  We study the distribution of fluctuations over a time scale $\Delta t$ (i.e.,
the returns) of the S&P 500 index by analyzing three distinct databases.
Database (i) contains approximately 1 million records sampled at 1 min
intervals for the 13-year period 1984-1996, database (ii) contains 8686 daily
records for the 35-year period 1962-1996, and database (iii) contains 852
monthly records for the 71-year period 1926-1996. We compute the probability
distributions of returns over a time scale $\Delta t$, where $\Delta t$ varies
approximately over a factor of 10^4 - from 1 min up to more than 1 month. We
find that the distributions for $\Delta t \leq$ 4 days (1560 mins) are
consistent with a power-law asymptotic behavior, characterized by an exponent
$\alpha \approx 3$, well outside the stable L\'evy regime $0 < \alpha < 2$. To
test the robustness of the S&P result, we perform a parallel analysis on two
other financial market indices. Database (iv) contains 3560 daily records of
the NIKKEI index for the 14-year period 1984-97, and database (v) contains 4649
daily records of the Hang-Seng index for the 18-year period 1980-97. We find
estimates of $\alpha$ consistent with those describing the distribution of S&P
500 daily-returns. One possible reason for the scaling of these distributions
is the long persistence of the autocorrelation function of the volatility. For
time scales longer than $(\Delta t)_{\times} \approx 4$ days, our results are
consistent with slow convergence to Gaussian behavior.

Statistical Mechanics;Disordered Systems and Neural Networks;Statistical Finance