Scaling and memory of intraday volatility return intervals in stock markets

Abstract

We study the return interval $\ensuremath{\tau}$ between price volatilities that are above a certain threshold $q$ for 31 intraday data sets, including the Standard and Poor's 500 index and the 30 stocks that form the Dow Jones Industrial index. For different threshold $q$, the probability density function ${P}_{q}(\ensuremath{\tau})$ scales with the mean interval $\overline{\ensuremath{\tau}}$ as ${P}_{q}(\ensuremath{\tau})={\overline{\ensuremath{\tau}}}^{\ensuremath{-}1}f(\ensuremath{\tau}∕\overline{\ensuremath{\tau}})$, similar to that found in daily volatilities. Since the intraday records have significantly more data points compared to the daily records, we could probe for much higher thresholds $q$ and still obtain good statistics. We find that the scaling function $f(x)$ is consistent for all 31 intraday data sets in various time resolutions, and the function is well-approximated by the stretched exponential, $f(x)\ensuremath{\sim}{e}^{\ensuremath{-}a{x}^{\ensuremath{\gamma}}}$, with $\ensuremath{\gamma}=0.38\ifmmode\pm\else\textpm\fi{}0.05$ and $a=3.9\ifmmode\pm\else\textpm\fi{}0.5$, which indicates the existence of correlations. We analyze the conditional probability distribution ${P}_{q}(\ensuremath{\tau}\ensuremath{\mid}{\ensuremath{\tau}}_{0})$ for $\ensuremath{\tau}$ following a certain interval ${\ensuremath{\tau}}_{0}$, and find ${P}_{q}(\ensuremath{\tau}\ensuremath{\mid}{\ensuremath{\tau}}_{0})$ depends on ${\ensuremath{\tau}}_{0}$, which demonstrates memory in intraday return intervals. Also, we find that the mean conditional interval $⟨\ensuremath{\tau}\ensuremath{\mid}{\ensuremath{\tau}}_{0}⟩$ increases with ${\ensuremath{\tau}}_{0}$, consistent with the memory found for ${P}_{q}(\ensuremath{\tau}\ensuremath{\mid}{\ensuremath{\tau}}_{0})$. Moreover, we find that return interval records, in addition to having short-term correlations as demonstrated by ${P}_{q}(\ensuremath{\tau}\ensuremath{\mid}{\ensuremath{\tau}}_{0})$, have long-term correlations with correlation exponents similar to that of volatility records.

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