Ludovic Cales, Apostolos Chalkis, Ioannis Z. Emiris, Vissarion Fisikopoulos
arXiv 15 Mar 2018 · cs.CG · 10 citations (OpenAlex)
arXiv:1803.05861 · PDF · DOI · OpenAlex · Extracted main text
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. 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; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple; 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; 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. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | I.Z. Emiris and V. Fisikopoulos (2014) Practical polytope volume approximation self | 0.737 | 3 | 2 | 100% |
| 2 | O. Ledoit and M. Wolf (2004) Honey, I shrunk the sample covariance matrix | 0.644 | 2 | 2 | 100% |
| 3 | L. Lovász (1999) Hit-and-run mixes fast | 0.644 | 2 | 2 | 100% |
| 4 | A.M. Mathai (2007) On linear combinations of independent exponential variables | 0.644 | 2 | 2 | 100% |
| 5 | M. Billio, M. Getmansky, and L. Pelizzon (2012) Dynamic risk exposures in hedge funds | 0.511 | 2 | 1 | 100% |
| 6 | J. Lawrence (1991) Polytope volume computation | 0.511 | 2 | 1 | 100% |
| 7 | I. Pouchkarev (2005) Performance evaluation of constrained portfolios | 0.511 | 2 | 1 | 100% |
| 8 | Y. Abbasi-Yadkori, P.L. Bartlett, V. Gabillon, and A. Malek (2017) Hit-and-run for sampling and planning in non-convex spaces | 0.405 | 1 | 1 | 100% |
| 9 | M. Maswood Ali (1973) Content of the frustum oa a simplex | 0.405 | 1 | 1 | 100% |
| 10 | M. Billio, L. Calès, and D. Guéguan (2011) A cross-sectional score for the relative performance of an allocation | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 34 scored citations.