13-Fat Tails-Scaling-Stabl Page 390 Wednesday, February 4, 2004 1:00 PM
390 The Mathematics of Financial Modeling and Investment ManagementIn the area of stock price returns at short time horizons, initial find-
ings by Mantegna and Stanley^31 seemed to indicate truncated inverse
power laws with exponents in the range 1.4–1.6, well within the scaling
regime. More recent findings by Plerou et al^32 point to an exponent 3
without truncation, well outside the Levy stable regime. Johanson and
Sornette^33 suggest that market crashes are not the fat tails of return dis-
tributions, but outliers. Still other studies, for instance Laherre and Sor-
nette,^34 found that returns are better described by a function rather than
by a single exponent, thus creating multifractal distributions.
Applying the notion of stable laws to stock price returns raises addi-
tional questions. The infinite variance property of stable laws is some-
what in contrast with empirical findings about stock returns, most of
which seem to indicate finite variance, though higher order moments
might become infinite. This is in agreement with the use of volatility as a
key parameter in financial risk management. Stable laws, on the other
hand, would require abandoning the notion of volatility. It seems fair to
conclude that stable laws are not a good approximation to stock
returns, though inverse power laws with exponent >2 might still hold.
As noted above, the fundamental practical importance of the pres-
ence of stable laws in economic and financial phenomena is that they
would render risk management and financial decision-making difficult:
If variables are governed by stable laws, there is no possibility of diver-
sifying risk. Modeling with fat-tailed distributions has the status of a
theoretical hypothesis as it implies extrapolating that the future will
bring unbounded innovation. In the insurance industry, for example, the
assumption of scaling is appropriate in domains such as catastrophe
insurance, where there is no natural bound to the size of catastrophes
and where experience has shown that very large catastrophic events do
indeed occur.(^31) R. N. Mantegna and H.E. Stanley, “Scaling Behavior in the Dynamics of an Eco-
nomic Index,” Nature 46 (1995), p. 376.
(^32) V. Plerou, P. Gopikrishnan, L.A.N. Amaral, M. Meyer, and H.E. Stanley, “Scaling
of the Distribution of Price Fluctuations of Individual Companies,” Physical Review
E 60, no. 6, Part A (December 1999), pp. 6519–6529
(^33) A. Johansen and D. Sornette, “Stock Market Crashes Are Outliers,” European
Physical Journal B 9, no. 1 (February 1998), pp. 141–143.
(^34) J. Laherre and D. Sornette, “Stretched Exponential Distributions in Nature and
Economy: ‘Fat Tails’ with Characteristic Scales,” European Physical Journal B 2
(1998), p. 525.