Solution of Stochastic Differential Equations in Finance

Numerical Solution of Stochastic Differential Equations in Finance


Timothy Sauer


Department of Mathematics George Mason University Fairfax, VA 22030 [e-mail]


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- nancial quantities such as asset prices, interest rates, and their derivatives. Un- like 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 stochas- tic 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 nu- merical methods. For full details, the reader may consult Klebaner (1998); Oksendal (1998); Steele (2001).

A set of random variables X_t indexed by real numbers t ≥ 0 is called a continuous-time stochastic process. Each instance, or realization of the stochas- tic process is a choice from the random variable X_t for each t, and is therefore a function of t.

Any (deterministic) function f (t) can be trivially considered as a stochastic process, with variance V (f (t)) = 0. An archetypal example that is ubiquitous in models from physics, chemistry, and finance is the Wiener process W_t, a continuous-time stochastic process with the following three properties:
Property 1. For each t, the random variable W_t is normally distributed with mean 0 and variance t.
Property 2. For each t₁ < t₂, the normal random variable W_t2 −W_t1 is indepen- dent of the random variable W_t1 , and in fact independent of all W_t, 0 ≤ t ≤ t₁.
Property 3. The Wiener process W_t can be represented by continuous paths.
The Wiener process, named after Norbert Wiener, is a mathematical con- struct that formalizes random behavior characterized by the botanist Robert Brown in 1827, commonly called Brownian motion. It can be rigorously de- fined as the scaling limit of random walks as the step size and time interval between steps both go to zero. Brownian motion is crucial in the modeling of stochastic processes since it represents the integral of idealized noise that is in- dependent of frequency, called white noise. Often, the Wiener process is called upon to represent random, external influences on an otherwise deterministic system, or more generally, dynamics that for a variety of reasons cannot be deterministically modeled.
A typical diffusion process in finance is modeled as a differential equation involving deterministic, or drift terms, and stochastic, or diffusion terms, the latter represented by a Wiener process, as in the equation