arXiv2008
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 $τ$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $τ$, $P_q(τ)$, assuming a stretched exponential function, $P_q(τ) \sim e^{-τ^γ}$. We find that the exponent $γ$ 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 $γ$ depends on four essential factors, capitalization, risk, number of trades and return. We show that $γ$ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that $γ$ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of $τ$, $μ_m \equiv <(τ/<τ>)^m>^{1/m}$, in the range of $10 < <τ> \le 100$ by a power-law, $μ_m \sim <τ>^δ$. The exponent $δ$ 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 $γ$. Moreover, we show that $δ$ decreases with $γ$ 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.