Quasi-Monte Carlo Methods in Finance

This second reference Mathieu Gerber gave me in the quest for educating myself about QMC, is paper by Pierre L’Ecuyer from the Winter Simulation Conference in 2004. It was much clearer as a tutorial (for me) as compared to the Art Owen paper. Maybe because it didn’t contain so much ANOVA. Or maybe because I was more used to ANOVA from Arts paper.

This paper specifically and quite transparently treats different constructions for low discrepancy point sets, in particular digital nets and their special cases. On the other hand, randomization procedures are discussed, which sometimes seem to be very specialized to the sequence used. One seemingly general transform after randomization called the baker transformation results in surprisingly high variance reduction of order O(n^{-4+\epsilon}). The transformation being to replace the uniform coordinate u \in [0,1) by 2u for u\leq 0.5 and 2(1-u) else.

In the examples L’Ecuyer mentions that using an Eigenzerlegung of covariance matrices (i.e. PCA) results in much higher variance reductions as compared to using Cholesky factors. Which he attributes to dimension reduction – a naming I find odd, as the complete information is retained (as opposed to, e.g. tossing the components with lowest Eigenvalue). My intuition is that maybe the strong empirical gains with PCA might rather be attributed to the fact that Eigenvectors are orthogonal, making this decomposition as close as possible to QMCs beloved unit hypercube.

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