Modelling Extremal Events

Embrechts, Kluppelberg and Mikosch wrote the standard graduate reference on extreme value theory for insurance and finance. It presents the mathematics of the tails of distributions: the behaviour of maxima, the theory of regular variation, heavy-tailed distributions, point process methods, and the estimation of tail parameters from data.

The two central results are the Fisher-Tippett theorem, which establishes that the maximum of a sample, appropriately normalised, converges to one of three distributions collectively described by the generalised extreme value family, and the Pickands-Balkema-de Haan theorem, which establishes that exceedances over a high threshold converge to the generalised Pareto distribution. Together these give a principled way to estimate the probability of events beyond anything observed in the sample, which is what risk measurement requires and what standard distributional assumptions cannot supply.

The book covers the peaks-over-threshold method, the Hill estimator and its instability, the difficulty of choosing a threshold, and the estimation problems that arise from having, by construction, very few observations in the region of interest. The applications are to insurance claims, operational risk and financial returns.

It is a mathematics text and requires probability theory at graduate level. Its practical value to risk practitioners is real, and it is the reference behind most tail risk measurement in insurance and banking. The authors are explicit about the method's limits, particularly that extreme value theory assumes independent identically distributed observations, which financial returns violate through volatility clustering, and that parameter uncertainty in the tail is irreducibly large.