We derive upper and lower bounds on the expectation of f (S) under dependence uncertainty, i.e. when the marginal distributions of the random vector S = (S1, . . . , S d ) are known but their dependence structure is partially unknown. We solve the problem by providing improved Fréchet-Hoeffding bounds on the copula of S that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of [0, 1] d , the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show that, in contrast to the twodimensional case, the bounds are quasi-copulas but fail to be copulas if d > 2. Thus, in order to translate the improved Fréchet-Hoeffding bounds into bounds on the expectation of f (S), we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators. By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved Fréchet-Hoeffding bounds to bounds on the expectation of f (S). Finally, we apply these results to compute model-free bounds on the prices of multi-asset options that take partial information on the dependence structure into account, such as correlations or market prices of other traded derivatives. The numerical results show that the additional information leads to a significant improvement of the option price bounds compared to the situation where only the marginal distributions are known.

Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for these Fréchet classes that extend previous results in the related literature. These proofs are based on representation results for increasing convex functionals and the explicit computation of the conjugates. We show that the dual transport problem admits an explicit solution for the function f = 1B, where B is a rectangular subset of R d , and provide an intuitive geometric interpretation of this result. The improved Fréchet-Hoeffding bounds provide ad-hoc upper bounds for these Fréchet classes. We show that the improved Fréchet-Hoeffding bounds are pointwise sharp for these classes in the presence of uncertainty in the marginals, while a counterexample yields that they are not pointwise sharp in the absence of uncertainty in the marginals, even in dimension 2. The latter result sheds new light on the improved Fréchet-Hoeffding bounds, since Tankov [30] has showed that, under certain conditions, these bounds are sharp in dimension 2.

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We derive bounds on the distribution function, therefore also on the Value-at-Risk, of ϕ(X) where ϕ is an aggregation function and X = (X1, . . . , X d ) is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of X, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Fréchet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of X is known on a subset of its domain, and finally we consider the case where the copula of X lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Fréchet-Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Fréchet-Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.