Long-time fluctuations in a dynamical model of stock market indices.

Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. Recent empirical studies of stock market indices examined whether the distribution P(r) of returns r(tau) after some time tau can be described by a (truncated) Lévy-stable distribution L(alpha)(r) with some index 0<alpha< or =2. While the Lévy distribution cannot be expressed in a closed form, one can identify its parameters by testing the dependence of the central peak height on tau as well as the power-law decay of the tails. In an earlier study [R. N. Mantegna and H. E. Stanley, Nature (London) 376, 46 (1995)] it was found that the behavior of the central peak of P(r) for the Standard & Poor 500 index is consistent with the Lévy distribution with alpha=1.4. In a more recent study [P. Gopikrishnan et al., Phys. Rev. E 60, 5305 (1999)] it was found that the tails of P(r) exhibit a power-law decay, with an exponent alpha congruent with 3, thus deviating from the Lévy distribution. In this paper we study the distribution of returns in a generic model that describes the dynamics of stock market indices. For the distributions P(r) generated by this model, we observe that the scaling of the central peak is consistent with a Lévy distribution while the tails exhibit a power-law distribution with an exponent alpha>2, namely, beyond the range of Lévy-stable distributions. Our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results.