution X of the given SDE These methods are therefore suitable

for 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 Y Δ

con-verges weakly with order β to X at time T , if for each g ∈ C 2(β+1)

for each Δ ∈ (0, Δ0) Here C 2(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 for

for all y ∈  d and any partial derivative ∂ j y i g(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 such

as 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 Y Δ be a discrete-time approximation, constructed on a time

discretization (t)Δ, with strong order of convergence γ, see (0.0.1) Consider

a function g :  d →  satisfying the Lipschitz condition

), the above result implies that if a discrete-time approximation Y Δachieves

an order γ of strong convergence, then it also achieves at least an order β = γ

of 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 order

γ = 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, thus

enhancing the order of strong convergence from γ = 0.5 to γ = 1.0 less, the order of weak convergence of the Milstein scheme equals β = 1.0,

Nonethe-which is not an improvement over the order of weak convergence of the ler scheme Therefore, to price a European call option, for example, the Eulerscheme is often computationally more efficient than the Milstein scheme, since

Eu-it 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 X T at a final time T , there exist alternative numerical methods Under

suitable conditions, the pricing function u(x, t) = E(g(X T)|X t = x) can

be expressed as a solution of a partial integro differential equation (PIDE).

Therefore, an approximation of the pricing function u(x, t) can be obtained by

solving 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 incorporate

early 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 problems

in finance The monograph is divided into three parts The first part, 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

cover-Motivation and Brief Survey XXVII

and pure jump SDEs It also includes some discussions on parameter 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 simulation

estima-of hedge ratios, efficient variance reduction techniques, Markov chain imations and finite difference methods are discussed in the context of weakapproximation

approx-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 ered we may refer the reader to various well-written books, including Ikeda

cov-& Watanabe (1989), Niederreiter (1992), Elliott, Aggoun cov-& Moore (1995),Milstein (1995a), Embrechts, Kl¨uppelberg & Mikosch (1997), Bj¨ork (1998),Karatzas & Shreve (1998), Mikosch (1998), Kloeden & Platen (1999), Shiryaev(1999), Bielecki & Rutkowski (2002), Borodin & Salminen (2002), J¨ackel(2002), Joshi (2003), Sch¨onbucher (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 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

Fi-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 the end of the book Some readers may find the Author Index at the end of the book useful Each chapter finishes with some Exercises with Solutions given

in 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 tion and theoretical understanding of numerical methods for stochastic differ-ential equations with jumps are still under development This book shall stim-

applica-ulate interest and further work on such methods The Bibliographical Notes

at the end of this research monograph may be of assistance

Stochastic Differential Equations with Jumps

Stochastic differential equations (SDEs) with jumps provide the most flexible,numerically accessible, mathematical framework that allows us to model theevolution of financial and other random quantities over time In particular,feedback effects can be easily modeled and jumps enable us to model events.This chapter introduces SDEs driven by Wiener processes, Poisson processesand Poisson random measures We also discuss the Itˆo formula, the Feyman-Kac formula and the existence and uniqueness of solutions of SDEs Thesetools and results provide the basis for the application and numerical solution

of stochastic differential equations with jumps

1.1 Stochastic Processes

Stochastic Process

If not otherwise stated, throughout the book we shall assume that there

ex-ists a common underlying probability space (Ω, A, P ) consisting of the sample

space Ω, the sigma-algebra or collection of events A, and the probability

mea-sure P , see for instance Shiryaev (1984) One typically observes a collection of random variables Xt0, X t1, , which describe the evolution of financial quan-

tities, for instance, daily closing prices of an index at the observation times

t0 < t1 < The collection of random variables is indexed by the time t,

and we call T the time set The state space of X is here the d-dimensional

Euclidean space  d , d ∈ N = {1, 2, }, or a subset of it.

