1.6 Financial applications of stable laws
21
Table 1.3: Fits to 4444 JPY/USD exchange rate returns from the period De-
cember 1, 1978 - January 31, 1991. Test statistics and the corre-
sponding p-values (in parentheses) are also given.
|
Parameters: |
ασ |
β______μ |
|
α-stable fit |
1.3274 0.0020 |
-0.1393 -0.0003 |
|
Gaussian fit |
___________0.0049 |
-0.0001 |
|
Tests: |
Anderson-Darling |
Kolmogorov |
|
α-stable fit |
4.7833 |
1.4520 |
|
(<0.005) |
(<0.005) | |
|
Gaussian fit |
91.7226 |
6.7574 |
|
( <0.005) |
(<0.005) |
θ STFstab08.xpl
law, in many cases the test statistics and p-values suggest that the fit is not
as good as for these two data sets. This can be seen in Figure 1.8 and Table
1.3, where the calibration results for 4444 daily returns of the Japanese yen
against the US dollar (JPY/USD) exchange rate from December 1, 1978 to
January 31, 1991 are presented. The empirical distribution does not exhibit
power-law tails and the extreme tails are largely overestimated by the stable
distribution. For a risk manager who likes to play safe this may not be a bad
idea, as the stable law overestimates the risks and thus provides an upper limit
of losses. However, from a calibration perspective other distributions, like the
hyperbolic or truncated stable, may be more appropriate for many data sets
(Weron, 2004).