Before revision:  Our results can be used either to estimate the probability of tail events directly, or to construct efficient variance reduction procedures for precise estimation of these probabilities by Monte Carlo methods. They lead to important insights concerning the behavior of individual stocks and portfolios during market downturns in the multidimensional Black-Scholes model .
After revision:  Our results can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one .