Paper: arXiv 2406.10648

Authors: Hélène Cossette, Etienne Marceau, Alessandro Mutti, Patrizia Semeraro

Abstract

Building on the one-to-one relationship between generalized FGM copulas and multivariate Bernoulli distributions, we prove that the class of multivariate distributions with generalized FGM copulas is a convex polytope. Therefore, we find sharp bounds in this class for many aggregate risk measures, such as value-at-risk, expected shortfall, and entropic risk measure, by enumerating their values on the extremal points of the convex polytope. This is infeasible in high dimensions. We overcome this limitation by considering the aggregation of identically distributed risks with generalized FGM copula specified by a common parameter $p$. In this case, the analogy with the geometrical structure of the class of Bernoulli distribution allows us to provide sharp analytical bounds for convex risk measures.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of generalized FGM copulas and convex bounds using multivariate Bernoulli distributions and convex polytopes. While mathematically sophisticated with solid proofs, it lacks real-world empirical validation or extensive backtesting on historical market data.

Research Flowchart

  flowchart TD
    A[Goal: Find Sharp Bounds for Aggregate Risk Measures under Generalized FGM Copulas] --> B[Map Generalized FGM Copulas to Multivariate Bernoulli Distributions]
    B --> C[Establish Geometric Structure: Class forms a Convex Polytope]
    C --> D{High Dimension?}
    D -- Yes: Infeasible --> E[Assume Identically Distributed Risks with Common Parameter p]
    D -- No --> F[Enumerate Values at Extremal Points of Polytope]
    E --> G[Derive Analytical Bounds for Convex Risk Measures: VaR, ES, Entropic]
    F --> G
    G --> H[Outcome: Tractable & Sharp Bounds for Multi-Asset Aggregate Risk]

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0 of 58 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.