This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value-at-risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method. KEY WORDS: model uncertainty, scenario aggregation, expected shortfall, value-at-risk, statistical divergence, Swiss Solvency Test. INTRODUCTIONThe last decades have seen strong developments in the statistical measurement of risk. The quantitative methods used by banks and insurance companies for risk management serve many purposes such as enterprise risk management, pricing, capital allocation, or reporting to regulators. The latter have required regulated institutions to implement and document internal models that should be used to report their amount of required capital which is bearing the risk and to show that they would remain solvent in case of extreme scenarios. Although the risk modeling methodology of an institution is reported and subject to approval from the regulators or internal model validation, uncertainty on the validity of the model remains inherent and should therefore be challenged. The risk of inappropriate modeling can usually be raised at many levels. For example, in a factor model, one could question a specific choice of risk factors, the marginal distribution of these risk factors, or the dependence structure between them (see, e.g., Embrechts, Puccetti, and Rüschendorf 2013; Embrechts, Wang, and Wang 2014 for the last aspect).We thank Matthias Aellig, David Babbel, Alexis Bailly, Rama Cont, Freddy Delbaen, Kabir Dutta, Paul Embrechts, Jan Friedrich, Hansjörg Furrer, Jean-Francois Guérin, Andreas Haier, Stefan Jaschke, Thorsten Pfeiffer, Alexander Schied, Michael Schmutz, Ruodu Wang, three anonymous referees, the editor (Jerome Detemple), and participants at the In many examples of risk management processes, the model will be used to estimate risk measures that greatly depend on the tail of the loss distribution and one should therefore check that this part of distribution is appropriately modeled.This paper provides a coherent method for incorporating external views on scenarios into an internal model. This scenario aggregation method aims at supporting regulatory purposes, such as stress testing, and serves as a device to address internal model uncertainty.A clear distinction has been made in the literature, in the footsteps 1 of Knight (1921), between the notions of risk and uncertainty. The former relates to the unknowns with respect to future events for which probabilities are known with certainty, while the latter notion happens whenever these probabilities are unknown. Although the wording model risk is broadly used in both the academic and industry li...

The present work addresses the question how sampling algorithms for commonly applied copula models can be adapted to account for quasi-random numbers. Besides sampling methods such as the conditional distribution method (based on a one-to-one transformation), it is also shown that typically faster sampling methods (based on stochastic representations) can be used to improve upon classical Monte Carlo methods when pseudo-random number generators are replaced by quasi-random number generators. This opens the door to quasi-random numbers for models well beyond independent margins or the multivariate normal distribution. Detailed examples (in the context of finance and insurance), illustrations and simulations are given and software has been developed and provided in the R packages copula and qrng.

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