Scaling and memory of intraday volatility return intervals in stock markets

We study the return interval 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(tau) scales with the mean interval (tau) over bar as P-q(tau)=(tau) over bar (-1)f(tau/(tau) over bar), 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)similar to e(-ax gamma), with gamma=0.38 +/- 0.05 and a=3.9 +/- 0.5, which indicates the existence of correlations. We analyze the conditional probability distribution P-q(tau parallel to tau(0)) for tau following a certain interval tau(0), and find P-q(tau parallel to tau(0)) depends on tau(0), which demonstrates memory in intraday return intervals. Also, we find that the mean conditional interval increases with tau(0), consistent with the memory found for P-q(tau parallel to tau(0)). Moreover, we find that return interval records, in addition to having short-term correlations as demonstrated by P-q(tau parallel to tau(0)), have long-term correlations with correlation exponents similar to that of volatility records.