Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?

Abstract

Many problems in mathematical finance can be formulated as high-dimensional integrals, where the large number of dimensions arises from small time steps in time discretization and/or a large number of state variables. Quasi--Monte Carlo (QMC) methods have been successfully used for approximating such integrals. To understand this success, this paper focuses on investigating the special features of some typical high-dimensional finance problems, namely, option pricing and bond valuation. We provide new insight into the connection between the effective dimension and the efficiency of QMC, and we present methods to analyze the dimension structure of a function. We confirm the observation of Caflisch, Morokoff, and Owen that functions from finance are often of low effective dimension, in the sense that they can be well approximated by their low-order ANOVA (analysis of variance) terms, usually just the order-1 and order-2 terms. We explore why the effective dimension is small for many integrals from finance. By deriving explicit forms of the ANOVA terms in simple cases, we find that the importance of each dimension is naturally weighted by certain hidden weights. These weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher-order ANOVA terms. We study thevariance ratios captured by low-order ANOVA terms and their asymptotic properties as the dimension tends to infinity, and we show that with the increase of dimension the lower-order terms continue to play a significant role and the higher-order terms tend to be negligible. This provides some insight into high-dimensional problems from finance and explains why QMC algorithms are efficient for problems of this kind.