# Financial Mathematics 3

## Unit aims

This unit provides an introduction to the mathematical ideas underlying modern financial mathematics. The aim of the course is to understand the pricing of financial derivatives and apply these ideas to a variety of option contracts. In particular, the course will give a derivation of the Black-Scholes option pricing formula.

## Unit description

In 1973 Black and Scholes solved the problem of pricing a basic financial derivative (a product based on an underlying asset), the European call option. They assumed that the market had no arbitrage, and hence determined a unique fair price of the option. This course develops the sophisticated mathematics required by the subsequent explosion of trade in increasingly complex derivatives.

We first analyse a very simple model with just two time points where trading is possible. All basic ideas are already explained in this setting, including the notion of a risk-neutral probability measure. The theory is then extended to general discrete models with an arbitrary number of periods using martingales. In the second half of the course we model asset prices in continuous time by exponential Brownian motion, and informally introduce stochastic calculus. The final part of the course will consider the pricing of derivatives and the Black-Scholes formula.

## Relation to other units

The units Financial Mathematics and Queuing Networks apply probabilistic methods to problems arising in various fields. This course develops and applies rigorous mathematical techniques, and requires a good understanding of probability theory.

## Learning objectives

At the end of the course the student should be able to

- describe the difference between common financial instruments
- express financial problems in a mathematical framework
- calculate prices of simple financial instruments
- do calculations with martingales and Brownian motion.

## Transferable skills

Ability to compute prices of basic financial instruments

Mathematical modelling skills

Problem solving

## Syllabus

Introduction to financial terminology. Forwards; Options, European and American; arbitrage.

Elementary probability ideas, conditional expectation, filtration, introduction to discrete martingales, Optional stopping theorem.

The relationship between arbitrage and martingales, risk neutral measures. Discrete option pricing in binomial tree models. Discussion of American options.

Introduction to Brownian motion. Simple calculations with Brownian motion. Geometric Brownian motion.

Continuous martingales, the basic tools of stochastic calculus, Ito formula, Girsanov theorem.

Application to option pricing, Black-Scholes formula.

## Reading and References

There is no one set text. The course will use the following books

For detailed financial applications:

1. S.R. Pliska, I*ntroduction to Mathematical Finance: Discrete Time Models*, Blackwell Publishers (1997) [main resource for the first half of the course]

2. N.H. Bingham and R. Kiesel, *Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives*, Springer (1998).

3. D. Lamberton and B. Lapeyre, *Introduction to stochastic calculus applied to finance*, Chapman \& Hall (1996).

For mathematics behind the subject:

4. R. Durrett, *Essentials of Stochastic Processes,* Springer (1999)

5. Bhattacharya & Waymire, *Stochastic Processes With Applications*, Wiley (1991)

For less technical background material:

6. J.C Hull, *Options, futures and other derivatives*, Prentice Hall (1997).

7. M. Baxter and A. Rennie, *Financial Calculus*, Cambridge University Press (1996).

**Unit code:** MATH35400

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturers:** Dr Nick Whiteley and Dr Jia Wei Lim

## Pre-requisites

MATH11300 Probability 1, MATH11400 Statistics 1 and MATH20008 Probability 2.

## Co-requisites

None

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators of an approved type (non-programmable, no text facility) are allowed.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.