9.4 Comparison of Implied and Historical SPD

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 $ 500$. The $ 3$ month IBT implied SPD for Friday, September $ 17$, $ 1999$ is slightly positively skewed. It may be due to the fact that in the months preceeding June $ 1999$, the month in which the $ 3$ month implied SPD was estimated, the DAX index stayed within a quite narrow horizontal range of index values after a substantial downturn in the $ 3$rd quarter of $ 1998$ (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($ f^*$)$ >$skew($ g^*$).

Figure 9.3: Comparison of Skewness time series for $ 30$ periods.
\includegraphics[width=1.4\defpicwidth]{SkewCompTSPS.ps}

Figure 9.4: Comparison of Kurtosis time series for $ 30$ periods.
\includegraphics[width=1.4\defpicwidth]{KurtCompTSPS.ps}

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 $ 500$. The $ 3$ month IBT implied SPD for Friday, October $ 16$, $ 1998$ 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 $ 1998$ after a period of $ 8$ 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.