We study the volatility time series of 1137 most traded stocks in the U.S. stock markets for the two-year period 2001–2002 and analyze their return intervals $\tau$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $\tau$, $P_q\left(\tau \right)$, assuming a stretched exponential function, $P_q\left(\tau \right)\sim e^{-{\tau }^{\gamma }}$. 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 $\tau$, ${\mu }_m\equiv {\mathbf{⟨}{(\tau ∕⟨\tau ⟩)}^m\mathbf{⟩}}^{1∕m}$, in the range of $10<⟨\tau ⟩⩽100$ 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 increasing $\gamma$ 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.

• Received 22 August 2008

DOI:https://doi.org/10.1103/PhysRevE.79.016103