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.

中文翻译：

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关于极端水平下风险价值和预期缺口的估计

极端水平（例如 0.1%）的风险估计对于捕捉市场低迷时期（例如全球金融危机和 COVID-19 市场崩盘）期间的损失至关重要。对于许多现有模型来说，估计极端水平的风险具有挑战性。为了改进这种估计，我们开发了一个框架，通过扩展单因素 GAS 模型和混合 GAS/GARCH 模型来估计风险价值和预期短缺，从而在极端水平上估计风险价值和预期短缺同时针对两个水平，即极端水平和更常见水平（例如10%）。我们的模拟结果表明，所提出的模型在样本内和样本外损失值以及回测拒绝率方面优于 GAS 模型基准。我们将所提出的模型应用于石油期货（WTI、布伦特原油、柴油和取暖油），并将其与一系列参数、非参数和半参数替代方案进行比较。结果表明，我们提出的模型通常优于替代方案。