Polynomial processes and their applications to mathematical Finance

TL;DR

This paper introduces m-polynomial Markov processes that simplify moment calculations, encompassing many models in finance, and demonstrates their applications in statistical estimation, option pricing, and variance reduction techniques.

Contribution

It defines a new class of polynomial processes that unify and extend existing models, enabling efficient computation of moments for financial applications.

Findings

Includes affine and Lévy-driven processes within the m-polynomial class.

Facilitates efficient moment calculations via matrix exponentials.

Enhances Monte Carlo methods with variance reduction techniques.

Abstract

We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as L\'evy-driven SDEs with affine vector fields. Thus, many popular models such as exponential L\'evy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.