# BSc Mathematics with Placement Year

Year of entry: 2024

## Course unit details:Mathematical Modelling in Finance

Unit code MATH39032 10 Level 3 Semester 2 No

### 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. It is assumed that students will be familiar with financial terms, stochastic differential equations, and partial differential equations. It is shown that mathematical methods developed to solve Partial Differential Equations can be powerful tools in solving financial problems once suitable models have been devised.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Introduction to Financial Mathematics MATH20912 Pre-Requisite Compulsory

### Aims

The unit aims to give students 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 both derive and solve for basic financial derivatives contracts.

### Learning outcomes

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

1. Recognise the role that financial derivatives play in reducing risk

2. Derive boundary conditions for financial contracts priced under the Black-Scholes model

3. Construct a PDE to price financial contracts, using the concepts of stochastic calculus and hedging

4. Apply transformations and similarity solution techniques to PDEs such as Black-Scholes equation and derive analytic solutions.5. Use the analytic formulae to evaluate fair prices for European options

6. Extend the basic European option model (to include dividends, stochastic volatility, stochastic interest rates, early exercise and barriers) and where possible solve the resulting models analytically

### Syllabus

1. Introduction to options, futures, no arbitrage principle [3]2. Models for stock prices, basics of stochastic calculus and Ito's lemma. [3]3. Deriving the the pricing partial differential equation, and the assumptions behind it. Formulating the mathematical problem. Analytic solutions and Implied volatility. 34. 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. [3]5. Extension to consider options on assets paying dividends. [2]6. American options and free boundary problems. [2]7. Interest-rate models and bonds. [2]8. Multi Factor models and Barrier options. [3]

### Teaching and learning methods

There are 3 or 4 videos released per week, delivering content from the course .Students are expected to watch the videos, fill in the gaps in the notes, as well as reading and reviewing the notes. Each lecture has a formative assessment attached to test the student's understanding of the lecture. A 1 hour review session highlight some of the more important material from the videos and goes through some of the examples sheets together. A 1 hour feedback tutorial provides an opportunity to work on problems in class, answers and partial solutions will be revealed in class. Finally a coursework test provides an opportunity for students to receive feedback on how well they understand the first half of the course. Students can also get feedback on their understanding directly from the lecturer, either using the Piazza forum or by arranging a meeting during the lecturer's office hour.

Method Weight
Written exam 80%
Set exercise 20%

### 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.

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

Wilmott, P., 2001: Paul Wilmott Introduces Quantitative Finance, 2nd Edition, Wiley. ISBN: 0471498629. Wilmott, P., 2000: Paul Wilmott on Quantitative Finance, Wiley. ISBN: 0471874388

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

### Teaching staff

Staff member Role
Paul Johnson Unit coordinator