The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 387 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 387

sample by looking at its distribution; there is absence of scale. An exam-

ple from finance comes from price patterns. If a price pattern is generated

by a process with the scaling property, the plots of average daily and

monthly prices will appear to be perfectly similar in distribution; looking

at the plot, it’s impossible to tell if it refers to daily or monthly prices.

Self-similarity is another way of expressing the same concept. A

process is self-similar if a portion of the process is similar to the entire

process. As we are considering a random environment, self-similarity

applies to distributions, not to the actual realization of a process. Let’s

now make these concepts more precise.

A stochastic process X(t) is said to be self-similar (ss) of index H (H-

ss) if all its finite-dimensional distributions obey the scaling relationship:

D

(Xkt ... ... k

1

,Xkt

2

, , Xkt ) = k –H(Xt

1

,Xt

2

, , Xt ) > ∀ 0

m m

0 < < H 1 , t 1 ,, , t 2 ...tm > 0

The above expression means that the scaling of time by the factor k

scales the variables X by the factor kH. It gives precise meaning to the

notion of self-similarity applied to stochastic processes.

There is a wide variety of self-similar processes that cannot be charac-

terized in a simple way as scaling laws: The scaling property of stochastic

processes might depend upon the shape of distributions as well as the

shape of correlations. Let’s restrict our attention to processes that are self-

similar with stationary increments (sssi) and with index H (H-sssi). These

processes can be either Gaussian or non-Gaussian. Note that a Gaussian

process is a process whose finite-dimensional distributions are all Gaussian.

Gaussian H-sssi processes might have independent increments or

exhibit long-range correlations. The only Gaussian H-sssi process with

independent increment is the Brownian motion, but there are an infinite

number of fractional Brownian motions, which are Gaussian H-sssi pro-

cesses with long-range correlations. Thus there are an infinite variety of

Gaussian self-similar processes. Among the many non-Gaussian H-sssi

processes with independent increments are the stable Levy processes,

which are random walks whose increments follow a stable distribution.^24

There is another definition of self-similarity for stochastic processes

which makes use of the concept of aggregation; it is closer, at least in

spirit, to the theory of renormalization groups. Consider a stationary

(^24) See G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes
(New York: Chapman & Hall, 1994).