Scaling and memory in volatility return intervals in financial markets

Abstract

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\bar {{\tau}}}\end{equation*}\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}P_{q}({\tau})={\bar {{\tau}}}^{-1}f({\tau}/{\bar {{\tau}}})\end{equation*}\end{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.

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