Analysis of Financial Time Series, Third Edition

7.5 Extreme Value Theory

In this section, we review some extreme value theory in the statistical literature. Denote the return of an asset, measured in a fixed time interval such as daily, by rt. Consider the collection of n returns, {r1, … , rn}. The minimum return of the collection is r(1), that is, the smallest order statistic, whereas the maximum return is r(n), the maximum order statistic. Specifically, r(1) = min1≤j≤n{rj} and r(n) = max1≤j≤n{rj}. Following the literature and using the loss function in VaR calculation, we focus on properties of the maximum return r(n). However, the theory discussed also applies to the minimum return of an asset over a given time period because properties of the minimum return can be obtained from those of the maximum by a simple sign change. Specifically, we have , where with the superscript c denoting sign change. The minimum return is relevant to holding a long financial position. As before, we shall use negative log returns, instead of the log returns, to perform VaR calculation for a long position.

7.5.1 Review of Extreme Value Theory

Assume that the returns rt are serially independent with a common cumulative distribution function F(x) and that the range of the return rt is [l, u]. For log returns, we have . Then the CDF of r(n), denoted ...