The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 356 Wednesday, February 4, 2004 1:00 PM

356 The Mathematics of Financial Modeling and Investment Management

that belong to the class L but not to the class S. Distributions that have

both properties are called subexponential as it can be demonstrated

that, as all distributions in L, they satisfy the property:

lim Fx()e λx = ∞ , λ∀ > 0

x ∞ →

Note, however, that the class of distributions that satisfies the latter

property is broader than the class of subexponential distributions; this

is because the former includes, for instance, the class L.^6

Subexponential distributions do not have finite exponential

moments of any order, that is, Ee

sX

[ ] = ∞ for every s ≥ 0. They may or

may not have a finite mean and/or a finite variance. Consider, in fact,

that the class of subexponential distributions includes both Pareto and

Weibull distributions. The former have infinite variance but might have

finite or infinite mean depending on the index; the latter have finite

moments of every order (see below).

The key indicators of subexponentiality are (1) the equivalence in

the distribution of the tail between a variable and a sum of independent

copies of the same variable and (2) the fact that a sum is dominated by

its largest term. The importance of the largest terms in a sum can be

made more quantitative using measures such as the large claims index

introduced in Embrechts, Kluppelberg, and Mikosch that quantifies the

ratio between the largest p terms in a sum and the entire sum.

The class of subexponential distributions is quite large. It includes

not only Pareto and stable distributions but also log-gamma, lognormal,

Benkander, Burr, and Weibull distributions. Pareto distributions and sta-

ble distributions are a particularly important subclass of subexponential

distributions; these will be described in some detail below.

Power-Law Distributions

Power-law distributions are a particularly important subset of subexpo-

nential distributions. Their tails follow approximately an inverse power

law, decaying as x –α. The exponent α is called the tail index of the distri-

bution. To express formally the notion of approximate power-law decay,

we need to introduce the class ℜ(α), equivalently written as ℜα of regu-

larly varying functions.

A positive function f is said to be regularly varying with index α or f

∈ℜ(α) if the following condition holds:

(^6) See Sigman, “A Primer on Heavy-Tailed Distributions.”