Regular Variation and Extremal Dependence of GARCH Residuals with Application to Market Risk Measures

Stock returns exhibit heavy tails and volatility clustering. These features, motivating the use of GARCH models, make it difficult to predict times and sizes of losses that might occur. Estimation of losses, like the Value-at-Risk, often assume that returns, normalized by the level of volatility, are Gaussian. Often under ARMA-GARCH modeling, such scaled returns are heavy tailed and show extremal dependence, whose strength reduces when increasing extreme levels. We model heavy tails of scaled returns with generalized Pareto distributions, while extremal dependence can be reduced by declustering data.