Paul Johnson - Department of Mathematics

Course Materials

Please use the Blackboard space for course materials this year.

Brief Description

This course unit is primarily concerned with the valuation of financial instruments known as derivatives. To achieve this, a mathematical model is developed and then solved for different types of problems. It is shown that mathematical methods can be powerful tools in solving financial problems once suitable models have been devised.

Aims

Students should gain an insight into both the development and solution of the mathematical models used to describe the value of financial derivatives. As a result they should be able to find the value of basic derivatives.

Syllabus

Introduction to options, futures and the no arbitrage principle - using this to calculate fair delivery prices for futures.
Model for the movements of stock prices, efficient markets, Brownian motion and geometric Brownian motion. Stochastic and deterministic processes.
Basics of stochastic calculus and Ito's lemma.
The Black-Scholes analysis. Derivation of the Black-Scholes partial differential equation, the assumptions behind it. Formulating the mathematical problem, determining boundary conditions for option pricing problems.
Solving the Black-Scholes equation. Connection with the heat conduction equation, solution of the heat conduction equation - similarity solutions and the Dirac delta function. Derivation of the price of European options.
Extension to consider options on assets paying dividends and American options; free boundary problems.
Interest-rate models and bonds.
Barrier options.

Intended Learning Outcomes

recognise the role that financial derivatives play in reducing risk;
construct payoff diagrams for standard options (and portfolios of options);
construct a PDE, using the concepts of stochastic calculus and hedging;
solve analytically the standard Black-Scholes equation;
use the Black-Scholes formulae to evaluate fair prices for European options;
extend the basic European option model (to include dividends and/or early exercise) and where possible to solve the resulting models analytically.

MATH39032: Mathematical Modelling in Finance