Indication of multiscaling in the volatility return intervals of stock markets

Abstract

The distribution of the return intervals $\tau$ 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 ${\mu }_m\equiv <(\tau /<\tau >{)}^m{>}^{1/m}$, which show a certain trend with the mean interval $<\tau >$. 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 $<\tau >$. 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 $<\tau >$, 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 $10<<\tau >\le 100$, and find the exponent $\alpha$ from the power law fitting ${\mu }_m\sim <\tau {>}^{\alpha }$ has a narrow distribution around $\alpha \ne 0$ 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.

19 pages, 6 figures

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