Numerical solution of stochastic differential equations in finance

Abstract. 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 con-cepts, 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 simu-lation 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 fi