Flexible Dependence Modeling for VaR Using Bernstein’s Copula

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Résumé

In this work, we establish theoretical and numerical links between the Bernstein copula, the Bernstein polynomial, and Value at Risk (VaR) in the context of dependence approximation in financial risk. Thanks to the smoothing induced by this copula, we express the portfolio VaR as the quantile of an approximate joint distribution, highlighting the effect of Bernstein smoothing on the estimation of extreme risk. In this work, we establish theoretical and numerical links between the Bernstein copula, the Bernstein polynomial, and Value at Risk (VaR) in the context of dependence approximation in financial risk. Thanks to the smoothing induced by this copula, we express the portfolio VaR as the quantile of an approximate joint distribution, highlighting the effect of Bernstein smoothing on the estimation of extreme risks. The relationship is further refined by representing the Bernstein copula as a morphism of these polynomials, allowing us to exploit their approximation properties to analyze the copula’s characteristics, such as its regularity and convergence. Finally, using the Bienaym´e-Tchebychev inequality, we bound the infinite norm of the difference between the target continuous function and the corresponding Bernstein polynomial.

Mots-clés

copulas, Bernstein copula; multivariate dependence; degree of discor- dance; value-at-Risk; Bienaymé-Tchebychev inequality.