Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises

Abstract

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined by the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be estimated by computing volumes of convex bodies.

We design and implement practical algorithms in the exact and approximate setting, and experimentally juxtapose them in order to study the trade-off of exactness and accuracy for speed. We also experimentally find an efficient parameter-tuning to achieve a sufficiently good estimation of the probability density of each copula. Our C++ software, based on Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.