The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 360 Wednesday, February 4, 2004 1:00 PM

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360 The Mathematics of Financial Modeling and Investment Management

where μ, σare respectively the expected value and standard deviation of X,

and Φthe standard normal distribution.

If the tail index α > 1, variables have finite expected value and the

SLNN holds. If the tail index α > 2, variables have finite variance and

the CLT in the previous form holds. If the tail index α≤ 2, then vari-

ables have infinite variance: The CLT in the previous form does not

hold. In fact, variables with α≤2 belong to the domain of attraction of

a stable law of index α. This means that a sequence of properly normal-

ized and centered sums tends to a stable distribution with infinite vari-

ance. In this case, the CLT takes the form

Sn – nμ D

----------------------→Gα, if 1 < α≤ 2

1

n α

Sn D

------→Gα, if 0 < α≤ 1

1

n α

where G are stable distributions as defined below. Note that the case α=

2 is somewhat special: variables with this tail index have infinite vari-

ance but fall nevertheless in the domain of attraction of a normal vari-

able, that is, G 2. Below the threshold 1, distributions have neither finite

variance nor finite mean. There is a sharp change in the normalization

behavior at this tail-index threshold.

Stable Distributions

Stable distributions are not, in their generality, a subset of fat-tailed dis-

tributions as they include the normal distribution. There are different,

equivalent ways to define stable distributions. Let’s begin with a key

property: the equality in distribution between a random variable and

the (normalized) independent sum of any number of identical replicas of

the same variable. This is a different property than the closure property

of the tail insofar as (1) it involves not only the tail but the entire distri-

bution and (2) equality in distribution means that distributions have the

same functional form but, possibly, with different parameters. Normal

distributions have this property: The sum of two or more normally dis-

tributed variables is again a normally distributed variable. But this

property holds for a more general class of distributions called stable dis-