Multiplicative point process as a model of financial markets

Signals consisting of a sequence of pulses show that inherent origin of the 1/f noise is a Brownian fluctuation of the average interevent time between subsequent pulses of the pulse sequence. In this paper we generalize the model of interevent time to reproduce the variety of self-affine time series exhibiting power spectral density S(f) scaling as a power of the frequency f . Furthermore, we analyze a relation between the power-law correlations and the origin of the power-law probability distribution of the signal intensity. We introduce the stochastic multiplicative model for the time intervals between point events and analyze the statistical properties of the signal numerically and analytically. Such model system exhibits power-law spectral density S(f) ∼ 1/f for various values of β, including β = 1/2; 1 and 3/2. Explicit expressions for the power spectra in a low frequency limit and for the distribution density of the interevent time are obtained. The counting statistics of the events is analyzed numerically and analytically, as well. The specific interest of our analysis is related with the financial markets, where long-range correlations of price fluctuations largely depend on the number of transactions. We analyze the spectral density and counting statistics of the number of transactions. The model reproduces spectral properties of the real markets and explains the mechanism of power law distribution of trading activity. The study provides evidence that the statistical properties of the financial markets are enclosed in the statistics of the time interval between trades. Multiplicative point process serves as a consistent model for this statistics.