Abstract
Stock price change in financial market occurs through transactions, in analogy with diffusion in stochastic physical systems. The analysis of price changes in real markets shows that long-range correlations of price fluctuations largely depend on the number of transactions. We introduce the multiplicative stochastic model of time interval between trades and analyze spectral density and correlations 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. Our study provides an evidence that statistical properties of financial markets are enclosed in the statistics of the time interval between trades. Multiplicative stochastic diffusion may serve as a consistent model for this statistics.
Highlights
Complex collective phenomena usually are responsible for power-laws which are universal and independent of the microscopic details of the phenomenon
Our study provides an evidence that statistical properties of financial markets are enclosed in the statistics of the time interval between trades
We have introduced multiplicative stochastic model of the time interval τ between trades in the financial markets Eq (7) with the diffusion restriction in the interval 0 ≤ τk ≤ 1
Summary
Complex collective phenomena usually are responsible for power-laws which are universal and independent of the microscopic details of the phenomenon. The generalized Lotka Volterra dynamics developed by S.Solomon and P.Richmond is in the use for various systems including financial markets [1] These models generically lead to the non-universal exponents and do not explain the power-law correlations in financial time series [8]. The idea to transfer long time correlations into stochastic process of the time interval between trades or time series of trading activity is in consistence with the detailed studies of the empirical financial data [5, 6] and fruitfuly reproduces spectral properties of financial time series [13]. We show the consistence of the approach with the results of statistical analyzes of the empirical financial time series This sheds light on the relation between the power-law probability distribution and the power-law correlations in financial time series
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