Modelling Extremal Events

The basic question each author should pose him/herself, preferably in the future tense before starting, is

For most of the problems treated in insurance mathematics, risk theory still provides the quintessential mathematical basis. The present chapter will serve a similar purpose for the rest of this book. The basic risk theory models will be introduced, stressing the instances where a division between small and large claims is relevant. Nowadays, there is a multitude of textbooks available treating risk theory at various mathematical levels. Consequently, our treatment will not be encyclopaedic, but will focus more on those aspects of the theory where we feel that, for modelling extremal events, the existing literature needs complementing. Readers with a background in finance rather than insurance may use this chapter as a first introduction to the stochastic modelling of claim processes.

This chapter is concerned with classical extreme value theory and consequently it is fundamental for many results in this book. The central result is the Fisher—Tippett theorem which specifies the form of the limit distribution for centred and normalised maxima. The three families of possible limit laws are known as extreme value distributions. In Section 3.3 we describe their maximum domains of attraction and derive centring and normalising constants. A short summary is provided in Tables 3.4.2–3.4.4 where numerous examples are to be found.

After having investigated in Chapter 3 the behaviour of the maximum, i.e. the largest value of a sample, we now consider the joint behaviour of several upper order statistics. They provide information on the right tail of a df.

Point process techniques give insight into the structure of limit variables and limit processes which occur in the theory of summation (see Chapter 2), in extreme value theory (see Chapters 3 and 4) and in time series analysis (see Chapter 7).

In the previous chapters we have introduced a multitude of probabilistic models in order to describe, in a mathematically sound way, extremal events in the one—dimensional case. The real world however often informs us about such events through statistical data: major insurance claims, flood levels of rivers, large decreases (or indeed increases) of stock market values over a certain period of time, extreme levels of environmental indicators such as ozone or carbon monoxide, wind—speed values at a certain site, wave heights during a storm or maximal and minimal performance values of a portfolio. All these, and indeed many more examples, have in common that they concern questions about extreme values of some underlying set of data. At this point it would be utterly foolish (and indeed very wrong) to say that all such problems can be cast into one or the other probabilistic model treated so far: this is definitely not the case! Applied mathematical (including statistical) modelling is all about trying to offer the applied researcher (the finance expert, the insurer, the environmentalist, the biologist, the hydrologist, the risk manager, ...) the necessary set of tools in order to deduce scientifically sound conclusions from data. It is however also very much about reporting correctly: the data have to be presented in a clear and objective way, precise questions have to be formulated, model—based answers given, always stressing the un-derlying assumptions. The whole process constitutes an art: statistical theory plays only a relatively small, though crucial role here.

In this chapter we present some recent research on time series with large fluctuations, relevant for many financial time series. We approach the problem starting from classical time series analysis presented in such a way that many standard results can also be used in the heavy-tailed case.

In Chapters 3–6 we presented a wealth of material on extremes. In most cases we restricted ourselves to iid observations. However, in reality extremal events often tend to occur in clusters caused by local dependence in the data. For instance, large claims in insurance are mainly due to hurricanes, storms, floods, earthquakes etc. Claims are then linked with these events and do not occur independently. The same can be observed with financial data such as exchange rates and asset prices. If one large value in such a time series occurs we can usually observe a cluster of large values over a short period afterwards.