The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 388 Wednesday, February 4, 2004 1:00 PM

388 The Mathematics of Financial Modeling and Investment Management

infinite sequence of independent and identically distributed variables Xi,

i ≥ 1. Create consecutive nonoverlapping blocks of m variables and

define the corresponding aggregated sequence of level m averaging over

each block as follows:

km

Xk() m

1

= ----- ∑ Xi

mi = (k – 1 )m + 1

A sequence is called exactly self-similar if, for any integer m the fol-

lowing relationship holds:

D ()

X = m

1 – H

X

m

A stationary sequence is called asymptotically self-similar if the above

relationship holds only for m → ∞.

When we apply the notion of scaling to stochastic processes—the

natural setting for economics and finance—we have to abandon the sim-

ple characterization of scaling as inverse power laws. Though the scal-

ing property is in itself characterized through simple power laws, the

scaling processes are complex and rich mathematical structures entail-

ing a variety of distributions and correlation functions. In particular, the

long-range correlation structure of the process plays a role as important

as the distribution of its variables.

EVIDENCE OF FAT TAILS IN FINANCIAL VARIABLES

To appreciate the applicability of scaling laws, let’s first look at the range of

variation of the economic and financial variables with which they are gen-

erally associated. Variables such as income, personal wealth, corporate size,

and market capitalization span many orders of magnitude. Large insurance

claims cover at least three orders of magnitude, with the largest claims

reaching billions of dollars.^25 Bankruptcies cover a similarly broad range of

orders of magnitude.^26 Daily stock returns span some two orders of magni-

tude. However, economic variables such as interest rates or GNP rates span

a smaller set of values. Obviously the range of variables is not in itself a

(^25) See Embrechts, Kluppelberg, and Mikosch, Modelling Extremal Events for Insur-
ance and Finance.
(^26) For empirical evidence on the Japanese experience, see H. Aoyama, Y. Nagahara,
M. P. Okazaki, W. Souma, H. Takayasu, and M. Takayasu, “Pareto’s Law for In-
come of Individuals and Debt of Bankrupt Companies,” Cond-Mat 0006038, 2000.