Closed-form approximations of moments and densities of continuous–time Markov models

Abstract

This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump–diffusions. The proposed expansions extend the ones in Kristensen and Mele (2011) to cover general Markov processes, and nest transition density and option price expansions recently developed in the literature, thereby connecting seemingly different ideas in a unified framework. We show how the general expansion can be implemented for fully general jump–diffusion models. We provide a new theory for the validity of the expansions which shows that series expansions are not guaranteed to converge as more terms are added in general once the time span of interest gets larger than some model–specific threshold. Thus, these methods should be used with caution when applied to problems with a larger time span of interest, such as long-term options or data observed at a low frequency. At the same time, the numerical studies in this paper demonstrate good performance of the proposed implementation in practice when applied to pricing options with time to maturity below three months. Thus, our expansions are particularly well suited for pricing ultra-short-term (such as “zero–day”) options.