Research in Mathematical Finance

Mathematical finance is one of the fastest growing areas of mathematics, and is now used heavily in the modern banking and corporate world. We now have a range of ongoing activities with a particular interest in option pricing theory.

The simplest type of option is a European call option, which is a contract between two parties (the holder and the underwriter) such that, on a specified date in the future, the holder may purchase a prescribed asset for a prescribed amount from the underwriter. Since the holder has the choice whether or not to exercise the option, an option has value - the key question is how much?

The work of Black & Scholes (1973) provided a deterministic valuation of options of this type. Since problems of this class generally lead to parabolic type partial differential equations, many classical techniques of applied mathematics can be employed.

Many other types of option exist - for example with American options the holder has the right to exercise the option at any time during the lifetime of the option. Because the holder then has additional rights, the value of these options is always worth more than corresponding European options. From a mathematical point of view, American options are challenging, and lead to a non-linear (free-boundary) problem, which must generally be tackled computationally.

The group's scale of activities has grown sharply in recent years, and there are a number of important collaborations especially with the Manchester Business School. The range of interests include mathematical modelling (e.g. of interest rates, credit risk, bond pricing, foreign exchange), developing and improving relevant numerical methods (trees, Monte Carlo, Finite Difference and highly accurate quadrature methods) and analysis. As mentioned, the group places significant emphasis on employing powerful applied mathematical ideas and tools to finance problems.

An example of this has been the successful use of singular perturbation techniques to option and bond pricing problems, for which it has been shown that the concept of boundary and shear layers is applicable.

Any queries, corrections, omissions or comments regarding the maintenance of the Mathematical Finance microsite should be directed to either Edwin Broni-Mensah or Sebastian Law.