Definition 1.1.1. We call a family X = {X t , t ∈ T } of random variables

X t ∈  d a d-dimensional stochastic process, where the totality of its dimensional distribution functions

E Platen, N Bruti-Liberati, Numerical Solution of Stochastic Differential

Equations with Jumps in Finance, Stochastic Modelling

and Applied Probability 64, DOI10.1007/978-3-642-13694-8 1,

© Springer-Verlag Berlin Heidelberg 2010

1

2 1 SDEs with Jumps

We set the time set to the intervalT = [0, ∞) if not otherwise stated On

some occasions the time set may become the bounded interval [0, T ] for T ∈

(0, ∞) or a set of discrete time points {t0, t1, t2, }, where t0< t1< t2<

One can distinguish between various classes of stochastic processes ing to specific properties First, we aim to identify stochastic processes thatare suitable as basic building blocks for realistic models From a quantitativepoint of view it is essential to select stochastic processes that allow explicitformulas or at least fast and efficient numerical methods for calculating mo-ments, probabilities, option prices or other quantities In the following we set

accord-usually the dimension d equal to one However, most concepts and results we

will introduce generalize to the multi-dimensional case

eco-eling equilibria is that of stationary processes since they allow us to express

probabilistically a form of equilibrium For instance, interest rates, dividendrates, inflation rates, volatilities, consumption rates, hazard rates and creditspreads are likely to be modeled by stationary processes since they typicallyexhibit in reality some type of equilibrium

Definition 1.1.2. We say that a stochastic process X = {X t , t ≥ 0} is

sta-tionary if its joint distributions are all invariant under time displacements,

that is if

F X t1+h ,X t2+h , ,X tn+h = F X t1 ,X t2 , ,X tn (1.1.5)

for all h > 0, t i ≥ 0, i ∈ {1, 2, , n} and n ∈ N

The random values Xt of a stationary process X have the same distribution for all t ∈ T Therefore, means, variances and covariances satisfy the equations μ(t) = μ(0), v(t) = v(0) and C(s, t) = c(t − s) (1.1.6)

1.1 Stochastic Processes 3

Fig 1.1.1 Sample path for the Vasicek interest rate model, T = 10

for all s, t ≥ 0, where c :  →  is a function However, to ensure this property

its initial value needs to be modeled as an approximate random variable.Therefore, in practice we may typically model equilibrium type dynamics

by employing a stationary process conditioned on an observed initial value.The concept of stationarity extends in a straightforward manner to multi-dimensional stochastic processes

For illustration, in Fig.1.1.1we display a trajectory of a stationary

contin-uous Gaussian process with mean μ(t) = 0.05, variance v(t) = 0.1 and initial value X0 = 0.05 We may interpret it as the sample path of some interest

rate obtained under the, so called, Vasicek interest rate model, see Vasicek(1977) The process fluctuates around a reference level and appears to revertback towards its mean As a stationary process we may continue to observe itsmovements over longer time horizons with its mean, variance and covariancesnot changing

Filtration as Information Structure

As we will see later, in finance the notion of a stochastic process for whichits last observed value provides the best forecast for its future values, plays

a fundamental role Forecasting is primarily based on current available mation Financial markets are strongly influenced by information Essentially,

infor-it is information that drives the dynamics of markets Therefore, in financialmodeling a precise definition of the information structure available is impor-tant

As already indicated previously, our modeling is based on a given

probabil-ity space (Ω, A, P ), which consists of the sample space Ω, the sigma-algebra

A that is generated by all events and the given probability measure P On

4 1 SDEs with Jumps

such a probability space we consider dynamics, typically of a financial ket model, that is based on the observation of a continuous time stochastic

mar-vector process X = {X t ∈  d , t ≥ 0}, d ∈ N We denote by ˆ A t the time t

information set, which is the sigma-algebra generated by the events that are

known at time t ≥ 0, see Shiryaev (1984) Our interpretation of ˆ A t is that

it represents the information available at time t, which is obtained from the

observed values of the vector process X up to time t More precisely, it is the

sigma-algebra

ˆ

A t = σ {X s : s ∈ [0, t]}

generated from all observations of X up to time t Since information is not

lost, the increasing family

ˆ

A = { ˆ A t , t ≥ 0}

of information sets ˆA t satisfies, for any sequence 0 ≤ t1 < t2 < < ∞ of

observation times, the relation ˆA t1 ⊆ ˆ A t2 ⊆ ⊆ ˆ A ∞=∪ t ≥0 Aˆt

For technical reasons one introduces the information set A t as the

aug-mented sigma-algebra of ˆ A t for each t ≥ 0 It is augmented by every null

set in ˆA ∞ such that it belongs to A0, and also to each ˆA t One also says

that A t is complete Define A t+ =∩ ε>0 A t+ε as the sigma-algebra of events

immediately after t ∈ [0, ∞) The family A = {A t , t ≥ 0} is called right tinuous if A t = A t+ holds for every t ≥ 0 Such a right-continuous family

con-A = {con-A t , t ≥ 0} of information sets one calls a filtration Such a filtration

can model the evolution of information as it becomes available over time Wedefine for our purposes A as the smallest sigma-algebra that contains A ∞

= ∪ t ≥0 A t From now on, if not stated otherwise, we always assume in this book a filtered probability space (Ω, A, A, P ) to be given.

