On Stochastic Partial Differential Equations and their applications to Derivative Pricing through a conditional Feynman-Kac formula

TL;DR

This paper develops a novel SPDE framework for derivative pricing using a conditional Feynman-Kac formula, addressing irregularities with new techniques and proposing mixed Monte-Carlo PDE methods.

Contribution

It introduces a conditional SPDE approach for derivative pricing, incorporating backward filtration and irregularity handling, along with new numerical methods.

Findings

Derived a conditional SPDE for derivative pricing.

Addressed irregularities using novel techniques.

Proposed mixed Monte-Carlo PDE numerical methods.

Abstract

The price of a financial derivative can be expressed as an iterated conditional expectation, where the inner term conditions on the future of an auxiliary process. We show that this inner conditional expectation solves an SPDE (a 'conditional Feynman-Kac formula'). The problem requires conditioning on a backward filtration generated by the noise of the auxiliary process and enlarged by its terminal value, leading us to search for a backward Brownian motion here. This adds a source of irregularity to the SPDE which we tackle with new techniques. Lastly, we establish a new class of mixed Monte-Carlo PDE numerical methods.