The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 361 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 361

tributions or Levy-stable distributions. Normal distributions are thus a

special type of stable distributions.

The above can be formalized as follows: Stable distributions can be

defined as those distributions for which the following identity in distri-

bution holds for any number n ≥ 2:

n

= CnXD+ n

i = 1

∑ Xi

D

where Xi are identical independent copies of X and the Cn, Dn are con-

stants. Alternatively, the same property can be expressed stating that

stable distributions are distributions for which the following identity in

distribution holds:

D

AX 1 + BX 2 = CX + D

Stable distributions are also characterized by another property that

might be used in defining them: a stable distribution has a domain of

attraction (i.e., it is the limit in distribution of a normalized and cen-

tered sum of identical and independent variables). Stable distributions

coincide with all variables that have a domain of attraction.

Except in the special cases of Gaussian (α = 2), symmetric Cauchy

(α = 1, β = 0) and stable inverse Gaussian (α = ¹⁄₂, β = 0) distributions,

stable distributions cannot be written as simple formulas; formulas have

been discovered but are not simple. However, stable distributions can be

characterized in a simple way through their characteristic function, the

Fourier transform of the distribution function. In fact, this function can

be written as

α

ΦX() t = exp{iγtc – t [ 1 – iβsign ()tzt( α , )]}

where t ∈ R, γ∈ R, c > 0, α∈ (0,2), β∈ [–1,1], and

(

πα

ztα , ) = tan-------if α ≠ 1

2

zt( α , ) = –2log t if α = 1

It can be shown that only distributions with this characteristic function

are stable distributions (i.e., they are the only distributions closed under