Risk Management with Extreme Value Theory

Abstract

In this paper we review certain aspects around the Value-at-Risk, which is nowadays the industry benchmark risk measure. As a small quantile (usually 1%) Value-at-Risk is closely related to extreme value theory. We explain an estimation method based on extreme value theory. Since the variance of the estimated Value-at-Risk may depend on the dependence structure of the data, we investigate the extreme behaviour of some of the most prominent time series models in finance, continuous as well as discrete time models. We also determine optimal portfolios, when risk is measured by the Value-at-Risk. Again we use realistic models, moving away from the traditional Black-Scholes model to the class of Levy processes. This paper is the contribution to a book by several authors on Extreme Value Theory, which will appear by CRC/Chapman and Hall.