Statistical properties of the volatility of price fluctuations.

Liu Y, Gopikrishnan P, Cizeau P, Meyer M, Peng CK, Stanley HE.

We study the statistical properties of volatility, measured by locally
averaging over a time window T, the absolute value of price changes
over a short time interval deltat. We analyze the S&P 500 stock index
for the 13-year period Jan. 1984 to Dec. 1996. We find that the
cumulative distribution of the volatility is consistent with a
power-law asymptotic behavior, characterized by an exponent mu
approximately 3, similar to what is found for the distribution of
price changes. The volatility distribution retains the same functional
form for a range of values of T. Further, we study the volatility
correlations by using the power spectrum analysis. Both methods
support a power law decay of the correlation function and give
consistent estimates of the relevant scaling exponents. Also, both
methods show the presence of a crossover at approximately 1.5 days. In
addition, we extend these results to the volatility of individual
companies by analyzing a data base comprising all trades for the
largest 500 U.S. companies over the two-year period Jan. 1994 to
Dec. 1995.