On the estimation of Value-at-Risk and Expected Shortfall at extreme levels

The estimation of risk at extreme levels (such as 0.1%) can be crucial to capture the losses during market downturns, such as the global financial crisis and the COVID-19 market crash. For many existing models, it is challenging to estimate risk at extreme levels. In order to improve such estimation, we develop a framework to estimate Value -at -Risk and Expected Shortfall at an extreme level by extending the one -factor GAS model and the hybrid GAS/GARCH model to estimate Value -at -Risk and Expected Shortfall for two levels simultaneously, namely for an extreme level and for a more common level (such as 10%). Our simulation results indicate that the proposed models outperform the GAS model benchmarks in terms of in -sample and out -of -sample loss values, as well as backtest rejection rates. We apply the proposed models to oil futures (WTI, Brent, gas oil and heating oil) and compare them with a range of parametric, nonparametric, and semiparametric alternatives. The results show that our proposed models are generally superior to the alternatives.