Probabilistic Error Bounds for Simulation Quantile Estimators

Quantile estimation has become increasingly important, particularly in the financial industry, where value at risk (VaR) has emerged as a standard measurement tool for controlling portfolio risk. In this paper, we analyze the probability that a simulation-based quantile estimator fails to lie in a prespecified neighborhood of the true quantile. First, we show that this error probability converges to zero exponentially fast with sample size for negatively dependent sampling. Then we consider stratified quantile estimators and show that the error probability for these estimators can be guaranteed to be 0 with sufficiently large, but finite, sample size. These estimators, however, require sample sizes that grow exponentially in the problem dimension. Numerical experiments on a simple VaR example illustrate the potential for variance reduction.