Multifactor analysis of multiscaling in volatility return intervals

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(tau), assuming a stretched exponential function, P-q(tau)similar to 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)equivalent to <(tau/<tau >)(m)>(1/m), in the range of 10 <<tau ><= 100 by a power law, mu(m)similar to <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.