Adapting extreme value statistics to financial time series: dealing with bias and serial dependence

We handle two major issues in applying extreme value analysis to financial time series, bias and serial dependence, jointly. This is achieved by studying bias correction methods when observations exhibit weak serial dependence, in the sense that they come from β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta$\end{document}-mixing series. For estimating the extreme value index, we propose an asymptotically unbiased estimator and prove its asymptotic normality under the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta$\end{document}-mixing condition. The bias correction procedure and the dependence structure have a joint impact on the asymptotic variance of the estimator. Then we construct an asymptotically unbiased estimator of high quantiles. We apply the new method to estimate the value-at-risk of the daily return on the Dow Jones Industrial Average index.