Kernel quantile estimators for nested simulation with application to portfolio value-at-risk measurement

Nested simulation has been widely used in portfolio risk measurement in recent years. We focus on one risk measure, value at risk (VaR), and study a kernel quantile estimator (KQE) for nested simulation to estimate this risk. We analyze the bias, variance, and mean squared error (MSE), based on which we show that the variance is reduced in the lower-order terms, while in some cases bias could be reduced in the dominant term. For practical implementation, we propose an efficient bootstrap-based algorithm to guide kernel bandwidth selection and budget allocation in nested simulation. We also conduct numerical experiments to show that KQE works quite well at different significance levels compared with the widely used sample quantile.