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.

10 pages, 3 figures

References in corpus (7)

• Inverse Cubic Law for the Probability Distribution of Stock Price Variations
• Statistical Properties of Share Volume Traded in Financial Markets
• Power Laws are Logarithmic Boltzmann Laws
• Modeling 1/f noise
• Autoregressive model of 1/f noise
• Dynamics of the Number of Trades of Financial Securities
• Multiplicative Stochastic Model of the Time Interval between Trades in Financial Markets

Cited by in corpus (19)

• Point process model of 1/f noise versus a sum of Lorentzians
• 1/f noise from nonlinear stochastic differential equations
• Modeling scaled processes and 1/f^b noise by the nonlinear stochastic differential equations
• Nonlinear stochastic models of 1/f noise and power-law distributions
• A long-range memory stochastic model of the return in financial markets
• Tsallis distributions and 1/f noise from nonlinear stochastic differential equations
• Herding model and 1/f noise
• Modeling long-range memory trading activity by stochastic differential equations
• Three-state herding model of the financial markets
• The class of nonlinear stochastic models as a background for the bursty behavior in financial markets
• Trading activity as driven Poisson process: comparison with empirical data
• Long-range memory model of trading activity and volatility
• Intermittency in relation with 1/f noise and stochastic differential equations
• Binary versus non-binary information in real time series: empirical results and maximum-entropy matrix models
• Understanding the nature of the long-range memory phenomenon in socioeconomic systems
• Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes
• Agent-based Versus Macroscopic Modeling of Competition and Business Processes in Economics and Finance
• Nonlinear Stochastic Model of Return matching to the data of New York and Vilnius Stock Exchanges
• Modeling interaction of trading volume in financial dynamics