Implementing quasi-Monte Carlo simulations with linear transformations

Abstract

The curse of dimensionality limits the accuracy of the quasi-Monte Carlo (QMC) method in high-dimensional problems. Imai and Tan (Proceedings of the 2002 winter simulation conference, 2002; Monte Carlo and Quasi-Monte Carlo Methods 2002, pp 275–292, Springer, Berlin, 2004; J Comput Finance 10(2):129–155, 2007) have proposed a dimension reduction technique, named linear transformation (LT), aiming to improve the efficiency of the QMC method. We investigate this approach in detail and make it more convenient. We implement a faster QR decomposition that considerably reduces the computational burden. The efficacy of our algorithm is illustrated by considering two high-dimensional option pricing problems: Asian basket options in the Black–Scholes (BS) model and Asian options in the Cox–Ingersoll–Ross (CIR) model. We employ a QMC generator only for the components selected by the LT construction and use Latin hypercube sampling (LHS) for all the others. Finally, we compare our results to those obtained by different random number generators and standard algorithms; subsequently, we benchmark our computational times against those presented in Imai and Tan (J Comput Finance 10(2):129–155, 2007).