Applied Mathematics Seminar

Closed-form option prices are known for only a few simple models, including the classical Black-Scholes formulas for vanilla options (European call and put options). However no analytic formulas are known for many non-vanilla flavoured options (e.g. American, exotic and path-dependent options) and also for vanilla options driven by stochastic processes other than the simple geometrical Brownian motion of the Black-Scholes framework.
It is well-known that geometrical Brownian motion can be represented as as the limit of a simple random walk on a multiplicative binomial tree. Binomial trees or B-trees, provided they are recombinant, are easy to compute and can readily model many of the non-vanilla options. But can they also model more complicated processes than geometrical Brownian motion ?
The seminar explores this question in detail. By introducing a new generalised binomial jump process it is shown that any Ito process can be represented as the limit of a generalised recombinant B-tree. This provides for the development of a simple numerical algorithm to compute both vanilla and non-vanilla option prices for arbitrary Ito processes.
This algorithm is described and used to price, as a demonstration of the method, American options driven by a CEV (constant elasticity of variance) stock process.