Scaling and memory in volatility return intervals for the 500 stocks belonging to the S&P 500 index
Center for Polymer Studies and Department of Physics, Boston
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
H. Eugene Stanley
Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 USA
Last modified: May 25, 2007
We investigate the return intervals τ between volatilities larger than a given threshold q for 500 intraday datasets, each of them is one stock of the Standard & Poor's 500 index (S&P 500). For q≤6, we find that the probability density function (PDF) Pq(τ) scales with the mean interval <τ> as Pq(τ)=<τ>-1f(τ/<τ>). We find the scaling function f(x) is similar for all 500 stocks, and is well approximated by a stretched exponential, f(x)≌c•exp(-a•xγ), with exponent γ=0.42±0.07, while the parameters c and a are determined by γ. We also examine the return intervals for very large q by aggregating all 500 stocks. We find their PDF’s exhibit power law decays for small τ, as well as stretched exponential decays for large τ. Further, we study the time ordering of return intervals. We investigate the conditional PDF Pq(τ|τ0) for a return interval τ following a given interval τ0. For different bins of τ0, we find the conditional PDF’s significantly differ and therefore memory exists in the time ordered sequence of intervals. By examining mean intervals after a certain interval τ0, we show that the memory is persistent for a very long time. Moreover, we study the connection between correlations in the volatility and in the return intervals. To both series, we employ the autocorrelation function and Detrended Fluctuation Analysis (DFA) method. We show similar correlations, which suggests the correlations in return intervals arise from correlations in the volatility.