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

TL;DR

This paper introduces power series expansions for moments, transition densities, and option prices of continuous-time Markov processes, including jump-diffusions, unifying previous approaches and demonstrating practical effectiveness with caution on convergence issues.

Contribution

It extends existing series expansion methods to general Markov processes, including jump-diffusions, and provides a unified theoretical framework connecting prior models.

Findings

Series expansions perform well with few terms in practice.

Convergence of the series is not guaranteed, requiring careful application.

Unified framework links different existing expansion methods.

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. We demonstrate that the class of expansions nests the transition density and option price expansions developed in Yang, Chen, and Wan (2019) and Wan and Yang (2021) as special cases, 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. Thus, these methods should be used with caution. At the same time, the numerical studies in this paper demonstrate good performance of the proposed implementation in practice when a small number of terms are included.