A brief analysis of certain numerical methods used to solve stochastic differential equations

Abstract

Stochastic differential equations (SDE’s) are used to describe systems which are influenced by randomness. Here, randomness is modelled as some external source interacting with the system, thus ensuring that the stochastic differential equation provides a more realistic mathematical model of the system under investigation than deterministic differential equations. The behaviour of the physical system can often be described by probability and thus understanding the theory of SDE’s requires the familiarity of advanced probability theory and stochastic processes. SDE’s have found applications in chemistry, physical and engineering sciences, microelectronics and economics. But recently, there has been an increase in the use of SDE’s in other areas like social sciences, computational biology and finance. In modern financial practice, asset prices are modelled by means of stochastic processes. Thus, continuous-time stochastic calculus plays a central role in financial modelling. The theory and application of interest rate modelling is one of the most important areas of modern finance. For example, SDE’s are used to price bonds and to explain the term structure of interest rates. Commonly used models include the Cox-Ingersoll-Ross model; the Hull-White model; and Heath-Jarrow-Morton model. Since there has been an expansion in the range and volume of interest rate related products being traded in the international financial markets in the past decade, it has become important for investment banks, other financial institutions, government and corporate treasury offices to require ever more accurate, objective and scientific forms for the pricing, hedging and general risk management of the resulting positions. Similar to ordinary differential equations, many SDE’s that appear in practical applications cannot be solved explicitly and therefore require the use of numerical methods. For example, to price an American put option, one requires the numerical solution of a free-boundary partial differential equation. There are various approaches to solving SDE’s numerically. Monte Carlo methods could be used whereby the physical system is simulated directly using a sequence of random numbers. Another method involves the discretisation of both the time and space variables. However, the most efficient and widely applicable approach to solving SDE’s involves the discretisation of the time variable only and thus generating approximate values of the sample paths at the discretisation times. This paper highlights some of the various numerical methods that can be used to solve stochastic differential equations. These numerical methods are based on the simulation of sample paths of time discrete approximations. It also highlights how these methods can be derived from the Taylor expansion of the SDE, thus providing opportunities to derive more advanced numerical schemes.