The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 366 Wednesday, February 4, 2004 1:00 PM

366 The Mathematics of Financial Modeling and Investment Management

the concept of the Maximum Domain of Attraction (MDA) of an

extreme value distribution H or MDA(H).

A random variable X is said to belong to the MDA(H) of the extreme

value distribution H if there exist constants cn > 0, dn ∈R such that

• 1 D
  cn (Mn– dn ) →H

Two distribution functions F, G are said to be tail equivalent if they

have the same right endpoints and the following condition holds:

Fx()

lim ------------- = c , 0 < c < ∞

x → ∞ Gx()

Tail equivalence is an important concept for characterizing MDAs. In

fact, it can be demonstrated that every MDA(H) is closed with respect

to tail equivalence (i.e., if two distribution functions F and G are tail

equivalent F ∈MDA(H) if and only if G ∈MDA(H)). Tail equivalence

allows for a powerful characterization of the three MDAs.

Let’s first define the quantile function. Given a distribution function

F, the quantile function of F, written F←(x), is defined as follows:

F←(x) = inf[s ∈R: F(s) ≥x], 0 < x < 1

The MDA of the Frechet Distribution

The Frechet distribution is written as Φα()x = exp(–x –α). Let’s start by

observing that the tail of the Frechet distribution decays as an inverse power

law. In fact, we can write 1 – Φα()x = 1 – exp(–x –α) ≈x –α for x → ∞.

It can be demonstrated that a distribution function F belongs to the

MDA of a Frechet distribution Φα()x , α> 0 if and only if there is a

slowly varying function L such that Fx()= x –αLx(). In this case, the

constants assume the values

c = ( 1 ⁄F

←

n )()n , dn = 0

We can rewrite this condition more compactly as follows:

F ∈MDA(Φα) F∈ ⇔ R–α

From the above definitions it can be demonstrated that the follow-

ing five distributions belong to the MDA of the Frechet distribution: (1)

Pareto; (2) Cauchy; (3) Burr; (4) Stable laws with exponent α< 2; or (5)

log-gamma distribution.