Estimating The Effective Dimension Of High-Dimensional Finance Problems Using Sobol’ Sensitivity Indices

Problems in many disciplines, such as physics, chemistry, and finance, can be modelled as integrals of high dimensions (hundreds or even thousands). Quasi-Monte Carlo (QMC) methods, which perform sampling using a more uniform point set than that used in MC, have been successfully used to approximate multivariate integrals with an error bound of size O((logN)kN−1) or even O((logN)kN−3/2), where N is the size of the sample and k depends on the dimension of the problem. This suggests an outperformance over the standard MC whose error bound is only O(N−1/2). But for high dimensional problems, this outperfomance might not appear at feasible sample sizes due to the dependence of the QMC convergence rate on the dimension of the problem. However, around 1993, it was found by researchers at Columbia University that QMC provides better convergence rates than MC for very high-dimensional problems in finance and in physics as well. This may be explained by the fact that the integrands in these problems have low “effective dimension” properties that interact positively with the properties of the point set used by the QMC method. To understand the efficiency of QMC, this paper uses Sobol’ method for global sensitivity analysis to investigate features of specific finance problems, digital option pricing and mortgage-backed securities, in dimensions as high as 360. Using Sobol’ sequences, we estimate the low-order Sobol’ sensitivity indices of these problems and estimate their effective dimensions accordingly. We also examine the efficiency of the Brownian Bridge technique in reducing the estimated effective dimension.