Accurate modeling of extremal price changes is vital to financial risk management. We examine the small sample properties of adaptive tail index estimators under the class of student-t marginal distribution functions including GARCH and propose a model-based bias-corrected estimation approach. Our simulation results indicate that bias strongly relates to the underlying model and may be positively as well as negatively signed. The empirical study of daily exchange rate changes reveals substantial differences in measured tail-thickness due to small sample bias. As a consequence, high quantile estimation may lead to a substantial underestimation of tail risk.

This paper reconsiders return-volume dependence for the U.S. and six international equity markets. We contribute to previous work by proposing surprise volume as a new proxy for private information flow and apply extreme value theory in studying dependence for large volume and return, i.e. under situations of market stress. Results from a GARCH-M model indicate that surprise volume is superior in explaining conditional variance and reveals a positive market risk premium. Under conditions of market stress, the return-volume dependence is weaker, albeit mostly significant. The results for the U.S. market are most pronounced in that surprise volume explains ARCH- as well as leverage-effects and, under market stress, the return-volume dependence remains significant and symmetric. For the European and Asian markets, however, the dependence is weaker with asymmetry under market stress, i.e. small minimal returns show lower volume dependence than large maximal returns. We argue that our results are more consistent with a Gennotte and Leland (1990) misinterpretation hypothesis for market crashes than with cascade or behavioral explanations which associate high volume with steep price declines.

Estimation of the tail index of stationary, fat-tailed return distributions is non-trivial since the well-known Hill estimator is optimal only under iid draws from an exact Pareto model. We provide a small sample simulation study of recently suggested adaptive estimators under ARCH-type dependence. The Hill estimator's performance is found to be dominated by a ratio estimator. Dependence increases estimation error which can remain substantial even in larger data sets. As small sample bias is related to the magnitude of the tail index, recent standard applications may have overestimated (underestimated) the risk of assets with low (high) degrees of fat-tailedness.