Numerical Valuation of Derivatives in High-Dimensional Settings Via Partial Differential Equation Expansions

Abstract

We propose a new numerical approach to solving high-dimensional partial differential equations (PDEs) that arise in the valuation of exotic derivative securities. The proposed method is extended from the work of Reisinger and Wittum and uses principal component analysis 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 decompositions and is expected to be accurate whenever the covariance matrix has one or few dominating eigenvalues. We give a careful analysis of the numerical accuracy and computational complexity compared with state-of-the-art Monte Carlo methods, using Bermudan swaptions and ratchet floors, which are considered difficult benchmark problems, as examples. We demonstrate that, for problems with medium to high dimensionality and moderate time horizons, the PDE method presented delivers results comparable in accuracy to the Monte Carlo methods considered here in a similar or (often significantly) faster run time.