1.6 Financial applications of stable laws
17
Table 1.1: Fits to 2000 Dow Jones Industrial Average (DJIA) index returns
from the period February 2, 1987 - December 29, 1994. Test statis-
tics and the corresponding p-values based on 1000 simulated samples
(in parentheses) are also given.
|
Parameters: |
ασ |
β______μ |
|
α-stable fit |
1.6411 0.0050 |
-0.0126 0.0005 |
|
Gaussian fit |
___________0.0111 |
0.0003 |
|
Tests: |
Anderson-Darling |
Kolmogorov |
|
α-stable fit |
0.6441 |
0.5583 |
|
(0.020) |
(0.500) | |
|
Gaussian fit |
+∞ |
4.6353 |
|
( <0.005) |
(<0.005) |
θ STFstab06.xpl
1.6 Financial applications of stable laws
Many techniques in modern finance rely heavily on the assumption that the
random variables under investigation follow a Gaussian distribution. However,
time series observed in finance - but also in other applications - often deviate
from the Gaussian model, in that their marginal distributions are heavy-tailed
and, possibly, asymmetric. In such situations, the appropriateness of the com-
monly adopted normal assumption is highly questionable.
It is often argued that financial asset returns are the cumulative outcome of
a vast number of pieces of information and individual decisions arriving al-
most continuously in time. Hence, in the presence of heavy-tails it is natural
to assume that they are approximately governed by a stable non-Gaussian dis-
tribution. Other leptokurtic distributions, including Student’s t, Weibull, and
hyperbolic, lack the attractive central limit property.
Stable distributions have been successfully fit to stock returns, excess bond
returns, foreign exchange rates, commodity price returns and real estate returns
(McCulloch, 1996; Rachev and Mittnik, 2000). In recent years, however, several
studies have found, what appears to be strong evidence against the stable model
(Gopikrishnan et al., 1999; McCulloch, 1997). These studies have estimated the