GARCH-UGH: a bias-reduced approach for dynamic extreme Value-at-Risk estimation in financial time series

The Value-at-Risk (VaR) is a widely used instrument in financial risk management. The question of estimating the VaR of loss return distributions at extreme levels is an important question in financial applications, both from operational and regulatory perspectives; in particular, the dynamic estimation of extreme VaR given the recent past has received substantial attention. We propose here a new two-step bias-reduced estimation methodology for the estimation of one-step ahead dynamic extreme VaR, called GARCH-UGH (Unbiased Gomes-de Haan), whereby financial returns are first filtered using an AR-GARCH model, and then a bias-reduced estimator of extreme quantiles is applied to the standardized residuals. Our results indicate that the GARCH-UGH estimates of the dynamic extreme VaR are more accurate than those obtained either by historical simulation, conventional AR-GARCH filtering with Gaussian or Student-t innovations, or AR-GARCH filtering with standard extreme value estimates, both from the perspective of in-sample and out-of-sample backtestings of historical daily returns on several financial time series.