Scaling and memory in volatility return intervals in financial markets

doi:10.1073/pnas.0502613102

Scaling and memory in volatility return intervals in financial markets

^*Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215; ^†Department of Environmental Sciences, Tokyo University of Information Sciences, Chiba 265-8501, Japan; ^‡Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel; and ^¶Institut für Theoretische Physik III, Justus-Liebig-Universität, D-35392 Giessen, Germany

Contributed by H. Eugene Stanley, April 1, 2005

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 $\text{[math]}$ as $\text{[math]}$ . 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 τ₀ 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.

Footnotes

↵ § To whom correspondence should be addressed. E-mail: havlin{at}ophir.ph.biv.ac.il.
Author contributions: S.H. and H.E.S. designed research; K.Y., L.M., S.H., and H.E.S. performed research; A.B. contributed new reagents/analytic tools; A.B. analyzed data; and S.H. wrote the paper.
Abbreviations: pdf, probability density function; S&P 500, Standard and Poor's 500 Index; USD, U.S. dollar; JPY, Japanese yen; SEK, Swedish krona.