MPhys Physics with Astrophysics

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.

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.

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.

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]

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.

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

The independent study hours will normally comprise the following. During each week of the taught part of the semester:
 
•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests
 
Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.
 
The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.