  We study the statistical properties of volatility---a measure of how much the
market is likely to fluctuate. We estimate the volatility by the local average
of the absolute price changes. We analyze (a) the S&P 500 stock index for the
13-year period Jan 1984 to Dec 1996 and (b) the market capitalizations of the
largest 500 companies registered in the Trades and Quotes data base,
documenting all trades for all the securities listed in the three major stock
exchanges in the US for the 2-year period Jan 1994 to Dec 1995. For the S&P 500
index, the probability density function of the volatility can be fit with a
log-normal form in the center. However, the asymptotic behavior is better
described by a power-law distribution characterized by an exponent 1 + \mu
\approx 4. For individual companies, we find a power law asymptotic behavior of
the probability distribution of volatility with exponent 1 + \mu \approx 4,
similar to the S&P 500 index. In addition, we find that the volatility
distribution scales for a range of time intervals. Further, we study the
correlation function of the volatility and find power law decay with long
persistence for the S&P 500 index and the individual companies with a crossover
at approximately 1.5 days. To quantify the power-law correlations, we apply
power spectrum analysis and a recently-developed modified root-mean-square
analysis, termed detrended fluctuation analysis (DFA). For the S&P 500 index,
DFA estimates for the exponents characterizing the power law correlations are
\alpha_1=0.66 for short time scales (within \approx 1.5 days) and \alpha_2=0.93
for longer time scales (up to a year). For individual companies, we find
\alpha_1=0.60 and \alpha_2=0.74, respectively. The power spectrum gives
consistent estimates of the two power-law exponents.
