## Mathematical Modelling in Finance

 Unit code: MATH39032 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

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#### 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.

#### Overview

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. No previous background in finance is necessary. It is shown that mathematical methods can be powerful tools in solving financial problems once suitable models have been devised.

#### Learning outcomes

On successful completion of the course, students will be able to:

• 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.

#### Assessment Further Information

End of semester examination: two hours weighting 100%

#### Syllabus

1. Introduction to options, futures and the no arbitrage principle - using this to calculate fair delivery prices for futures. [4 lectures]
2. Model for the movements of stock prices, efficient markets, Brownian motion and geometric Brownian motion. Stochastic and deterministic processes. [2]
3. Basics of stochastic calculus and Ito's lemma. [2]
4. 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. [5]
5. 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. [6]
6. Extension to consider options on assets paying dividends and American options; free boundary problems. [5]

• Wilmott, O., Howison, S., Dewynne, J., The Mathematics of Financial Derivatives, Cambridge University Press 1995. ISBN 0521497892

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Sergei Fedotov - Unit coordinator