Abstract

We construct a flexible and numerically tractable class of asset models by firstly choosing a bivariate diffusion process $(U,Y)$, and then defining the price of the asset at time $t$ to be the value of $Y$ when $U$ first exceeds $t$. Such price processes will typically have jumps; conventional pricing methodologies would try to solve a PIDE , which can be numerically problematic, but using the fact that the pricing problem is embedded in a two-dimensional diffusion, we are able to exploit efficient methods for two-dimensional diffusion equations to find prices. Models with time dependence (that is, where the bivariate diffusion is $U$-dependent) are no more difficult in this approach.

Pricing a European option for a model in this class consists of solving a linear second order elliptic PDE . Models in this class range from the most parsimonious, with few parameters, to those which can match the observed term structure of implied volatility. This allows flexibility. We construct an example model which accounts for so-called volatility events, caused by the scheduled release of pertinent information, such as unemployment figures, inflation rates and economic growth rates.