The checkerboard copula and dependence concepts

TL;DR

This paper introduces the checkerboard copula, a maximum entropy copula that preserves dependence structure, useful for modeling in scenarios with non-continuous marginals and in calculating co-risk measures.

Contribution

It proposes the checkerboard copula as a novel approach for dependence modeling when marginals are not continuous, emphasizing its maximum entropy property and preservation of dependence information.

Findings

Checkerboard copula has the largest Shannon entropy among all copulas.

It preserves the dependence structure of the original data.

Numerical results show benefits in co-risk measure calculations.

Abstract

We study the problem of choosing the copula when the marginal distributions of a random vector are not all continuous. Inspired by four motivating examples including simulation from copulas, stress scenarios, co-risk measures, and dependence measures, we propose to use the checkerboard copula, that is, intuitively, the unique copula with a distribution that is as uniform as possible within regions of flexibility. We show that the checkerboard copula has the largest Shannon entropy, which means that it carries the least information among all possible copulas for a given random vector. Furthermore, the checkerboard copula preserves the dependence information of the original random vector, leading to two applications in the context of diversification penalty and impact portfolios. The numerical and empirical results illustrate the benefits of using the checkerboard copula in the calculation of co-risk measures.