Correlators of Polynomial Processes

TL;DR

This paper derives an explicit formula for computing cross-moments of polynomial jump-diffusion processes, facilitating applications in financial pricing and risk management by enabling explicit sensitivity analysis.

Contribution

It introduces a linear combination formula involving exponentials of the generator matrix for correlators in polynomial processes, extending existing moment formulas.

Findings

Explicit correlator formula for polynomial jump-diffusions

Application to path-dependent options and stochastic volatility models

Closed-form expressions for Greeks in financial models

Abstract

In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for polynomial processes. The developed framework can, for example, be applied in financial pricing, such as for path-dependent options and in a stochastic volatility models context. In applications to options, having closed and compact formulations is attractive for sensitivity analysis and risk management, since Greeks can be derived explicitly.