Numerical Solution of Stochastic Differential Equations in Finance

This chapter is an introduction and survey of numerical solution methods for stochastic differential equations. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial systems. We include a review of fundamental concepts, a description of elementary numerical methods and the concepts of convergence and order for stochastic differential equation solvers. In the remainder of the chapter we describe applications of SDE solvers to Monte-Carlo sampling for financial pricing of derivatives. Monte-Carlo simulation can be computationally inefficient in its basic form, and so we explore some common methods for fostering efficiency by variance reduction and the use of quasi-random numbers. In addition, we briefly discuss the extension of SDE solvers to coupled systems driven by correlated noise, which is applicable to multiple asset markets. 1 Stochastic differential equations Stochastic differential equations (SDEs) have become standard models for financial quantities such as asset prices, interest rates, and their derivatives. Unlike deterministic models such as ordinary differential equations, which have a unique solution for each appropriate initial condition, SDEs have solutions that are continuous-time stochastic processes. Methods for the computational solution of stochastic differential equations are based on similar techniques for ordinary differential equations, but generalized to provide support for stochastic dynamics. We will begin with a quick survey of the most fundamental concepts from stochastic calculus that are needed to proceed with our description of numerical methods. For full details, the reader may consult [16, 30, 40]. A set of random variables Xt indexed by real numbers t ≥ 0 is called a continuous-time stochastic process. Each instance, or realization of the stochastic