Synopsis for B10b: Mathematical Models of Financial Derivatives

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2B49)

Recommended Prerequisites:

Topology is helpful but not essential. B10a would be good background. For B10b, Part A Probability is a prerequisite.

Overview

The course aims to introduce students to mathematical modelling in financial markets. At the end of the course the student should be able to formulate a model for an asset price and then determine the prices of a range of derivatives based on the underlying asset using arbitrage free pricing ideas.

Learning Outcomes

Students will have a familiarity with the mathematics behind the models and analytical tools used in Mathematical Finance. This includes being able to formulate a model for an asset price and then determining the prices of a range of derivatives based on the underlying asset using arbitrage free pricing ideas.

Synopsis

Introduction to markets, assets, interest rates and present value; arbitrage and the law of one price: European call and put options, payoff diagrams. Probability spaces, random variables, conditional expectation, discrete-time martingales. The binomial model; European and American claim pricing.

Introduction to Brownian motion and its quadratic variation , continuous-time martingales, informal treatment of Itô's formula and stochastic differential equations. Discussion of the connection with PDEs through the Feynman–Kac formula.

The Black–Scholes analysis via delta hedging and replication, leading to the Black–Scholes partial differential equation for a derivative price. General solution via Feynman–Kac and risk neutral pricing, explicit solution for call and put options.

Extensions to assets paying dividends, time-varying parameters. Forward and future contracts, options on them. American options, formulation as a free-boundary problem. Simple exotic options. Weakly path-dependent options including barriers, lookbacks and Asians.

1. S.E Shreve, Stochastic Calculus for Finance, vols I and II, (Springer 2004).
2. T. Bjork, Arbitrage Theory in Continuous Time (Oxford University Press, 1998).
3. P. Wilmott, S. D. Howison and J. Dewynne, Mathematics of Financial Derivatives (Cambridge university Press, 1995).
4. A. Etheridge, A Course in Financial Calculus (Cambridge University Press, 2002).

Background on Financial Derivatives

1. J. Hull, Options Futures and Other Financial Derivative Products (4th edition, Prentice Hall, 2001).