Luís A. Nunes Amaral

We study the distribution of fluctuations of the S&P 500 index over a time scale Delta t by analyzing three distinct databases. Database (i) contains approximately 1200000 records, sampled at 1-min intervals, for the IS-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 one month. We find that the distributions for Delta t less than or equal to 4 d (1560 min) are consistent with a power-law asymptotic behavior, characterized by an exponent alpha approximate to 3, well outside the stable Levy 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 18-year period 1984-1997, and database (v) contains 4649 daily records of the Hang-Seng index for the Is-year period 1980-1997. 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)(x) approximate to 4 d, our results are consistent with a slow convergence to Gaussian behavior.