The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 368 Wednesday, February 4, 2004 1:00 PM

368 The Mathematics of Financial Modeling and Investment Management

The MDA of a Gumbel distribution encompasses a large class of dis-

tributions that includes the exponential distribution, the normal distribu-

tion, and the lognormal distribution. Though the Gumbel distribution

has exponential tails, its MDA includes subexponential distributions

such as the Berktander distribution, as explained in Goldie and Resnick.^7

Max-Stable Distributions

Stable distributions remain unchanged after summation; max-stable dis-

tributions remain unchanged after taking maxima. A non-degenerate

random variable X and the relative distribution is called max-stable if

there are constants cn > 0, dn ∈ R such that the following conditions are

satisfied

D

max(X 1 , ..., Xn) = c Xdn + n

where X, X 1 , ..., Xn are IID variables.

It can be demonstrated that the class of max-stable distributions

coincides with the class of possible limit laws for normalized and cen-

tered maxima. In view of the previous discussions, the max-stable laws

are the three possible limit laws: Frechet, Weibull, and Gumbel.

Generalized Extreme Value Distributions

The three extreme value distributions, Frechet, Weibull, and Gumbel,

can be represented as a one-parameter family of distributions through

the Standard Generalized Extreme Value Distribution (GEV) of Jenkin-

son and Von Mises. Define the distribution function Hξ as follows:

exp[–( 1 + ξx) –^1 ⁄ ξ] for ξ ≠ 0

Hξ = 

exp(–exp(–x)) for ξ =^0

where 1 + ξx > 0. One can see from the definition that ξ = α–1 > 0 corre-

sponds to the Frechet distribution, ξ = 0 corresponds to the Gumbel dis-

tribution, and ξ = –α–1 < 0 corresponds to the Weibull distribution. We

can now introduce the related location-scale dependent family Hξ;μ,ψ by

replacing the argument x with (x – μ)/ψ.

(^7) C.M. Goldie and S. Resnick, “Distributions that are Both Subexponential and in
the Domain of Attraction of an Extreme-Value Distribution,” Advanced Applied
Probability, 20 (1988), pp. 706–718.