The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 386 Wednesday, February 4, 2004 1:00 PM

386 The Mathematics of Financial Modeling and Investment Management

However, scaling can be defined without making reference to frac-

tals. In its simplest form, the notion of scaling entails a variable x and

an observable A which is a function of A = A(x). If the observable obeys

a scaling relationship, there is a constant factor between x and A in the

sense that A(λx) = λsA(x), where s is the scaling exponent that does not

depend on x. The only function A(x) that satisfies this relationship is a

power law. In the three-dimensional Euclidean space, volume scales as

the third power of linear length and surface as the second power, while

fractals scale according to their fractal dimension.

The same ideas can be applied in a random context, but require

careful reasoning. A power-law distribution has a scaling property as

multiplying the variable by a factor multiplies probabilities by a con-

stant factor, regardless of the level of the variable. This means that the

ratio between the probability of the events X > x and X > ax depends

only on a power of a, not on x. As an inverse power law is not defined

at zero, scaling in this sense is a property of the tails. The probabilistic

interpretation of this property is the following: the probability that an

observation exceeds ax conditional on the knowledge that the observa-

tion exceeds x does not depend on x but only on a.

There are, however, other meanings attached to scaling and these

might be a source of confusion. In the context of physical phenomena,

scaling is often intended as identity of distribution after aggregation. The

same idea is also behind the theory of groups of renormalization and the

notion of self-similarity applied to structures such as coastlines. In the lat-

ter case, the intuitive meaning of self-similarity is that if one aggregates

portions of the coastline, approximating their shape with a straight line,

and then rescales; the resulting picture is qualitatively similar to the origi-

nal. The same idea applies to percolation structures: By aggregating

“sites” (i.e., points in a percolation lattice) into supersites and carefully

redefining links, one obtains the same distribution of connected clusters.

Applying the idea of aggregation in a random context, self-similar-

ity seems to mean that, after rescaling, the distribution of the sum of

independent copies of a random variable maintains the same shape of

the distribution of the variable itself. Note that this property holds only

for the tails of subexponential distributions—and it holds strictly only

for stable laws that have tails in the (0,2) range but whose shape is not a

power law except, approximately, in the tails. It also holds for Gaussian

distributions that do not have power-law tails.

Scaling acquires yet another meaning when applied to stochastic pro-

cesses that are functions of time. The most common among the different

meanings is the following: A stochastic process is said to have a scaling

property if there is no natural scale for looking at its paths and distribu-

tions. Intuitively, this means that it is not possible to gauge the scale of a