Risk Management with Extreme Value Theory

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 Lévy processes.
This paper is the contribution to a book by several authors on
Extreme Value Theory, which will appear by CRC/Chapman and Hall.