Projection of Functionals and Fast Pricing of Exotic Options

TL;DR

This paper introduces a novel approach using the Karhunen-Loève expansion and path signatures to efficiently approximate path functionals, significantly improving the speed and accuracy of exotic options pricing.

Contribution

It presents the KLMC algorithm and explores path signatures as new tools for fast, accurate pricing of path-dependent derivatives.

Findings

KLMC enables rapid, accurate exotic option pricing.

Path signatures offer an effective alternative for functional approximation.

The methods outperform traditional techniques in computational efficiency.

Abstract

We investigate the approximation of path functionals. In particular, we advocate the use of the Karhunen-Lo\`eve expansion, the continuous analogue of Principal Component Analysis, to extract relevant information from the image of a functional. Having accurate estimate of functionals is of paramount importance in the context of exotic derivatives pricing, as presented in the practical applications. Specifically, we show how a simulation-based procedure, which we call the Karhunen-Lo\`eve Monte Carlo (KLMC) algorithm, allows fast and efficient computation of the price of path-dependent options. We also explore the path signature as an alternative tool to project both paths and functionals.