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 $P\left(r\right)\mathrm{}$ of returns $r\left(\tau \right)$ after some time $\tau$ can be described by a (truncated) Lévy-stable distribution $L_{\alpha }\mathrm{}\left(r\right)\mathrm{}$ with some index $0<\mathrm{}\alpha <~2.$ 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 $P\left(r\right)\mathrm{}$ for the Standard & Poor 500 index is consistent with the Lévy distribution with $\alpha =\mathrm{1.4.}\mathrm{}$ In a more recent study [P. Gopikrishnan et al., Phys. Rev. E $60,$ 5305 (1999)] it was found that the tails of $P\left(r\right)\mathrm{}$ exhibit a power-law decay, with an exponent $\alpha \cong 3,$ 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 $P\left(r\right)\mathrm{}$ 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.