Extreme Events in Finance

Chapter 4Extreme Value Theory: An Introductory Overview

Isabel Fraga Alves¹ and Cláudia Neves²

¹CEAUL, University of Lisbon, Portugal

²CEAUL, Portugal and Department of Mathematics and Statistics, University of Reading, United Kingdom

“It seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis.”

–Emil Julius Gumbel (1891–1966)

4.1 Introduction

In this chapter we give an introduction to the most important results in extreme value theory (EVT) with a flavor of how they can be applied in practice. EVT is the theory underpinning the study of the asymptotic distribution of extreme or those rare events, which can be considered huge relatively to the bulk of observations. Relying on well-founded theory on which parametric or semiparametric statistical models are built for handling with rare events, EVT is the adequate theory for modeling and measuring events which occur with a very small probability. EVT has proven to be a powerful and useful tool to describe atypical situations that may have a significant impact in many application areas, where knowledge of the behavior of the tail of the actual distribution is in demand. The main objective is to tackle the problem of modeling rare phenomena with large magnitude, hence lying outside the range of the available observations (out-of-sample).

The typical question we would like to answer is

If things go wrong, how wrong can they go?

which in a certain sense is the mitigation attitude ...