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 [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {}{-69pt} {document} {equation*}{ {{τ}}}{equation*}{document} as [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {}{-69pt} {document} {equation*}Pq({τ})={ {{τ}}}⁻¹f({τ}/{ {{τ}}}){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.