Linear models for the impact of order flow on prices. II. The Mixture Transition Distribution model

Abstract

Modelling the impact of the order flow on asset prices is of primary importance to understand the behaviour of financial markets. Part I of this paper reported the remarkable improvements in the description of the price dynamics which can be obtained when one incorporates the impact of past returns on the future order flow. However, impact models presented in Part I consider the order flow as an exogenous process, only characterised by its two-point correlations. This assumption seriously limits the forecasting ability of the model. Here we attempt to model directly the stream of discrete events with a so-called Mixture Transition Distribution (MTD) framework, introduced originally by Raftery [J. R. Stat. Soc. Ser. B, 1985, 528–539]. We distinguish between price-changing and non price-changing events and combine them with the order sign in order to reduce the order flow dynamics to the dynamics of a four-state discrete random variable. The MTD represents a parsimonious approximation of a full high-order Markov chain. The new approach captures with adequate realism the conditional correlation functions between signed events for both small and large tick stocks and signature plots. From a methodological point of view, constraining the MTD within the class of ergodic Markov models, and exploiting the buy–sell symmetry of the data, we propose a weak restriction on the transition matrices which solves the problem of identifiability of mixture models. In spite of the large number of parameters, this translates into a feasible and robust estimation procedure. Out-of-sample analyses demonstrate that the model does not overfit the data.

Keywords:

• Market microstructure
• Price formation
• Markov chain
• Forecasting ability

AMS Subject Classifications:

• G1
• C25

Acknowledgements

We acknowledge several interesting comments from two anonymous referees. We want to thank Z. Eisler, J. Donier and I. Mastromatteo for very useful discussions. D. E. Taranto acknowledges CFM for supporting his extended visit at CFM where part of this research was done.

Notes

No potential conflict of interest was reported by the authors.

1 More recent modelling in continuous time makes use of Hawkes processes (Bacry and Muzy 2014), which bear some degree of similarity with the models considered in the present paper.

2 We remind the reader that large tick stocks have the property that the ratio between tick size and price is relatively high and as a consequence spread is almost always equal to one tick.

3 The case considered in Taranto et al. (2016) corresponds to a MTD(p) model with transition matrices that are the same for all g, and(24)

In the stationary condition the two states have the same probability, as can be verified solving the left eigenvalue problem .