Market Mill Dependence Pattern in the Stock Market: Multiscale Conditional Dynamics

arXiv:0810.4409v2 [q-fin.ST] 3 Nov 2008 MARKET MILL DEPENDENCE PATTERN IN THE STOCK MARKET: MULTISCALE CONDITIONAL DYNAMICS Sergey Zaitsev(a), Alexander Zaitsev(a), Andrei Leonidov(b,a,c)1,2, Vladimir Trainin(a), (a) Letra Group, LLC, Boston, Massachusetts, USA (b) Theory Department, P.N. Lebedev Physics Institute, Moscow, Russia (c) Institute of Theoretical and Experimental Physics, Moscow, Russia Abstract Market Mill is a complex dependence pattern leading to nonlinear corre- lations and predictability in intraday dynamics of stock prices. The present paper puts together previous efforts to build a dynamical model reflecting the market mill asymmetries. We show that certain properties of the con- ditional dynamics at a single time scale such as a characteristic shape of an asymmetry generating component of the conditional probability distribution result in the ”elementary” market mill pattern. This asymmetry generat- ing component matches the empirical distribution obtained from the market data. We discuss these properties as a mixture of trend-preserving and con- trarian strategies used by market agents. Three basic types of asymmetry patterns characterizing individual stocks are outlined. Multiple time scale considerations make the resulting ”composite” mill similar to the empirical market mill patterns. Multiscale model also reflects a multi-agent nature of the market. 1Corresponding author. E-mail leonidov@lpi.ru 2Supported by the RFBR grant 06-06-80357 1 1 Introduction The present paper continues a series of papers studying the complex depen- dence patterns in high frequency stock price dynamics [1, 2, 3, 4, 5]. The most important of them, the market mill asymmetries [2, 3, 4, 5], correspond to specific probabilistic interrelations between consequent price increments. The term ”market mill” refers to a mill-like asymmetric four-blade depen- dence pattern [2], see Fig. 1. The main emphasis of [2, 3, 4] was on systematic phenomenological description of the market mill asymmetries and other re- lated properties of high frequency stock price dynamics. In [5] a causal conditional dynamics model leading to the market mill asymmetries and nonlinear dependence of expectation value of a future price increment y (”response”) on the value of a realized price increment x (”push”) was suggested. The model described probabilistic relation between the push x and the response y in terms of the three - component conditional distribution P(y|x). The distribution P(y|x) was described as an x-dependent additive superposition of the symmetric contribution P0(y|x) and the asymmetry- generating components P+(y|x) and P−(y|x) characterized by a bias towards trend-preserving and contrarian strategies correspondingly. The model of [5] referred to a single time scale. It is however well known that a description of certain features of stock price dynamics requires accounting for multiple time scales, at the level of both price increments (returns) [6, 7, 8, 9, 10, 11, 12, 13] and microscopic long- memory properties of order flow and trades [14, 15, 16]. In particular, the presence of several distinct time scales in volatility dynamics was explicitly demonstrated in [9]. In [2] the empirical market mill patterns were shown to exist at different time scales ranging from minutes to hours. In the present paper we incorporate the idea of multiple time scales into the market mill model. First we introduce an elementary market mill mecha- nism at a fixed time scale. We describe an easier way of specifying the elemen- tary market mill by reformulating the model of [5] in such a way that y is a sum of noise and non-random asymmetry generating components. Introduc- ing specific features of the non-random component based on empirical data we come up with the market mill pattern. Then we build a multiscale com- posite mill as a weighted superposition of elementary asymmetry-generating mechanisms operating at different timescales. The outline of the paper is as follows. We start with a description of generic features of the market mill asym- 2 metries in paragraph 2.1. Particular emphasis is put onto formulating a version stock price dynamics with an additive superposition of noise and asymmetry-generating mechanisms. In paragraph 2.2 we describe the condi- tional distribution allowing to reproduce all observed market mill asymme- tries. The properties of explicit dynamical model giving rise to a single time scale elementary market mill asymmetries are discussed in paragraph 2.3. The composite multiscale dynamics allowing to reproduce all the properties of the market mill asymmetries is described in paragraph 2.4. The section 3 contains a discussion of the origin of the market mill asymmetries in terms of three basic strategies, market mill, trend-following and contrarian, used by market participants. We demonstrate that appropriately weighted super- positions of these basic strategies allows to describe various two-dimensional asymmetry patterns

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