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 12ptminimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs -69pt document equation* {{}}equation*document as 12ptminimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs -69pt document equation*Pₐ () = {{}}^-1f (/ {{}}) equation*document. 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|τ 0) depends on the previous return interval τ 0 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.