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Of fundamental importance to science, finance and engineering, are processes with random fluctuations. The series of prices of a financial instrument such as an equity, bond, or contract is an ideal and extremely important example. Some graduates will work in financial and commercial applications of mathematics where stochastic differential equations (SDEs) are of fundamental importance. SDEs also apply in many other areas in science and engineering and have many features that distinguish them from other mathematical models. Developing technical communication is also essential as preparation for the workplace which is addressed in this course.

This course begins by investigating models of economic activity and the financial and economic strategies which are used to stimulate economic activity. After these models of financial processes, such as equity prices, interest rates, bond yields, and so on are considered. Simulation models of such processes are developed and used in experiments using scripts written in R and scilab which are supplied on the course web page (students may choose whether to use R or scilab - it is not necessary to use both).

The theory of Stochastic differential equations is introduced and studied by simulation and in theory. Techniques for solving such equations by means of Ito's formula are explained and applied. This is applied to financial process problems and the Black-Scholes differential equation is formulated and solved. Binomial tree models are introduced and used to solve a variety of option pricing models. In the last part of the course the method for solving option pricing problems based on the equivalent martingale measure. The on-campus offering of this course is normally available only in odd numbered years. The external offering of this course is available yearly.

On completion of this course students will be able to:

1. examine how to make use of simple mathematical models of an economy
2. simulate stochastic processes of various types, using provided software, and interpret the results;
3. apply mathematical models of financial or economic activity to model risk;
4. solve and interpret stochastic differential equations (SDEs);
5. prepare, for a general audience (not just mathematicians), documents and presentations of technical material both individually and in collaboration with other students.