The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 352 Wednesday, February 4, 2004 1:00 PM

352 The Mathematics of Financial Modeling and Investment Management

SCALING, STABLE LAWS, AND FAT TAILS

Let’s begin with a review of the different but related concepts and prop-

erties of fat tails, power laws, and stable laws. These concepts appear

frequently in the financial and economic literature, applied to both ran-

dom variables and stochastic processes.

Fat Tails

Consider a random variable X. By definition, X is a real-valued function

from the set Ωof the possible outcomes to the set R of real numbers,

such that the set (X ≤x) is an event. Recall from Chapter 6 that if P(X ≤

x) is the probability of the event (X ≤x), the function F(x) = P(X ≤x) is a

well-defined function for every real number x. The function F(x) is called

the cumulative distribution function, or simply the distribution function,

of the random variable X. Note that X denotes a function Ω →R, x is a

real variable, and F(x) is an ordinary real-valued function that assumes

values in the interval [0,1]. If the function F(x) admits a derivative

fx()= dF x()

---------------

dx

The function f(x) is called the probability density of the random vari-

able X. The function Fx()= 1 – Fx()is the tail of the distribution F(x).

The function Fx()is called the survival function.

Fat tails are somewhat arbitrarily defined. Intuitively, a fat-tailed distri-

bution is a distribution that has more weight in the tails than some refer-

ence distribution. The exponential decay of the tail is generally assumed as

the borderline separating fat-tailed from light-tailed distributions. In the lit-

erature, distributions with a power-law decay of the tails are referred to as

heavy-tailed distributions. It is sometimes assumed that the reference distri-

bution is Gaussian (i.e., normal), but this is unsatisfactory; it implies, for

instance, that exponential distributions are fat-tailed because Gaussian tails

decay as the square of an exponential and thus faster than an exponential.

These characterizations of fat-tailedness (or heavy-tailedness) are not

convenient from a mathematical and statistical point of view. It would be

preferable to define fat-tailedness in terms of a function of some essential

property that can be associated to it. Several proposals have been

advanced. Widely used definitions focus on the moments of the distribu-

tion. Definitions of fat-tailedness based on a single moment focus either on

the second moment, the variance, or the kurtosis, defined as the fourth

moment divided by the square of the variance. In fact, a distribution is

often considered fat-tailed if its variance is infinite or if it is leptokurtic