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 or the corresponding number
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
respectively
, 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
is estimated given the past, which together with the
data results in an estimator for the innovations
. In
calculating the conditional quantile it is not assumed that the
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.