Scaling and memory in volatility return intervals in financial markets

For both stock and currency markets, we study the return intervals τ between the daily volatilities of the price changes that are above a certain threshold q. We find that the distribution function P_q(τ) scales with the mean return interval τ̄ as P_q(τ) = τ̄^-1f(τ/τ̄). The scaling function f(x) is similar in form for all seven stocks and for all seven currency databases analyzed, and f(x) is consistent with a power-law form, f(x) ∼ x ^-γ with γ ≈ 2. We also quantify how the conditional distribution P_q(τ/τ₀) depends on the previous return interval TQ and find that small (or large) return intervals are more likely to be followed by small (or large) return intervals. This "clustering" of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long-term correlations known to be present in the volatility.