Numerical Solution of Stochastic Differential Equations with Jumps in Finance

by Eckhard Platen and Nicola Bruti-Liberati, Springer (2010). ISBN 978-3642120572.

This book presents numerical methods for stochastic differential equations (SDEs) with jumps via simulation, with motivation and examples coming from applications in finance. I found it useful because I do a lot of simulations of SDEs in my own research in various areas of financial mathematics. My main take-away was that numerical SDE is a subtle subject, and much like numerical (deterministic) PDE, one should take the time to analyse the scheme she chose to numerically solve an SDE. Careful analysis of such schemes is the main theme in this book, along with contemporary applications in finance.

A simple Euler scheme of order one (or the Euler–Maruyama scheme as it is known in the literature) is the basic method for approximating solutions of SDEs, but there is significant interest in higher order schemes (such as the Milstein scheme) because of their faster rate of convergence. However, higher order schemes do not always produce better results. For instance, in finance we are often interested in simulating paths of an SDE and using them to compute risk-neutral expectations (what the authors call a ‘weak approximation’), but the convergence offered by the Milstein scheme is no higher than the convergence offered by the Euler scheme in this case. Of further importance is the issue of stability (i.e. the assurance that there is no propagation of uncontrolled approximation errors), which the authors point out should be of greater concern than the order of convergence. In fact, Chapter 14 is a nice exposition of how simple schemes for solving a simple SDE can be unstable.

Numerical SDEs were covered previously in Kloeden and Platen (1992), which is closely related to this book. The former had a more extensive introduction to probability theory, no jumps and no applications in finance. In this book, the authors have provided a more advanced and contemporary text on the matter. Comparatively, Glasserman (2003) might be a more general reference for numerical methods in finance, in which Chapter 6 covers numerical SDE, but Glasserman (2003) does not have the extensive analysis of issues that can arise due to instability of numerical schemes.

Let be a Poisson process with intensity . A basic SDE with jumps takes the form:

(1)

with , where the functions and are Lipschitz continuous in , with and satisfying the standard linear growth condition, and with satisfying an integrated growth condition (that we will specify shortly). Equation (1) is part of a more general class of SDEs that involve a Poisson random measure. Denote the set of possible jump sizes as , and consider the intensity measure

where is a measure on the Borel -algebra with

(2)

From the measures and , a Poisson random measure can be defined:

A Poisson random measure the with mean measure is a collection of random variables for subsets , such that and are independent when and are disjoint, and for each with ,

This random measure has a series of random marks, call them , which are mass-points occurring at points in such that

where denotes the Dirac-delta function. The condition set forth by equation (2) insures that does not have infinite activity, thereby causing positive time to elapse between marks (with probably one).

The random measure is used to write a general class of SDEs with jumps,

(3)

with the same conditions on and that were mentioned with equation (1), and with growth condition

for some constant and . Equation (3) is quite general and encompasses several frequently used models in finance such as Lévy processes, which is the case of equation (1) when , , and the coefficient is independent of the marks. The book’s primary purpose is to analyse numerical methods for equation (3), and Chapter 1 is a good introduction to the basic concepts (and also some other preliminaries including martingales, stopping times, the Itô calculus for jumps and various models from finance).

Numerical schemes for approximation of (3) are presented and analysed in Chapters 4 through 8. Many of the concepts are based on the generalized Wagner–Platen expansion, which is a stochastic generalization of the Taylor expansion that involves multiple integrals and is of higher order accuracy. For instance, for a time-step , a simple example of a Wagner–Platen expansion is

where and contains the remaining higher order terms (see pp. 207–208 in the book). Truncating from this Wagner–Platen expansion leads to the well-known Euler–Maruyama scheme,

where . A popular extension of the Euler–Maruyama scheme to higher order is the Milstein scheme,

which is derived by including one of the higher order terms in , namely, the higher order term is

Also discussed in these chapters are stability and accuracy of general higher order schemes, derivative-free schemes, implicit schemes and predictor-corrector schemes. Chapters 11–13 cover Monte Carlo and weak approximations. Overall, the book is a good reference just for these chapters alone because of the detailed and well-explained account of numerical methods for SDEs.

Chapter 14 is of particular interest because it is an example of how the field of numerical SDE is still developing. The chapter presents some results on stability of numerical solutions, and raises some important questions on the matter. The chapter is simulation-based and shows how a seemingly accurate numerical scheme can actually be unstable. The authors consider a linear, scalar SDE that has an explicit solution, and is a variant of geometric Brownian motion that is commonly used in finance:

(4)

where , and . Through Itô’s formula it can be verified that for

if and only if , in which case we say that is -stable. For a given numerical scheme and the sequence generated using step-size , we define and say that the scheme is -stable if

(5)

The set of triplets for which the scheme is -stable define the scheme’s stability region. The chapter has 3D plots of the stability regions for the family of predictor-corrector Euler schemes, from which it can be seen that non-implicit schemes (such as the basic Euler–Maruyama) have relatively small stability regions, whereas semi-implicit and and fully-implicit (Euler) schemes have larger stability regions. The main points that the authors emphasize are the following: stability should be of equal or greater concern than a scheme’s order of convergence, and that schemes with implicitness in both the drift and diffusion terms exhibit the largest stability regions. After reading this chapter, researchers working with numerical SDEs will take extra care when choosing schemes.

The remainder of the book is selective reading, for which the pages XV, XVI and XVII of the introduction have outlines of different curricula for readers with varying mathematical background. It is probably the case that readers will initially be interested in a certain subset of chapters. Indeed, the book covers a diverse array of interesting topics:

1.	Chapter 2 is a good introduction to simulation of some SDEs that are common in finance and that also have exact solutions that can be used in benchmarking. Simulations using copulas are presented, and the section on almost-exact solutions by conditioning is useful for understanding how to simulate stochastic volatility models.
2.	Chapter 3 on benchmarking has more examples from finance. Key concepts are introduced, such as market risk premium, diversification, and the growth optimal portfolios. One nice thing about this chapter is that it shows how to generalize some of the ideas of stochastic portfolio theory (see Fernholz (2002)) to include jumps.
3.	Chapter 15 approaches hedging of contingent claims from a numerical perspective, which I thought was an interesting treatment due to the fact that, in practice, hedges are maintained in discrete time. Technicalities such as non-smooth pay-off functions are covered, and so are exotic derivatives including the lookback option. One thing missing is a discussion of possible hedging strategies in the presence of jumps.
4.	Chapter 16 is an instructive chapter on variance reduction techniques for SDE-based Monte Carlo. Concepts such as control variates, measure transformation and the Heath-Platen (HP) method (which they show to be a type of control variate technique) are explained well. I liked how this chapter presented the material with derivative pricing as an underlying theme; I also thought the HP method was explained well. Comparatively, this chapter is complementary to other good texts on Monte Carlo such as Asmussen and Glynn (2007) and Glasserman (2003).

Overall, I think this book is a worthwhile investment for someone who works with numerical SDEs. Indeed, this book introduces a lot of ideas that were developed in the last decade and contains a broad spectrum of interesting applications and ideas. The previous book (Kloeden and Platen 1992) is good, but the addition of jumps is non-trivial and recent developments in finance add a lot to this new volume.

Acknowledgments

Work partially supported by NSF Grant DMS-0739195.