Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence

Abstract

Nelsen et al. [20] find bounds for bivariate distribution functions when there are constraints on the values of its quartiles. Tankov [25] generalizes this work by giving explicit expressions for the best upper and lower bounds for a bivariate copula when its values on a compact subset of [0 1] 2 are known. He shows that they are quasi-copulas and not necessarily copulas. Tankov [25] and Bernard et al. [3] both give sufficient conditions for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that both bounds are simultaneously copulas. Furthermore, we develop a novel application to quantitative risk management by computing bounds on a bivariate risk measure. This can be useful in optimal portfolio selection, in reinsurance, in pricing bivariate derivatives or in determining capital requirements when only partial information on dependence is available. Nelsen [18] derives best possible bounds when the copula is known at a specific point. Our objective in this paper is to extend this literature to the case when the copula is known in more than one point and to show how these bounds can be useful in quantifying dependence misspecification. Assuming that marginals are given and that the dependence is unspecified, or partly unspecified, bounds on copulas can be indeed used to quantify this type of model risk. ). Tankov shows that they are quasi-copulas and not necessarily copulas. In this paper, we focus on deriving bounds on copulas that are also copulas. The first section focuses on finding simple conditions to ensure that Tankov's bounds are copulas. When the bounds are not copulas, it is possible to approximate them by a copula. The second section illustrates a method to find the best copula for the uniform norm that approximates the * Keywords

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