Scaling and memory of intraday volatility return intervals in stock markets

We study the return interval τ 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 Pq (τ) scales with the mean interval τ̄ as Pq (τ) = τ̄ -1 f (τ/τ̄), 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) ∼ e-a xγ, with γ=0.38±0.05 and a=3.9±0.5, which indicates the existence of correlations. We analyze the conditional probability distribution Pq (τ/ τ0) for τ following a certain interval τ0, and find Pq (τ/ τ0) depends on τ0, which demonstrates memory in intraday return intervals. Also, we find that the mean conditional interval τ/ τ increases with τ0, consistent with the memory found for Pq (τ/ τ0). Moreover, we find that return interval records, in addition to having short-term correlations as demonstrated by Pq (τ/ τ0), have long-term correlations with correlation exponents similar to that of volatility records.