Any right-continuous stochastic process Y = {Y t , t ≥ 0} generates its ural filtration A Y ={A Y

nat-t , t ≥ 0}, which is the sigma-algebra generated by Y

up to time t For a given model with a vector process X we typically set

t , which means that it includes

all null events, allows us to conclude that for two random variables Z1 and

Z2, where Z1 = Z2 almost surely (a.s.) and Z1 isA X

or expectation E(X) is the coarsest estimate that we have for an integrable

random variable X, that is, for which E( |X|) < ∞, see Shiryaev (1984) If

we know that some event A has occurred we may be able to improve on this

1.1 Stochastic Processes 5

estimate For instance, suppose that the event A = {ω ∈ Ω : X(ω) ∈ [a, b]}

has occurred Then in evaluating our estimate of the value of X we need only

to consider corresponding values of X in [a, b] and weight them according to

their likelihood of occurrence, which thus becomes the conditional probabilitygiven this event, see Shiryaev (1984)

The resulting estimate is called the conditional expectation of X given the event A and is denoted by E(X | A) For a continuous random variable X with

a density function f X the corresponding conditional density is

which is conditioned on the event A and is, thus, a number.

More generally let (Ω, A, P ) be a given probability space with an

inte-grable random variable X We denote by S a sub-sigma-algebra of A, thus

representing a coarser type of information than is given byS ⊂ A We then

define the conditional expectation of X with respect to S, which we denote by E(X | S), as an S-measurable function satisfying the equation

for all Q ∈ S The Radon-Nikodym theorem, see Shiryaev (1984),

guaran-tees the existence and uniqueness of the random variable E (X | S) Note

that E (X | S) is a random variable defined on the coarser probability space

(Ω, S, P ) and thus on (Ω, A, P ) However, X is usually not a random variable

on (Ω, S, P ), but when it is we have

E

which is the case when X is S-measurable.

The following results are important when handling the evolution of tic processes in conjunction with an evolving information structure, as is oftenthe case in finance

stochas-For nested sigma-algebrasS ⊂ T ⊂ A and an integrable random variable

X we have the iterated conditional expectations

6 1 SDEs with Jumps

where X and Y are integrable random variables and α, β ∈  are deterministic

constants Furthermore, if X is S-measurable, then

The conditional expectation E(X | S) is in some sense obtained by

smooth-ing X over the events in S Thus, the finer the information set S, the more E(X | S) resembles the random variable X.

Wiener Process

In contrast to stationary processes we consider now stochastic processes with

stationary independent increments These basic processes have

mathemati-cal properties that make them suitable as fundamental building blocks instochastic modeling and, thus, in financial modeling The random increments

X t j+1 − X t j , j ∈ {0, 1, , n−1}, of these processes are independent for any

sequence of time instants t0< t1< < t n in [0, ∞) for all n ∈ N If t0= 0

is the smallest time instant, then the initial value X0 and the random

incre-ment X t j − X0for any other t j ∈ [0, ∞) are also required to be independent.

Additionally, the increments Xt+h − X tare assumed to be stationary, that is

X t+h − X t has the same distribution as Xh − X0for all h > 0 and t ≥ 0.

The most important continuous process with stationary independent

in-crements is the Wiener process Bachelier was the first who employed, already

in 1900, such a mathematical object in his modeling of asset prices at theParis Bourse, see Bachelier (1900) or Davis & Etheridge (2006) He did thiseven before Einstein used an equivalent mathematical construct that we call

now the Wiener process or Brownian motion, see Einstein (1905).

