Scaling of the distribution of fluctuations of financial market indices

Abstract

We study the distribution of fluctuations over a time scale $Δt$ (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 $Δt$, where $Δt$ varies approximately over a factor of 10^4 - from 1 min up to more than 1 month. We find that the distributions for $Δt \leq$ 4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent $α\approx 3$, well outside the stable Lévy regime $0 < α< 2$. 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 $α$ 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 $(Δt)_{\times} \approx 4$ days, our results are consistent with slow convergence to Gaussian behavior.

12 pages in multicol LaTeX format with 27 postscript figures (Submitted to PRE May 20, 1999). See http://polymer.bu.edu/~amaral/Professional.html for more of our work on this area