The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 380 Wednesday, February 4, 2004 1:00 PM

380 The Mathematics of Financial Modeling and Investment Management

The Law of Large Numbers tells that if individual processes are

independent and have finite variance, then phenomena average out in

aggregate and tend to an average limit. However, if individual processes

have fat tails, phenomena do not average out even in the infinite limit.

The weight of individual tails prevails and drives the aggregate process.

Philip W. Anderson, the corecipient of the 1997 Nobel Prize in Physics,

remarked:

Much of the real world is controlled as much by the

“tails” of distributions as by means or averages: by the

exceptional, not the mean; by the catastrophe, not the

steady drip; by the very rich, not the “middle class.” We

need to free ourselves from “average” thinking.^16

When and if fat-tailed drivers exist, they control the ensemble to

which they belong. But what generates these powerful drivers? Models

that generate fat tails from standard normal innovations attempt to

answer this question. Different types of models have been proposed.

One such category of models is purely geometric and exploits mathe-

matical theories such as percolation and random graph. Others exploit

phenomena of dynamic nonlinear self-reinforcing cascades of events.

Percolation models are based on the well known mathematical fact

that in regular spatial structures of nodes connected by links, a uniform

density of links produces connected subsets of nodes whose size is dis-

tributed according to power laws. Percolation models are time-transver-

sal models: They model aggregation at any given time. They might be

used to explain how fat-tailed IID sequences are generated.

Dynamic financial econometric models exploit cascading phenom-

ena due to nonlinearities, in particular multiplicative noise. In a deter-

ministic setting, it is well known that nonlinear chaotic models generate

sequences that, when analyzed statistically, exhibit fat-tailed distribu-

tions. The same happens when noise is subject to nonlinear transforma-

tion. In the next sections, we explore simple ARMA models, ARCH-

GARCH models, subordinated models, and state-space models, all

examples of dynamic financial econometric models.

Before doing this, however, let’s go back to the question of estima-

tion. As observed above, if variables are not IID but can be considered

generated by a DGP, the question of estimation is no longer the estima-

tion of a variable but that of estimating a model or a theory. The estima-

(^16) Philip W. Anderson, “Some Thoughts About Distribution in Economics,” in W.
B. Arthur, S. N. Durlaf, and D.A. Lane (eds.), The Economy as an Evolving Complex
System II (Reading, MA: Addison-Wesley, 1997).