Multifactor Analysis of Multiscaling in Volatility Return Intervals
  We study the volatility time series of 1137 most traded stocks in the US stock markets for the two-year period 2001-02 and analyze their return intervals $\tau$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $\tau$, $P_q(\tau)$, assuming a stretched exponential function, $P_q(\tau) \sim e^{-\tau^\gamma}$. We find that the exponent $\gamma$ depends on the threshold in the range between $q=1$ and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how $\gamma$ depends on four essential factors, capitalization, risk, number of trades and return. We show that $\gamma$ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that $\gamma$ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of $\tau$, $\mu_m \equiv <(\tau/<\tau>)^m>^{1/m}$, in the range of $10 < <\tau> \le 100$ by a power-law, $\mu_m \sim <\tau>^\delta$. The exponent $\delta$ is found also to depend on the capitalization, risk and return but not on the number of trades, and its tendency is opposite to that of $\gamma$. Moreover, we show that $\delta$ decreases with $\gamma$ approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization. 
Fengzhong Wang, Kazuko Yamasaki, Shlomo Havlin and H. Eugene Stanley