Figure 1.
The probability density function of absolute return volatility and realized volatility of TOPIX Core30 Index members drawn on a log-log plot.
Both of them follow power-law distribution. The slope of realized volatility is a bit larger than that of absolute return volatility
, which indicates that realized volatility has slightly larger fat tails than absolute return volatility. For realized volatility about 1996 of the 2500 power law fitness KS tests fail to reject the null while for absolute return volatility about 1482 of the 2500 power law fitness KS tests failed to reject the null. The results suggest that the power law distribution may fit both of them but realized volatility has better fit with power law compared to absolute return volatility. The power law fitness KS test details may refer [30], [31].
Figure 2.
The distribution peak (near 0) of realized volatility changes between neighboring days is much sharper than of absolute return volatility changes
.
The kurtosis of realized volatility is 105 which is much higher than the kurtosis of absolute return volatility which is 61. Furthermore since we had normalized the variance of both values to 1. The differ of kurtosis are mostly contributed by the relations between neighboring days. The result indicates that the realized volatility is much smoother than absolute return volatility. Black curve stands for absolute return volatility of 30 TOPIX Core30 Index members while red dash curve represents realized volatility.
Figure 3.
Short-term effect of realized volatility is stronger than that of absolute return volatility.
Shown is the mean conditional volatility and
for both absolute return volatility (black triangles) and realized volatility (red squares). Compared to absolute return volatility, realized volatility has stronger short-term effect because the red square line is above the black triangle line all the time except for the lower left points.
Figure 4.
The conditional probability density for the largest and smallest 1/6th portion of the absolute return volatility (black line) and realized volatility (blue dots).
The cross-over area (gray area) of absolute return volatility is much larger than the cross-over area (dark gray area) of realized volatility. Noted that we had normalized the variance of both values to 1, the results may mostly reflect that the neighboring days' memory of and
are significantly different.
Figure 5.
Long term memory effect in volatility subset clusters.
Shown is the mean conditional volatility of the absolute return volatility (black triangles) and the realized volatility (red squares) given consecutive values that are above (+) or below (−) the median of the entire volatility data set. The upper part of the curves is for + clusters while the lower part is for – clusters. For the + clusters, the mean conditional volatilities for both methods increase with the size of the cluster, behavior opposite to that for the – clusters, indicating the presence of long-term memory in both volatility methods.
Figure 6.
Hurst exponents of realized volatility (squares) are significant higher than the hurst exponent of absolute return volatility (triangles).
Additionally the Hurst exponent of realized volatility increases with the decreasing of sampling interval .
Figure 7.
The cross correlation between average realized volatility and average absolute return volatility is much higher than cross correlation between any separate realized volatility and absolute return volatility of each stock.
(a) shows an example time series, realized volatility and absolute return volatility
of the stock Nintendo, and the average correlation coefficients of all TOPIX Core30 components
; (b) shows the average
and
time series of all TOPIX Core30 components with the correlation coefficient between them is 0.65.
Figure 8.
Different multiscale entropy patterns for average realized volatility (squares) and average absolute return volatility
(triangles).
The values of entropy depend on the scale factor. For scale one, time series are assigned the much higher value of entropy than the entropy value for
time series. Following the increase of the scale, the value of entropy for
decrease, while the entropy value for
is increasing. Two entropy values become closer for lager scales.