4 Empirical results
4.4 Estimation of dynamic properties of the state variables with a model
This section contains a comparison between the estimates of some dynamic properties of the state variables obtained by using the kernel estimator ˆf and the full-information estimator ˆf?at each day of July and August 2008. As in Section4.1, the considered quantities are the historical conditional correlation function between the state variables, the conditional Sharpe ratio of an investment in
the IBM stock over the horizon of one day and the conditional skewness and kurtosis of the returns.
In the first four panels of Figure 7the estimates of the four quantities as functions of the date are displayed. They are computed for the contemporaneous value of the conditioning state variables.
For each quantity, the dotted line, labeled “Kernel”, indicates the estimate obtained by using ˆf , while the solid line, labeled “Tilted”, indicates the estimate obtained by using ˆf?. We can see the daily stock return and RV during the two considered months in the center and right lower panels.
The variation of the estimated quantities over time has potentially three causes: the changing values of the conditioning state variables, the statistical variability in the estimation and any possi-ble model misspecification. First, stock return and RV vary over time, as we can see in the center and right lower panels of Figure7, and the considered quantities are conditional on specific values of these variables. Second, the estimation of the model parameters is performed with different data samples. Third, the choices of the state variables and the parametrization of Equation (1) are assumptions. The contribution given by the two last points to the time variation of the estimated quantities is small. Concerning the different data samples, let us make two general considerations.
First, the application of the kernel estimator to time series with several hundreds observations that differ only for few of them most likely does not lead to statistically different results. Second, it seems reasonable to assume that every option mid-quote considered after applying the filters de-scribed in Section 3.1 carries similar information about the data generating process. Moreover, as shown in Section 4.2, the XMM estimates of the SDF parameter vector are quite stable over time and this stability supports the validity of the assumptions. Then, as a whole, a major part of the variation in the time series of the estimated quantities is caused by the changing value of the conditioning state variables and only a minor part is due to the statistical variability.
The range of variation over time of the estimates obtained by ˆf? is mainly smaller than the one of the estimates obtained by ˆf . Moreover, the range of variation of the latter estimates has two regimes: it is broader before 2008/07/17, when the state variables have extreme values, and lower afterwards, when the state variables assume very high but not extreme values. In the first period the stock return varies approximatively in the range [−0.03 : 0.03] and the RV does it in [0.01 : 0.025], while in the second period the return varies approximatively in the range [−0.015 : 0.015] and the RV does it in [0.01 : 0.015]. To visualize the different parts of the state variables space visited during these two periods, let us consider the upper panels of Figure 4. In the first period the realizations of stock return and RV are scattered in the region for a RV that is higher than 0.01. Differently, in the second period these realizations are concentrated in the part of the
area surrounded by the rectangle (i.e. the area considered in the lower panels) and with an higher RV than 0.01. While we observe two regimes of variability for the estimates obtained by ˆf , no clear separation in different regimes appear when the estimation is performed by ˆf?. Adopting an arbitrage-free pricing model leads to estimates of some dynamic properties of the state variables that are more stable over time.
We see in the left upper panel of Figure7the time series of the estimates of the conditional correlation between the state variables. The estimates obtained by ˆf? are almost always nega-tive, while the estimates obtained by ˆf vary approximatively in the range [−0.25 : 0.15] before 2008/07/17 and are negative afterwards. While the former estimates support the presence of a contemporaneous leverage effect, the latter does it only after 2008/07/17.
We see in the center upper panel the time series of the estimates of the conditional conditional Sharpe ratio of an investment in the IBM stock over the horizon of one day. The estimates ob-tained by ˆf and by ˆf? give very different results. The former are almost always positive, varying approximatively in the range [−0.02 : 0.14] before 2008/07/17 and in the range [−0.02 : 0.06]
afterwards. The latter are extremely stable over time with value at about −0.01. Then, while con-sidering only the time series of stock return and RV encourages, in terms of Sharpe ratio, almost always a long position in the IBM stock, taking into account this time series and the option data in an arbitrage-free pricing model provides an opposite suggestion.
We see in the right upper panel the time series of the estimates of the conditional skewness of the returns. The estimates obtained by ˆf vary approximatively in the range [−0.5 : 0.5] before 2008/07/17 and are almost always negative afterwards. Differently, the estimates obtained by ˆf? are almost always negative before 2008/07/17 and positive afterwards. Considering both the time series of the state variables and the option data under an arbitrage-free pricing model makes the shareholder fear the left tail risk only before 2008/07/17.
Finally, we see in the left lower panel the time series of the estimates of the conditional kurtosis of the returns. The estimates obtained by both the estimators of the transition density vary approx-imatively in the range [1.5 : 6] before 2008/07/17 and in the range [3 : 5.5] afterwards. In the second period, the transition density, estimated in both ways, is leptokurtic. Before 2008/07/17, considering state variables and option data under an arbitrage-free pricing model leads to take into account a greater fat tail risk, almost at any day. The opposite happens in the second period.
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