Time evolution of a financial market index as an effect of the joint action of Gaussian and Lévy fluctuations

We study the sequences of the Standard & Poor's 500 Index quotations by interpreting them as the spatial trajectories of a random walker. We interpret the resulting diffusion distribution by means of a fluctuation generator consisting in the joint action of two stochastic variables. These stochastic variables are independent but characterized by correlation functions with the same inverse power law. One variable is dichotomous and the other is gaussian. We prove that this makes it possible to account for both the rescaling properties of the distribution and the quenching at large distances of the tails of distribution. At intermediate distances the distribution exhibits tails corresponding to the Lévy processes generated by dichotomous fluctuations, and at larger distances a transition to tails with a faster decay is produced, in accordance with the experimental data (Mantegna and Stanley, Nature 1995; 376:47–49). We argue that the presence of these two distinct fluctuations reflects the actions of two different categories of financial operators. The Gaussian category establishes conditions favourable to the validity of the central limit theorems: a large number of price‐taking operators. The Lévy category implies the action of a small number of price‐making operators. Copyright © 1999 John Wiley & Sons, Ltd.