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 $r(\ensuremath{\tau})$ after some time $\ensuremath{\tau}$ can be described by a (truncated) L\'evy-stable distribution ${L}_{\ensuremath{\alpha}}(r)$ with some index $0<\ensuremath{\alpha}<~2.$ While the L\'evy distribution cannot be expressed in a closed form, one can identify its parameters by testing the dependence of the central peak height on $\ensuremath{\tau}$ 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 L\'evy distribution with $\ensuremath{\alpha}=1.4.$ 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 $\ensuremath{\alpha}\ensuremath{\cong}3,$ thus deviating from the L\'evy 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 L\'evy distribution while the tails exhibit a power-law distribution with an exponent $\ensuremath{\alpha}2,$ namely, beyond the range of L\'evy-stable distributions. Our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results.

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