Improved Fréchet–Hoeffding bounds on $d$-copulas and applications in model-free finance

Abstract

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.

Model-free bounds on Value-at-Risk using extreme value information and statistical distances

“…The improvement of the 'classical' Fréchet-Hoeffding bounds by using additional, partial information on the dependence structure has attracted some attention in the literature lately, see e.g. Nelsen [27], Tankov [43], Lux and Papapantoleon [24] and also Rachev and Rüschendorf [32].…”

“…it holds that C(x) = C * (x) for all x ∈ S and a reference copula C * . Applying results from Lux and Papapantoleon [24] and the improved standard bounds of Embrechts et al [16] and Embrechts and Puccetti [14] we derive bounds on VaR using the available information on the subset S. This relates to the trusted region in Bernard and Vanduffel [3], although the methods are different. The second type of dependence information corresponds to C lying in the vicinity of a reference copula C * as measured by a statistical distance D. In this case we establish improved Fréchet-Hoeffding bounds on the set of all (quasi-)copulas C in the δ-neighborhood of the reference model C * , i.e.…”

Value‐at‐Risk bounds with two‐sided dependence information

“…In addition, it is rarely the case that no information at all about the dependence structure of the risk vector is available, as partial information such as Kendall's tau or correlations between the risk factors can be estimated or inferred with sufficient accuracy. Such additional information can be translated into a lower and an upper bound on the copula of X ; see, for example, Rachev and Rüschendorf (), Nelsen (), and Tankov () for or Lux and Papapantoleon (, ) and Puccetti et al. () for .…”

“…Our main contribution is the development of a method to incorporate two‐sided bounds on the copula of X in order to obtain substantially improved VaR estimates in comparison to the case where only one‐sided information is available. For the derivation of two‐sided bounds on the copula from partial information about the distribution of the risks, we refer to Rachev and Rüschendorf (), Nelsen, Quesada‐Molina, Rodriguez‐Lallena, and Ubeda‐Flores (), Nelsen (, section 3.2.3), Tankov (), Lux and Papapantoleon (, ), and Puccetti et al. ().…”

Computation of Optimal Transport and Related Hedging Problems via Penalization and Neural Networks

“…[11,22,44]), or calculating worst case copula values and improved Fréchet-Hoeffding bounds (see e.g. [6,40]). Moreover, φ(f ) serves as a building block for several other problems, like generative adversarial networks (where additionally, the optimization includes generating a distribution, see e.g.…”