Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises

Abstract

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as 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 constructed by computing volumes of convex bodies.We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions, which are the method of choice for intersections with a single hyperplane; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple with additional structure; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed in the given body, such as a second-order cone program; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. * The views expressed are those of the authors and do not necessarily reflect official positions of the European Commission. 1 Calès acknowledges the financial support of the European Commission through its Proof-of-Concept program. 2 Emiris is partially supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No 734242 (Project LAMBDA).

Efficient computation of the volume of a polytope in high-dimensions using Piecewise Deterministic Markov Processes

“…In robotics, the computation of reachable sets for time-varying linear systems is based upon special polytopes called zonotopes [7]. In finance, the cross sectional score of a portfolio is defined from the intersection between a simplex (representing assets) and hyperplanes or ellipsoids [8]. In artificial intelligence, ReLU networks can be characterized by the conjunction of a set of linear inequalities which define a polytope in the input domain known as the activation condition [9].…”

Exchangeable Bernoulli distributions: high dimensional simulation, estimate and testing

“…For k-order moments the ratio of the volumes in Eq. (2.21) can be computed using an exact and iterative formula, see [6] and [7].…”

“…We recall that in this case the ratio of the volumes in Eq. (2.21) can be computed using an exact and iterative formula, see [6] and [7]. , where ∆ has been chosen equal to (max(µ 2 ) − min(µ 2 ))/10000.…”