Stable Distributions

1 Stable distributions

Table 1.2: Fits to 1635 Boeing stock price returns from the period July 1, 1997
- December 31, 2003. Test statistics and the corresponding p-values
based on 1000 simulated samples (in parentheses) are also given.

θ STFstab07.xpl

tail exponent directly from the tail observations and commonly have found α
that appears to be significantly greater than 2, well outside the stable domain.
Recall, however, that in Section 1.5.1 we have shown that estimating α only
from the tail observations may be strongly misleading and for samples of typical
size the rejection of the α-stable regime unfounded. Let us see ourselves how
well the stable law describes financial asset returns.

In this section we want to apply the discussed techniques to financial data. Due
to limited space we chose only one estimation method - the regression approach
of Koutrouvelis (1980), as it offers high accuracy at moderate computational
time. We start the empirical analysis with the most prominent example -
the Dow Jones Industrial Average (DJIA) index, see Table 1.1. The data set
covers the period February 2, 1987 - December 29, 1994 and comprises 2000
daily returns. Recall, that it includes the largest crash in Wall Street history
- the Black Monday of October 19, 1987. Clearly the 1.64-stable law offers
a much better fit to the DJIA returns than the Gaussian distribution. Its
superiority, especially in the tails of the distribution, is even better visible in
Figure 1.6.

To make our statistical analysis more sound, we also compare both fits through
Anderson-Darling and Kolmogorov test statistics (D’Agostino and Stephens,
1986). The former may be treated as a weighted Kolmogorov statistics which

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