MASCOS Lectures on Mathematical Finance

Date: 

Monday, 13th March 2006

In the week beginning March 12 Kostya Borovkov (University of Melbourne, and MASCOS Chief Investigator) will give four lectures on:

"Introduction to arbitrage-free pricing theory".

The lectures are aimed at mathematicians who may not be properly educated in the fundamentals, but who really have a need to know. They
are open to everyone in the School. There is a lot to cover, so the speed is likely to be considerable.

The formal requiremmnt is a knowledge of probability to second year leavel, and of analysis to the level of Lebesgue integration.

The topics to be covered are given below. The lectures will be held on

Mo 2-3 RC-4083
Tu 2-3 RC-3085
We 3-4 RC-3085
Th 2-3 RC-3085

Please let Ian Sloan (i.sloan@unsw.edu.au) know if you plan to attend, so that we have an idea of
numbers.

Four lectures:

1. Principles of no-arbitrage pricing in discrete markets.

Binomial asset pricing model. Contingent claims (derivative securities),options. Hedging and claim price. The replicating strategy and its value as an expectation under an "equivalent martingale measure". Finite single period markets: no-arbitrage pricing & completeness theorems. Multiperiod binomial markets: self-financing trading strategies, replication. Black-Scholes formula as a limit of the binomial one.

2. Conditional expectations and martingales.

Conditional expectations. Martingales. Stopping times. American derivative securities. Optional sampling theorem. Martingales and claims pricing.

3. The Brownian motion process and stochastic calculus.

Brownian motion (a.k.a. Wiener process) & Levy processes. Path properties. Markov property. Geometric Brownian motion. Ito integral. Ito formula. Stochastic differential equations (SDE's).

4. The Black-Scholes market.

The Black-Scholes SDE. The equivalent martingale measure. No-arbitrage & completeness of the market. Self-financing strategies & hedging.
Black-Scholes formula. Barrier options.