The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 389 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 389

sign of scaling or inverse power laws, but these variables cover a broad

enough range of values to make the scaling approximation meaningful.

The first example of scaling laws in economics is due to the econo-

mist Pareto in the nineteenth century. Pareto observed that, above some

threshold, the proportion of individuals with an income in excess of x is

inversely proportional to x. Generalizing, a distribution of the type

Fx()= PX ( >x)= ------^1 for x ≥ 1

α

x

is called a Pareto law.

The presence of scaling laws has also been researched in price

behavior. In 1963 Mandelbrot^27 observed self-similarity in economic

time series when he discovered that cotton price time series had approx-

imately the same shape at different time scales. Based on this empirical

discovery, Mandelbrot later proposed stable laws and fractional Brown-

ian motions as a model for price behavior.

Since Mandelbrot’s observations, researchers have been trying to

prove or disprove the existence of inverse power laws in the area of

asset returns. The jury is still out. A first remark is that scaling laws of

returns apply only to short-term (from one minute to a few days)

returns. Beyond this time horizon, returns exhibit complex behavior

that depends on the length and positioning of the observation periods.

One of the first systematic studies of the distribution of high-fre-

quency data was conducted by Zurich-based Olsen & Associates on

exchange rates.^28 Olsen researchers found that many exchange rates fol-

low scaling laws with exponents < 2. More recently, several as yet

unpublished studies have look at fat-tailed returns in less traded curren-

cies: Payaslioglu^29 used tail index estimation for the Turkish lira and

Chobanov, Mateev, Mittnik and Rachev^30 looked at the Bulgarian lev.

(^27) Benoit Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of
Business 36 (1963), pp. 394–419.
(^28) U.A. Muller, M.M. Dacorogna, and O.V. Pictet, “Heavy Tails in High Frequency
Financial Data,” in R. Adler, R. Feldman, and M.S. Taqqu (eds.) A Practical Guide
to Heavy Tails: Statistical Techniques for Analysing Heavy-Tailed Distributions
(Boston: Birkhauser, 1997).
(^29) Cem Payaslioglu, “Tail Behavior of Return Distributions of Exchange Rates under
Different Regimes: A Case Study for Turkey.”
(^30) G. Chobanov, P. Mateev, S. Mittnik, and S. Rachev, “Modeling the Distribution
of Highly Volatile Exchange-rate Time Series” in P.M. Robinson and M. Rosenblatt
(eds.), Athens Conference on Applied Probability and Time Series, Volume II: Time
Series Analysis (New York: Springer, 1996).