Multiplicative point process as a model of trading activity

Abstract

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 a variety of self-affine time series exhibiting power spectral density S(f) scaling as a power of the frequency f. Furthermore, we analyze the relation between the power-law correlations and the origin of the power-law probability distribution of the signal intensity. We introduce a stochastic multiplicative model for the time intervals between point events and analyze the statistical properties of the signal analytically and numerically. Such model system exhibits power-law spectral density S(f)~1/f**beta for various values of beta, including beta=1/2, 1 and 3/2. Explicit expressions for the power spectra in the low frequency limit and for the distribution density of the interevent time are obtained. The counting statistics of the events is analyzed analytically and numerically, 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. A multiplicative point process serves as a consistent model generating this statistics.Comment: 10 pages, 3 figure

Long-range memory model of trading activity and volatility

“…This approach reveals the structure of the power spectral density in wide range of frequencies and shows that the model exhibits not one but rather two separate power laws with the exponents β 1 = 0.33 and β 2 = 0.72. From many numerical calculations performed with the multiplicative point processes we can conclude that combination of two power laws of spectral density arise only when multiplicative noise is a crossover of two power laws, see (13) and (14). We will show in the next section that this may serve as an explanation of two exponents of the power spectrum in the empirical data of volatility for S&P 500 companies [19].…”

“…It has been shown analytically and numerically [12,14] that the point process with stochastic interevent time (1) may generate signals with the power-law distributions of the signal intensity and 1/f β noise. The corresponding Ito stochastic differential equation for the variable τ (t) as a function of the actual time can be written as…”

Solutions of nonlinear stochastic differential equations with 1/ƒ noise power spectrum

“…generating the power-law distributed P k (τ k ) ∼ τ α k , with α = 2γ/σ 2 τ − 2µ, sequence of the interevent times τ k and f −β , with β = 1 + α/(3 − 2µ), power spectral density of the signal [11]- [16]. Some motivations for equation (1) were given in papers [5], [9]- [19].…”

1/fnoise from nonlinear stochastic differential equations

“…When z 2 min ≪ t ≪ z 2 max we have the following lowest powers in the expansion of the approximate expression (39) for the autocorrelation function in the power series of t:…”

“…Recently nonlinear SDEs generating signals with 1/f noise were obtained in Refs. [29,30] (see also recent papers [5,31]), starting from the point process model of 1/f noise [27,[32][33][34][35][36][37][38][39].…”