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 τ, which are time intervals between volatilities above a given threshold q. We explore the probability density function of τ, Pq (τ), assuming a stretched exponential function, Pq (τ) ∼ e- τγ. 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, μm (τ/ τ) m 1/m, in the range of 10< τ ≤100 by a power law, μm ∼ τδ. 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 increasing 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.