Indication of multiscaling in the volatility return intervals of stock markets

Abstract

The distribution of the return intervals $τ$ between volatilities above a threshold $q$ 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 $μ_m \equiv <(τ/<τ>)^m>^{1/m}$, which show a certain trend with the mean interval $<τ>$. 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 $<τ>$. 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 $<τ>$, 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 $10<<τ>\leq100$, and find the exponent $α$ from the power law fitting $μ_m\sim<τ>^α$ has a narrow distribution around $α\neq0$ which depend on m for the 500 stocks. The distribution of $α$ for the surrogate records are very narrow and centered around $α=0$. This suggests that the return interval distribution exhibit multiscaling behavior due to the non-linear correlations in the original volatility.

19 pages, 6 figures