This module aims to demonstrate the advanced mathematical techniques underlying financial markets and the practical use of financial derivative products to analyse various problems arising in financial markets. Emphases are on the stochastic techniques, probability theory, Markov processes and stochastic calculus, together with the related applications

At the end of the module students should be able to do the following things.

1.    A critical awareness of current problems and research issues in the fields of probability and stochastic processes, stochastic analysis and financial mathematics.

2.    The ability to formulate stochastic cuclulus for the purpose of modelling particular financial questions.

3.    The ability to use appropriate mathematical tools and techniques in the context of a particular financial model. 

4.    The ability to read, understand and communicate research literature in the fields of probability, stochastic analysis and financial mathematics.

5.    The ability to recognise potential research opportunities and r esearch directions.

Brownian Motion: Definition of Brownian Motion; Brownian bridge; The Reflection Principle and Scaling.

Martingales: Change of measure, Radon-Nikodym derivative; conditional expectation (definition and properties); martingales (discrete and continuous time); stopping times; applications of stopping times; Doob’s inequality.

Stochastic calculus: Ito’s processes and stochastic differential. Definition and properties of Ito’s integral; Ito’s formula; integration by parts; stochastic Fubini theorem; Girsanov theorem; the Brownian martingale representation theorem.

Stochastic differential equations: Markov property; stochastic differential equations (SDEs); diffusions and the PDE connection; Feynman-Kac representation.

Applications:

·      Application to Optimal Control: optimal stopping problem; Hamilton-Jacob-Bellman equations.

·      Application to Mathematical Finance: risk-neutral pricing; exotic options; derivative securities.

·      Application to Biology: epidemic, competition and predation processes; population genetics process; expected time to extinction and first passage time.