Risk management with generalized hyperbolic distributions

Wembo Hu, Alec Kercheval

We examine certain Generalized Hyperbolic ($GH$) distributions for modeling equity returns, compared to usual Normal distributions. We describe these $GH$ distributions and some of their properties, and test them against six years of daily S+P500 index prices. We estimate Value-at-Risk from calibrated distributions, and show that the Normal distribution leads to $VaR$ estimates that significantly underestimate the realized empirical values, while the $GH$ distributions do not. Of several $GH$ distribution families considered, the most successful is the skewed-$t$ distribution.