Fast orthogonal transforms for pricing derivatives with quasi-Monte Carlo

There are a number of situations where, when computing prices of financial derivatives using quasi-Monte Carlo (QMC), it turns out to be beneficial to apply an orthogonal transform to the standard normal input variables. Sometimes those transforms can be computed in time O(nlog(n)) for problems depending on n input variables. Among those are classical methods like the Brownian bridge construction and principal component analysis (PCA) construction for Brownian paths. Building on preliminary work by Imai and Tan (2007) as well as Wang and Sloan (2011), where the authors try to find optimal orthogonal transform for given problems, we present how those transforms can be approximated by others that are fast to compute. We further present a new regression-based method for finding a Householder reflection which turns out to be very efficient for a wide range of problems. We apply these methods to several very high-dimensional examples from finance.