Prof Martin Anthony COL 3.13

This course is available on the BSc in Financial Mathematics and Statistics. This course is available as an outside option to students on other programmes where regulations permit. This course is available with permission to General Course students.

Students must have completed Measure Theoretic Probability (MA321).

This course provides the mathematical tools of stochastic calculus and develops the Black‐Scholes theory of financial markets. It covers the following topics. Continuous-time stochastic processes, filtrations, stopping times, super-, sub‐ and martingales, examples. Brownian motion, properties, Markov property. Construction of the stochastic/Ito integral, simple integrands, Derivation of Ito's isometry, Ito processes, Derivation of Ito's formula, stochastic differential equations. Changes of probability measure, Radon‐Nikodym derivative, Bayes' rule, Girsanov's theorem. Black‐Scholes model: self-financing portfolios, fundamental theorem of asset pricing, risk neutral measure, existence of replicating strategies via martingale representation, risk neutral valuation of European contingent claims, Black‐Scholes formula, Black‐Scholes PDE, Greeks, delta hedging. PDE techniques for pathwise derivatives: barrier and Asian options. Implied volatility, Dupire’s formula, local volatility, basic idea of calibration, variations of the Black-Scholes model.

22 hours of lectures and 10 hours of classes in the LT.

Students will be expected to produce 10 problem sets in the LT.

Lecture notes will be provided.

The following books may be useful.

T. Bjork, Arbitrage Theory in Continuous Time, Oxford Finance, 2004;

A. Etheridge, A Course in Financial Calculus, CUP, 2002;

M Baxter & A Rennie, Financial Calculus, CUP, 1996;

P. Wilmott, S. Howison & J. Dewynne, The Mathematics of Financial Derivatives, CUP, 1995;

J Hull, Options, Futures and Other Derivatives, 6th edition, Prentice-Hall, 2005.

D. Lamberton & B. Lapeyre, Introduction to stochastic calculus applied to finance, 2nd edition, Chapman & Hall, 2008.

S. E. Shreve, Stochastic Calculus for Finance. Volume I: The Binomial Asset Pricing Model. Springer, New York, 2004.

S. E. Shreve, Stochastic Calculus for Finance. Volume II: Continuous-Time Models. Springer, New York, 2004.

Other (100%).

Exam (100%) in the ST.