We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce the number of historical scenarios that potentially belong to the set of worst scenarios. The number of simulations increases as the number of candidate scenarios is reduced and the distance between them diminishes. For the most naive scheme, we show that the L p-error of the estimator of the Expected Shortfall is bounded by a linear combination of the probabilities of inversion of favorable and unfavorable scenarios at each step, and of the last step Monte Carlo error associated to each scenario. By using concentration inequalities, we then show that, for sub-gamma pricing errors, the probabilities of inversion converge at an exponential rate in the number of simulated paths. We then propose an adaptative version in which the algorithm improves step by step its knowledge on the unknown parameters of interest: mean and variance of the Monte Carlo estimators of the different scenarios. Both schemes can be optimized by using dynamic programming algorithms that can be solved off-line. To our knowledge, these are the first non-asymptotic bounds for such estimators. Our hypotheses are weak enough to allow for the use of estimators for the different scenarios and steps based on the same random variables, which, in practice, reduces considerably the computational effort. First numerical tests are performed.
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