Indication of multiscaling in the volatility return intervals of stock markets
Fengzhong Wang, Kazuko Yamasaki, Shlomo Havlin, and H. Eugene Stanley

The distribution of the return intervals $\tau$ between price volatilities above a threshold height $q$ for financial records has been approximated by a scaling behavior. To explore how accurate is the scaling and therefore understand the underlined non-linear mechanism, we investigate intraday datasets of 500 stocks which consist of the Standard \& Poor's 500 index. We show that the cumulative distribution of return intervals has systematic deviations from scaling. We support this finding by studying the $m$-th moment $\mu_m \equiv \langle(\tau/\langle\tau\rangle)^m\rangle^{1/m}$, which show a certain trend with the mean interval $\langle\tau\rangle$. We generate surrogate records using the Schreiber method, and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most range of $\langle\tau\rangle$. Those substantial differences suggest that non-linear correlations in the original volatility sequence account for the deviations from a single scaling law. We also find that the original and surrogate records exhibit slight tendencies for short and long $\langle\tau\rangle$, due to the discreteness and finite size effects of the records respectively. To avoid as possible those effects for testing the multiscaling behavior, we investigate the moments in the range $10&lt\langle\tau\rangle\leq100$, and find the exponent $\alpha$ from the power law fitting $\mu_m\sim\langle\tau\rangle^\alpha$ has a narrow distribution around $\alpha\neq0$ which depend on $m$ for the 500 stocks. The distribution of $\alpha$ for the surrogate records are very narrow and centered around $\alpha=0$. This suggests that the return interval distribution exhibit multiscaling behavior due to the non-linear correlations in the original volatility.

© 2008 The American Physical Society.