DEPARTMENT of  MATHEMATICS   Applied Research Areas 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 have a particular interest in option pricing theory.

Option Pricing Theory - some background

European call options

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?'

European put options

A European put option is a contract between two parties (the holder and the underwriter), such that:

On a specified date in the future, the holder may sell a prescribed asset for a prescribed amount to the underwriter.

Again, such an option has value.

The seminal work of Black & Scholes (1973) provided a deterministic valuation of options of this type. Since problems of this class generally lead to (backwards) parabolic type partial differential equations (indeed, in many instances the problem can be transformed to the heat conduction equation!), many classical techniques of Applied Mathematics can be employed in this new and exciting area.

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 nonlinear (free-boundary) problem, which must generally be tackled computationally. Fortunately, again, techniques widely employed in the past to solve (numerically) parabolic partial differential equations are widely applicable.

Members of staff involved:

Prof. Peter Duck

Dr. David Newton (Manchester Business School).

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