Market Mill Dependence Pattern in the Stock Market: Distribution Geometry, Moments and Gaussization

TL;DR

This study analyzes the geometric and statistical properties of the joint distribution of consecutive stock price changes, revealing asymmetries, invariances, and a transition towards Gaussian behavior with increasing push magnitude.

Contribution

It uncovers new geometrical features of the bivariate distribution of price increments and compares the conditional response dynamics to AR-ARCH regression models.

Findings

Distribution exhibits Market Mill dependence patterns.

Conditional response P(y|x) becomes more Gaussian with larger pushes.

Response volatility depends linearly on push magnitude.

Abstract

This paper continues a series of studies devoted to analysis of the bivariate probability distribution P(x,y) of two consecutive price increments x (push) and y (response) at intraday timescales for a group of stocks. Besides the asymmetry properties of P(x,y) such as Market Mill dependence patterns described in preceding paper [1], there are quite a few other interesting geometrical properties of this distribution discussed in the present paper, e.g. transformation of the shape of equiprobability lines upon growing distance from the origin of xy plane and approximate invariance of P(x,y) with respect to rotations at the multiples of $\pi/2$ around the origin of xy plane. The conditional probability distribution of response P(y|x) is found to be markedly non-gaussian at small magnitude of pushes and tending to more gauss-like behavior upon growing push magnitude. The volatility of P(y|,x) measured by the absolute value of the response shows linear dependence on the absolute value of the push, and the skewness of P(y|x) is shown to inherit a sign of the push. The conditional dynamics approach applied in this study is compared to regression models of AR-ARCH class.