Polynomial Approximation of Discounted Moments

TL;DR

This paper presents a polynomial-based approximation method for discounted moments in stochastic processes, improving accuracy in financial product pricing and validated through analytical and simulation comparisons.

Contribution

It introduces a high-order power series expansion approach for approximating discounted moments, connecting with polynomial process theory and applicable to various financial models.

Findings

Approximation error reduces to near machine precision in cases with analytical solutions.

Method effectively approximates moments for bond and credit derivative pricing.

Validation through Monte Carlo simulations confirms accuracy in non-analytical cases.

Abstract

We introduce an approximation strategy for the discounted moments of a stochastic process that can, for a large class of problems, approximate the true moments. These moments appear in pricing formulas of financial products such as bonds and credit derivatives. The approximation relies on high-order power series expansion of the infinitesimal generator, and draws parallels with the theory of polynomial processes. We demonstrate applications to bond pricing and credit derivatives. In the special cases that allow for an analytical solution the approximation error decreases to around 10 to 100 times machine precision for higher orders. When no analytical solution exists we tie out the approximation with Monte Carlo simulations.