# On extension of the Markov chain approximation method for computing Feynman--Kac type expectations

## TL;DR

This paper extends a Markov chain approximation method with Brownian bridge correction to efficiently compute Feynman--Kac expectations, especially for option pricing, demonstrating high convergence rates in numerical experiments.

## Contribution

The paper introduces an extension of the Markov chain approximation method to compute Feynman--Kac expectations for diffusion processes, applicable to option pricing.

## Key findings

- Convergence rate of $O(n^{-2})$ for smooth integrands.
- Effective extension to path-dependent functionals.
- Numerical experiments confirm high accuracy.

## Abstract

An efficient discrete time and space Markov chain approximation employing a Brownian bridge correction for computing curvilinear boundary crossing probabilities for general diffusion processes was recently proposed in Liang and Borovkov (2021). One of the advantages of that method over alternative approaches is that it can be readily extended to computing expectations of path-dependent functionals over the event of the process trajectory staying between two curvilinear boundaries. In the present paper, we extend the scheme to compute expectations of the Feynman--Kac type that frequently appear in option pricing. To illustrate our approximation scheme, we apply it in three special cases. For sufficiently smooth integrands, numerical experiments suggest that the proposed approximation converges at the rate $O(n^{-2})$, where $n$ is the number of steps on the uniform time grid used