Scaling of the distribution of fluctuations of financial market indices

Abstract

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 1 200 000 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 one month. We find that the distributions for $\Delta t<~$ 4 d (1560 min) are consistent with a power-law asymptotic behavior, characterized by an exponent $\alpha \approx 3,$ well outside the stable Lévy regime $0<\mathrm{}\alpha <2.\mathrm{}$ 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 1984–1997, and database (v) contains 4649 daily records of the Hang-Seng index for the 18-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 $(\Delta {t)}_{\times }\approx 4$ d, our results are consistent with a slow convergence to Gaussian behavior.