The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 362 Wednesday, February 4, 2004 1:00 PM

362 The Mathematics of Financial Modeling and Investment Management

summation). A stable law is characterized by four parameters: α, β, c, and

γ. Normal distributions correspond to the parameters: α = 2, β = 0, γ = 0.

Even if stable distributions cannot be written as simple formulas,

the asymptotic shape of their tails can be written in a simple way. In

fact, with the exception of Gaussian distributions, the tails of stable

laws obey an inverse power law with exponent α (between 0 and 2).

Normal distributions are stable but are an exception as their tails decay

exponentially.

For stable distributions, the CLT holds in the same form as for

inverse power-law distributions. In addition, the functions in the

domain of attraction of a stable law of index α < 2 are characterized by

the same tail index. This means that a distribution G belongs to the

domain of attraction of a stable law of parameter α < 2 if and only if its

tail decays as α. In particular, Pareto’s law belongs to the domain of

attraction of stable laws of the same tail index.

EXTREME VALUE THEORY FOR IID PROCESSES

In this section we introduce a number of important probabilistic con-

cepts that form the conceptual basis of Extreme Value Theory (EVT).

The objective of EVT is to estimate the entire tail of a distribution from

a finite sample by fitting to an appropriate distribution those values of

the sample that fall in the tail. Two concepts play a crucial role in EVT:

(1) the behavior of the upper order statistics (i.e., the largest k values in

a sample) and, in particular, of the sample maxima; and (2) the behavior

of the points where samples exceed a given threshold. We will explore

the limit distributions of maxima and the distribution of the points of

exceedances of a high threshold. Based on these concepts a number of

estimators of the tail index in sequences of independent and identically

distributed (IID) variables are presented.

Maxima

In the previous sections we explored the behavior of sums. The key result

of the theory of sums is that the behavior of sums simplifies in the limit of

properly scaled and centered infinite sums regardless of the shape of indi-

vidual summands. If sums converge, their limit distributions can only be

stable distributions. In addition, the normalized sums of finite-mean,

finite-variance variables always converge to a normal variable.

A parallel theory can be developed for maxima, informally defined

as the largest value in a sample. The limit distribution of maxima, if it

exists, belongs to one of three possible distributions: Frechet, Weibull,