| Unit name | Financial Mathematics |
|---|---|
| Unit code | MATH35400 |
| Credit points | 20 |
| Level of study | H/6 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Professor. Johnson |
| Open unit status | Not open |
| Pre-requisites |
Level 1 Analysis, Probability and Statistics |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
In 1973 Black and Scholes solved the problem of pricing a basic financial derivative, the European call option. Since then there has been an explosion of trade in and the different types of such financial instruments. These are financial products based on an underlying asset and by making assumptions about the market it is possible to determine a unique fair arbitrage free price. This course will develop the mathematical ideas which underly the problem of pricing options. We will model stock prices as stochastic processes and develop both continuous and discrete time models for option pricing. By developing the theory of martingales we will see how to express option pricing problems mathematically and see how to calculate prices. The aim is to understand the ideas at a practical level and detailed proofs of the more technical continuous time material will be omitted.
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.
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 and the lognormal distribution.
Continuous martingales, the basic tools of stochastic calculus, Ito formula, Girsanov theorem, without proofs.
Application to option pricing, 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.
At the end of the course the student should be able to
Transferable Skills:
Lectures, supplemented by directed reading and supported by examples sheets. There may also be problems classes as required.
The assessment mark for Financial Mathematics:
100% by means of a 2 ½-hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators of an approved type (non-programmable, no text facility) are allowed. Statistical tables will be provided.
There is no one set text. The course will use the following books
For detailed financial applications:
1. S.R. Pliska, Introduction 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).
