At this point it is time to compare implied and historical SPDs. Since by
construction, expectation and variance are adjusted, we focus the comparison on skewness
and kurtosis. Starting with skewness, we can extract from Figure 9.3
that except for one period the IBT implied SPD is systematically more negatively skewed
than the time series SPD, a fact that is quite similar to what Ait-Sahalia, Wang and Yared (2000)
already found for the S&P . The
month IBT implied SPD for Friday, September
,
is slightly positively skewed. It may be due to the fact that in the months
preceeding June
, the month in which the
month implied SPD was estimated, the
DAX index stayed within a quite narrow horizontal range of index values after a
substantial downturn in the
rd quarter of
(see Figure 9.11) and agents
therefore possibly believed index prices lower than the average would be more realistic
to appear. However, this is the only case where skew(
)
skew(
).
The kurtosis time series reveals a similar pattern as the skewness
time series. The IBT SPD has except for one period systematically
more kurtosis than the time series SPD. Again this feature is in
line with what Ait-Sahalia, Wang and Yared (2000) found for the S&P . The
month IBT implied SPD for Friday, October
,
has a
slightly smaller kurtosis than the time series SPD. That is,
investors assigned less probability mass to high and low index
prices. Note that the implied SPD was estimated in July
after a period of
months of booming asset prices (see Figure
9.11). It is comprehensible in such an environment that
high index prices seemed less realistic to appear. Since the
appearance of low index prices seemed to be unrealistic as well,
agents obviously expected the DAX move rather sideways.