Enhancing Quasi-Monte Carlo Simulation by Minimizing Effective Dimension for Derivative Pricing

摘要

Many problems in derivative pricing can be formulated as high-dimensional integrals. Many of them do not have closed-form solutions and have to be estimated by numerical integrations such as Monte Carlo or quasi-Monte Carlo (QMC) methods. Since the quasi-random points used for QMC simulation have perfect projections at the first few dimensions, reducing the effective dimension of the integrands can improve the efficiency of QMC. In this paper, based on the first-order Taylor approximations of the functions at Gaussian sample points, we propose a new general method based on principal component analysis (PCA) to reduce the effective dimensions of the functions. Rather than aiming at decomposing the covariance matrix of the Brownian motions as in the traditional PCA, the new method implements PCA on the gradients of the functions at sample points and then an orthogonal transformation is found to reduce the effective dimensions. Numerical experiments show that by using the new dimension reduction method, a significant efficient improvement of QMC can be achieved on pricing exotic options and mortgage-backed securities.