Market mill dependence pattern in the stock market: Modeling of predictability and asymmetry via multi-component conditional distribution

Recent studies have revealed a number of striking dependence patterns in high frequency stock price dynamics characterizing probabilistic interrelation between two consequent price increments x (push) and y (response) as described by the bivariate probability distribution P(x,y) [A. Leonidov, V. Trainin, A. Zaitsev, On collective non-gaussian dependence patterns in high frequency financial data, ArXiv:physics/0506072, A. Leonidov, V. Trainin, A. Zaitsev, S. Zaitsev, Market mill dependence pattern in the stock market: asymmetry structure, nonlinear correlations and predictability, arXiv:physics/0601098, A. Leonidov, V. Trainin, A. Zaitsev, S. Zaitsev, Market mill dependence pattern in the stock market: distribution geometry, moments and gaussization, arXiv:physics/0603103, A. Leonidov, V. Trainin, A. Zaitsev, S. Zaitsev, Market mill dependence pattern in the stock market: distribution geometry. Individual portraits, arXiv:physics/0605138]. There are two properties, the market mill asymmetries of P(x,y) and predictability due to nonzero z-shaped mean conditional response, that are of special importance. Main goal of the present paper is to put together a model reproducing both the z-shaped mean conditional response and the market mill asymmetry of P(x,y) with respect to the axis y=0. We develop a probabilistic model based on a multi-component ansatz for conditional distribution P(y|x) with push-dependent weights and means describing the both properties. In this paper we also introduce a quantitative measure of the relative weight of the asymmetric component of P(x,y) and show that the model reproduces a pattern observed in the market data. A relationship between the market mill asymmetry and predictability is discussed. A possible connection of the model to agent-based description of market dynamics is outlined.