18.5 Recommended Literature

Both of the basic theorems, Theorem 17.1 and Theorem 17.6, of this section go back to Fisher and Tippett (1928) respectively Pickands (1975). The essential notion of quantifying risk by coherent risk measures was introduced by Artzner et al. (1997).

A comprehensive summary of the modelling and statistical analysis of extreme results is given in the monograph from Embrechts et al. (1997). There one finds proofs as well as detailed mathematical and practical considerations of the content of this section and an extensive bibliography. Another actual and recommendable book on extreme value statistic is Reiss and Thomas (1997). A more in depth implementation of the method in the form of quantlets discussed in this last reference, which goes beyond the selection introduced in this section, can be found in Reiss and Thomas (2000).

A substantial problem that occurs when applying the methods of extreme value statistics such as the POT or the Hill estimators is the choice of the threshold value $ u$ or the corresponding number $ k$ of large order statistics. We have already mentioned how this choice can be made with the help of graphical representations. A more in depth discussion including the corresponding quantlet can be found in Reiss and Thomas (2000). Polzehl and Spokoiny (2003) and Grama and Spokoiny (2003) describe current procedures used for estimating the tail exponents, for which the choice of $ u$ respectively $ k$, given the available data, can be adaptively and thus automatically chosen.

The methods described in this chapter give estimators for the Value-at-Risk as unconditional quantiles. Often one wishes to include financial data from the recent past when estimating risk, for example in a GARCH(1,1) model the last observation and the last volatility. In this case the Value-at-Risk is a conditional quantile given the available information. One possibility of using extreme value statistics in such cases is based on the assumptions of a specific stochastic volatility model which is parametric as in McNeil and Frey (2000) or nonparametric as in Chapter 13.

Given the assumptions of the model a conditional volatility $ \sigma_t$ is estimated given the past, which together with the data results in an estimator for the innovations $ Z_t$. In calculating the conditional quantile it is not assumed that the $ Z_t$ are standard normally distributed, but instead the needed unconditional quantile of the innovations is estimated from the estimated innovations with, for example, the POT estimator. Alternatively one can estimate the conditional quantile also direct as nonparametric, in which the conditional distribution function is first estimated with a kernel estimator and then the inverse is taken. With moderately large quantiles, for example, with a 95% VaR, the method from Franke and Mwita (2003) gives good results, even for innovation distributions with heavy tails and infinite variance. For extreme quantiles such as the 99% VaR a semi-parametric method must be considered, as is the case with the POT method, in order to obtain useful results. Mwita (2003) estimates first a nonparametric, medium sized conditional quantile and modifies this estimator through the fitting of a Pareto distribution to the extreme excesses.