13-Fat Tails-Scaling-Stabl Page 351 Wednesday, February 4, 2004 1:00 PM

CHAPTER

13

Fat Tails, Scaling, and

Stable Laws

M

ost models of stochastic processes and time series examined thus far

assume that distributions have finite mean and finite variance. In

this chapter we describe fat tailed distributions with infinite variance.

Fat-tailed distributions have been found in many financial economic

variables ranging from forecasting returns on financial assets to model-

ing recovery distributions in bankruptcies. They have also been found in

numerous insurance applications such as catastrophic insurance claims

and in value-at-risk measures employed by risk managers.

In this chapter, we review the related concepts of fat-tailed, power-

law and Levy-stable distributions, scaling and self-similarity, as well as

explore the mechanisms that generate these distributions. We discuss the

key intuition relative to the applicability of fat-tailed or scaling pro-

cesses to finance: In a fat-tailed or scaling world (as opposed to an

ergodic world), the past does not offer an exhaustive set of possible con-

figurations. Adopting, as an approximation, a scaling description of

financial phenomena implies the belief that only a small space of possi-

ble configurations has been explored; vast regions remain unexplored.

We begin with the mathematics of fat-tailed processes, followed by

a discussion of classical Extreme Value Theory for independent and

identically distributed sequences. We then explore the consequences of

eliminating the assumption of independence and discuss different con-

cepts of scaling and self similarity. Finally, we present evidence of fat

tails in financial phenomena and discuss applications of Extreme Value

Theory.

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