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In this article, we propose a new numerical approach to high-dimensionalpartial differential equations (PDEs) arising in the valuation of exoticderivative securities. The proposed method is extended from Reisinger andWittum (2007) and uses principal component analysis (PCA) of the underlyingprocess in combination with a Taylor expansion of the value function intosolutions to low-dimensional PDEs. The approximation is related to anchoredanalysis of variance (ANOVA) decompositions and is expected to be accuratewhenever the covariance matrix has one or few dominating eigenvalues. A mainpurpose of the present article is to give a careful analysis of the numericalaccuracy and computational complexity compared to state-of-the-art Monte Carlomethods on the example of Bermudan swaptions and Ratchet floors, which areconsidered difficult benchmark problems. We are able to demonstrate that forproblems with medium to high dimensionality and moderate time horizons thepresented PDE method delivers results comparable in accuracy to the MC methodsconsidered here in similar or (often significantly) faster runtime.

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