Non-Affine Option Pricing

The original Black-Scholes (BS) model introduced financial economists to continuous-time mathematics. But evidence quickly accumulated that the BS constant volatility lognormal diffusion is not adequate to describe the behavior of real world stock prices or, especially, interest rates. Recent work by Duffie and Singleton, and others, extended the returns processes for which options could be valued in more or less closed form to the affine jump-diffusion (AJD) class, which allows an arbitrary number of stochastic factors and a broad range of possible behavior. And yet, such models still impose significant limits on the ways that volatility can be made stochastic. This article shows how to break out of the AJD class to a much broader range of price processes. Chourdakis’s approach is to construct a continuous-time Markov chain that can approximate nearly any plausible state process. Moreover, beyond allowing option pricing for a richer set of price processes, the technique also offers improved computational efficiency for AJD-class models, relative to alternative techniques.