Numerical Valuation of Derivatives in High-Dimensional Settings via PDE Expansions

TL;DR

This paper introduces a PDE-based numerical method for valuing high-dimensional derivatives, combining PCA and Taylor expansions to efficiently approximate solutions with accuracy comparable to Monte Carlo methods.

Contribution

It extends existing PDE techniques with PCA and Taylor expansions, providing a new approach for high-dimensional derivative valuation with improved computational efficiency.

Findings

Achieves comparable accuracy to Monte Carlo methods

Demonstrates faster runtime for medium to high-dimensional problems

Effective for complex derivatives like Bermudan swaptions and Ratchet floors

Abstract

In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and uses principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. The approximation is related to anchored analysis of variance (ANOVA) decompositions and is expected to be accurate whenever the covariance matrix has one or few dominating eigenvalues. A main purpose of the present article is to give a careful analysis of the numerical accuracy and computational complexity compared to state-of-the-art Monte Carlo methods on the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems. We are able to demonstrate that for problems with medium to high dimensionality and moderate time horizons the presented PDE method delivers results comparable in accuracy to the MC methods considered here in similar or (often significantly) faster runtime.