Definition 1.1.3. We define the standard Wiener process W = {W t ,

t ≥ 0} as an A-adapted process with Gaussian stationary independent ments and continuous sample paths for which

incre-W0= 0, μ(t) = E(W t ) = 0, Var(W t − W s ) = t − s (1.1.16)

for all t ≥ 0 and s ∈ [0, t].

1.1 Stochastic Processes 7

Fig 1.1.2 Sample path of a standard Wiener process, T = 10

Fig 1.1.3 Sample paths of a Wiener process, T = 10

The Wiener process is a continuous time stochastic process with dent Gaussian distributed increments generating continuous sample paths.This process is also known as Brownian motion, since it can model the mo-tion of a pollen grain under the microscope, as observed by Robert Brown inthe early 19th century

indepen-In Fig.1.1.2 we plot a sample path of a standard Wiener process Tovisualize more of its probabilistic properties we display 20 such trajectories inFig.1.1.3

The Wiener process has fundamental mathematical properties and is used

as a basic building block in many applications, in particular in finance TheWiener process is the most basic stochastic process that allows us to modelcontinuous uncertainty, as it arises for instance as trading noise Note that

8 1 SDEs with Jumps

the Wiener process is not a stationary process that is conditioned on its

known initial value, as can be seen from its increasing variance, see (1.1.16).However, its independent increments have the same distribution when takenover periods of the same time length and are therefore stationary in this sense.There exists also a multi-dimensional version of the above Wiener process

We call the vector process W = {W t = (W1

t , W2

t , , W m

t ) , t ≥ 0} an dimensional standard Wiener process if each of its components W j ={W j

m-t,

t ≥ 0}, j ∈ {1, 2, , m} is a scalar A-adapted standard Wiener process

and the Wiener processes W k and W j are independent for k

{1, 2, , m}.

This means that according to Definition 1.1.3, each random variable W t j

is Gaussian and A t-measurable with

a piecewise constant stochastic process with independent increments which

counts events This is the Poisson process which can be conveniently used to

model event driven uncertainty

Definition 1.1.4. A Poisson process N = {N t , t ≥ 0} with intensity λ > 0

is a piecewise constant process with stationary independent increments with initial value N0 = 0 such that Nt − N s is Poisson distributed with intensity

λ t −s , that is, with probability

P (N t − N s = k) = e

−λ (t−s) (λ (t − s)) k

for k ∈ {0, 1, }, t ≥ 0 and s ∈ [0, t].

1.1 Stochastic Processes 9

Fig 1.1.4 Trajectory of a standard Poisson process with intensity λ = 20

For the Poisson process N with intensity λ we have the mean

In Fig.1.1.4we plot a graph for a Poisson process with intensity λ = 20.

According to (1.1.21), we should expect on average 20 events to happen during

the time period [0, 1], which is about the case for this trajectory We size that also the Poisson process is not a stationary process conditioned on

empha-its known initial value However, empha-its increments have the same Poisson bution over time intervals of the same length and are in this sense stationary

distri-A Poisson process N counts events and generates an increasing sequence

of jump times τ1, τ2, related to each event that it counts Thus, N tequals

the number of events that occurred up until time t ≥ 0 For t ≥ 0 let the time

τ N t denote the last time that Ntmade a jump, that is,

for k ∈ N , t ≥ 0.

In some applications, as in the modeling of defaults, the intensity λ tthat

a certain type of event occurs may depend on the time t ≥ 0 This leads to a time transformed Poisson process N = {N t , t ≥ 0}, where

10 1 SDEs with Jumps

for k ∈ {0, 1, }, t ≥ 0 and s ∈ [0, t].

To obtain more flexibility in modeling we introduce the mark ξk of the

kth event which, for instance, could be the recovery rate of the kth default.

