Table 1.
The values of ,
,
,
,
and
for the six indices.
Figure 1.
With set to be −4, −3, −2, −1, 0, 1, 2, 3 and 4 respectively, time series
is simulated 100 times for
. The corresponding
is computed and averaged for each
. This plot shows a linear relation of
and
, i.e.,
, and this result remains robust for
between 0.9 and 1.1.
Figure 2.
The return-volatility correlation functions for the S&P 500 and Shanghai indices, and for the corresponding simulations.
The S&P 500 and Shanghai indices are simulated with and
, respectively. Dashed lines show an exponential fit
with
and
for the S&P 500 Index and the Shanghai Index.
Figure 3.
The return-volatility correlation functions for the four indices and the corresponding simulations.
The Nikkei 225, FTSE 100, Hangseng and DAX indices are simulated with ,
,
and
, respectively. Dashed lines show an exponential fit
with
for the Nikkei 225 Index,
for the FTSE 100 Index,
for the Hangseng Index and
for the DAX Index.
Table 2.
The values of and
of the exponential fit
for the six indices and the corresponding simulations.
Figure 4.
The auto-correlation functions of volatilities for the S&P 500 and Shanghai indices, and for the corresponding simulations.
For clarity, the curves for the S&P 500 Index have been shifted down by a factor of 10.
Figure 5.
The cumulative distributions of absolute returns for the S&P 500 and Shanghai indices, and for the corresponding simulations.
For clarity, the curves for the S&P 500 Index have been shifted left by a factor of 8.5.
Figure 6.
The return-volatility correlation functions for the simulated results of the S&P 500 and Shanghai indices, and for those of the controls.
The S&P 500 and Shanghai indices exhibit the leverage and anti-leverage effects, respectively. For the leverage effect, we consider two cases: is asymmetric;
is symmetric. The latter is the control. For the anti-leverage effect, we consider the following cases: both
and
are asymmetric; only
is asymmetric; only
is asymmetric; both
and
are symmetric. The last three cases are controls. For each case, the simulation is performed for 100 times, and the average
is displayed.