*Lecturer:* Prof H Van Der Hart

**AMA3007 Financial Mathematics (2**^{nd} semester)

Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

**Introduction**

Mathematical skills are highly sought after in the financial services industries, and this employment sector remains a favoured destination for graduates. Around 40% of Mathematics graduates entering employment across the UK (see www.prospects.ac.uk for recent data) go into financial services, which includes, accountancy, retail and investment banking, mergers and acquisitions, insurance and actuarial work, capital market trading, and hedge fund employment, and so on.

At the low end of this sector, retail banking for example, a degree in mathematics is certainly not essential. This work is mainly concerned with simple arithmetic operations. However, at the high end of financial services, in a hedge fund for example, employers expect to see PhD-level qualifications in mathematics from their applicants along with excellent software skills. These mathematicians are involved in the business of derivative pricing and trading and earn salaries well over 100k. Derivatives are financial products (instruments as they are called in the trade) derived from assets that have an unpredictable price. The total outstanding notional value of derivatives contracts today has grown beyond a quadrillion dollars (that's 10^{15} to you and me). It is a perilous and lucrative business!

Derivatives were originally devised to avoid risk by providing an insurance on a risky asset. Nowadays, they are an essential part of risk taking in capital markets. Indeed the speculation in buying and selling these instruments, specifically credit derivatives, precipitated the current credit crunch. Of course, this trade relies upon knowing the fair price of a derivative. Pioneering work by Black, Merton and Scholes, showed that, under certain assumptions for the unpredictability of the asset, the price of the derivative obeys a partial-differential equation. The construction of such equations and their solution is where mathematicians come in!

The objective of the course is to provide an introduction to the mathematical techniques which can be applied to pricing problems for financial derivatives. Specifically, our focus is on stochastic calculus and the theory and practice of pricing simple derivatives such as contracts and options.

We are grateful to *First Derivatives plc* for their support of this course and the provision of prizes for the best examination performance.

**Contents**

Introduction to financial derivatives: forwards, futures and options. Future markets and prices. Option markets. Binomial models and the risk-free portfolio. Stochastic calculus and random walks. Black-Scholes equation. Pricing European options. Various option pricing models. Interest-rate derivatives. Credit derivatives. Swaps.

**Assessment**

Exam 70% Report 20% Presentation 10%