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 [formula presented] of returns [formula presented] after some time [formula presented] can be described by a (truncated) Lévy-stable distribution [formula presented] with some index [formula presented] 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 [formula presented] as well as the power-law decay of the tails. In an earlier study [R. N. Mantegna and H. E. Stanley, Nature (London) [formula presented] 46 (1995)] it was found that the behavior of the central peak of [formula presented] for the Standard & Poor 500 index is consistent with the Lévy distribution with [formula presented] In a more recent study [P. Gopikrishnan et al., Phys. Rev. E [formula presented] 5305 (1999)] it was found that the tails of [formula presented] exhibit a power-law decay, with an exponent [formula presented] 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 [formula presented] 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 [formula presented] 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.