Let us assume for the moment that the kth mark ξk is deterministic,

k ∈ {1, 2, } We can then consider the process Y = {Y t , t ≥ 0} with

for t ≥ 0 If the intensity process λ = {λ t , t ≥ 0} is deterministic and the kth

mark ξk is deterministic for each k ∈ {1, 2, }, then it follows from (1.1.24)

that the mean μ(t) of Ytis given by the expression

where the above probabilities are expressed in (1.1.24) For instance, if one

chooses ξ1 = 1 and ξk = 0 for k ∈ {2, 3, }, then this simple process allows

us to model the credit worthiness Ct of an obligor at time t by setting

C t= 1− Y t

The credit worthiness may start at time zero with C0= 1 It declines to zero at

the time when the first default arises The expected value E(Ct) of the above credit worthiness at time t equals by (1.1.26) and (1.1.24) the expression

Compound Poisson Process

One often needs to differentiate between certain types of events, for instance,those with different recovery rates at a default A corresponding intensity forthe occurrence of each type of possible event is then required

To construct a process that models sequences of different types of events,

we consider a Poisson process N with intensity λ > 0, together with a sequence

of independent identically distributed (i.i.d.) random variables ξ1 , ξ2, that

are independent of N Here

denotes for z ∈  the corresponding value of the distribution function of ξ1.

This allows us to construct the compound Poisson process Y = {Y t , t ≥ 0},

where Y0= 0 and

1.1 Stochastic Processes 11

Fig 1.1.5 Compound Poisson process

Fig 1.1.6 Pairs of jump times and marks

for t ≥ 0 A compound Poisson process generates a sequence of pairs (τ k , ξ k)k ∈N

of jump times τ k and marks ξ k In Fig.1.1.5 we show the trajectory of a

compound Poisson process Y where the i.i.d random variables are uniformly distributed on the interval [0, 1], and N is as in Fig.1.1.4 In Fig.1.1.6 weplot the points generated by the pairs of jump times and marks of the path

of the compound Poisson process Y shown in Fig.1.1.5

A compound Poisson process is again a process with stationary dent increments Its trajectory is fully characterized by the sequence of pairs

indepen-(τk , ξ k)k∈N

12 1 SDEs with Jumps

Fig 1.1.7 Risk reserve process

The following simple, but important insurance model involves the above

compound Poisson process Y The Cram´ er-Lundberg model for the risk reserve

X t at time t of an insurance company can be described in the form

where the claim process Y = {Y t , t ≥ 0} is modeled as a compound Poisson

process with i.i.d claim sizes ξ k > 0, k ∈ N Here c denotes the premium

rate that describes the premium payment per unit of time that the insurancecompany collects We show in Fig.1.1.7the trajectory of a risk reserve process

X when choosing c = 10 and X0= 100

Poisson Measure

For the modeling of events that arise with high intensity and which have

different marks, the notion of a Poisson measure is useful A simple example

for a Poisson measure is given by a compound Poisson process with marks

that are uniformly distributed on [0, 1], as we will see below.

It is a fundamental feature of the Poisson process that due to the dence of its increments the location of the set of points in the time interval

indepen-[0, 1], see Fig.1.1.6, can be intuitively interpreted as if one has generated N1 independent uniformly U (0, 1) distributed random variables for obtaining the jump times On the other hand, the independent marks ξk of the compound

Poisson process Y , displayed in Fig.1.1.6, are also independent uniformly

distributed on [0, 1] Consequently, the pairs (τk , ξ k ), for k ∈ N , shown in

Fig.1.1.6, are independent uniformly distributed in the square [0, 1] × [0, 1].

Such a graph can be interpreted as the realization of some Poisson measure.The general notion of a Poisson measure is rather technical However, itrepresents simply a compound Poisson process as long as the total intensity

Here the element {0} is excluded which allows us to avoid conveniently in

our modeling jumps of size zero Let B(Γ ) denote the smallest sigma-algebra

containing all open sets of a set Γ Now, we consider on E × [0, ∞) a given intensity measure of the form

The corresponding Poisson measure pϕ( ·) on E × [0, ∞), see Protter (2005),

is assumed to be such that for T ∈ (0, ∞) and each set A from the

product-sigma-algebra ofB(E) and B([0, T ]) the random variable p ϕ(A), which counts the number of points in A ⊆ E × [0, ∞), is Poisson distributed with intensity

∈ {0, 1, } For disjoint sets A1, , A r, ⊆ E ×[0, T ], r ∈ N , the random

variables pϕ(A1), , pϕ(Ar) are assumed to be independent.

For example, the points displayed in Fig.1.1.6 can be interpreted as a

realization of a Poisson measure on [0, 1] × [0, T ] with ϕ(dv) = λdv, where

The following class of L´ evy processes generalizes the just mentioned processes