  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.
