A Fokker-Planck description for the queue dynamics of large tick stocks

TL;DR

This paper models the queue dynamics of large tick stocks using a two-dimensional Fokker-Planck equation, incorporating state dependence, jump events, and scale invariance, and finds universal behavior in rescaled volumes.

Contribution

It introduces a calibrated, scale-invariant Fokker-Planck model capturing complex drift structures and correlations in large tick stock queues.

Findings

Dynamical process is approximately scale invariant.

Rescaled volume dynamics are universal across stocks and time.

Drift has a stable, complex two-dimensional structure.

Abstract

Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation, that explicitly includes state dependence, i.e. the fact that the drift and diffusion depends on the volume present on both sides of the spread. "Jump" events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on best quotes. One of our central finding is the the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.