  Extreme value theory provides an asymptotically justified framework for
estimation of exceedance probabilities in regions where few or no observations
are available. For multivariate tail estimation, the strength of extremal
dependence is crucial and it is typically modeled by a parametric family of
spectral distributions. In this work we provide asymptotic bounds on exceedance
probabilities that are robust against misspecification of the extremal
dependence model. They arise from optimizing the statistic of interest over all
dependence models within some neighborhood of the reference model. A certain
relaxation of these bounds yields surprisingly simple and explicit expressions,
which we propose to use in applications. We show the effectiveness of the
robust approach compared to classical confidence bounds when the model is
misspecified. The results are further applied to quantify the effect of model
uncertainty on the Value-at-Risk of a financial portfolio.
