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Financial Mathematics (MATH 35400)

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Administrative Information

  1. Unit number and title: MATH 35400 Financial Mathematics
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 08/09
  5. First Given in this form: 2000
  6. Unit Organiser: Arne Kovac
  7. Lecturer: Dr A Kovac and Dr O Johnson
  8. Teaching block: 2
  9. Prerequisites: Probability 1, Statistics 1, and Applied Probability 2.

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.

General Description of the Unit

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.

Teaching Methods

Lectures, supplemented by directed reading and supported by examples sheets. There will also be problems classes as required.

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.

Assessment Methods

90% of the assessment mark for Financial Mathematics is calculated from 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. 10% of the final mark will come from a timed one-hour open-book in-class assessment, based on directed reading.

Award of Credit Points

Credit points are gained by:

  • either passing the unit (getting a final assessment mark of 40 or over),
  • or getting an assessment mark of 30 or over, and also handing in satisfactory attempts at seven homework assignments.

Transferable Skills

Ability to compute prices of basic financial instruments
Mathematical modelling skills
Problem solving

Texts

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).

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.