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 nonlinear mechanism, we investigate intraday data sets of 500 stocks which consist of 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 ${\mu }_m\equiv {⟨{\left(\tau /⟨\tau ⟩\right)}^m⟩}^{1/m}$, 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 ranges of $⟨\tau ⟩$. Those substantial differences suggest that nonlinear 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 that the exponent $\alpha$ from the power law fitting ${\mu }_m\sim {⟨\tau ⟩}^{\alpha }$ has a narrow distribution around $\alpha \ne 0$ which depends 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 exhibits multiscaling behavior due to the nonlinear correlations in the original volatility.

• Received 30 July 2007

DOI:https://doi.org/10.1103/PhysRevE.77.016109