Unit name | Financial Mathematics 34 |
---|---|

Unit code | MATHM5400 |

Credit points | 20 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Dr. Kovac |

Open unit status | Not open |

Pre-requisites |
MATH11300 Probability 1, MATH11400 Statistics 1 and MATH20008 Probability 2 |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**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 unit builds from and applies ideas from Probability 2 and complements Financial Risk Management.

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

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

**Recommended**

- Nicholas H. Bingham and Rüdiger Kiesel,
*Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives*, Springer, 1998 - Damien Lamberton and Bernard Lapeyre,
*Introduction to Stochastic Calculus Applied to Finance*, Chapman & Hall, 1996 - S.R. Pliska, I
*ntroduction to Mathematical Finance: Discrete Time Models*, Blackwell Publishers, 1997

**Further**

For mathematics behind the subject

- R.N. Bhattacharya and Edward C. Waymire,
*Stochastic Processes With Applications*, Wiley, 1991 - Richard Durrett,
*Essentials of Stochastic Processes,*Springer, 1999

For less technical background material

- Martin Baxter and Andrew Rennie,
*Financial Calculus*, Cambridge University Press, 1996 - John Hull,
*Options, Futures and Other Derivatives*, Prentice Hall, 1997