GREEN2	Instantons and the Financial Markets
Type	Theoretical
#students	1
Orientation	Why is the scientific problem of interest at all? The mathematical techniques of quantum mechanics - perturbation theory, Feynman diagrams etc - flow from deep physical understanding of the nature of quantum mechanical reality. Viewed mathematically, they may be understood as a set of tools for solving differential equations. It can sometimes prove useful to translate a particular problem that is unrelated to quantum mechanics to a form where it looks as though it is in order to make use of these techniques. In this problem we will investigate the application of some such techniques to problems in finance.
How	How is the research going to shed light on the given problem?. In recent years it has proven particularly useful to translate calculations of extreme events in population dynamics into the form of quantum mechanical instanton calculations - or tunnelling event probabilities. These techniques can be used to calculate the extinction probability for small populations and the stability of ecological niches entirely driven by fluctuations
What	What is the specific thing that the student will do, and how does it fit inside the overall project? In this project, we will attempt to apply these ideas to financial situations where rare events encode the risks of particular investments. We will study the bankruptcy probability of small businesses and investigate whether instanton calculations can be used to shed light upon Value at Risk (VaR) - an important measure of the risk associated with derivatives.
Special Knowledge	The project requires a good mastery of mathematical physics and particularly of quantum mechanics. It will be a combination of analytical and numerical, with simulations run in either mathematica or simple Fortran as preferred by the student.
Supervisor	Prof. Andrew Green andrew.green@ucl.ac.uk