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

Abstract (Summary)

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.
iv
University of Pretoria etd – Govender, N (2007)
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.
v
University of Pretoria etd – Govender, N (2007)
Bibliographical Information:

Advisor:

School:University of Pretoria/Universiteit van Pretoria

School Location:South Africa

Source Type:Master's Thesis

Keywords:mathematical models stochastic differential equations

ISBN:

Date of Publication: