Stationary distribution of the volume at the best quote in a Poisson order book model

TL;DR

This paper models the volume at the best quote in an electronic order book as a Markov process, deriving its stationary distribution and validating the model with empirical data.

Contribution

It introduces a Markovian framework for the volume at the best quote, incorporating order arrivals and removals, and derives its stationary distribution.

Findings

Derived explicit stationary distribution for the volume.

Empirical fitting confirms the model's relevance.

Compared different model variants with real data.

Abstract

In this paper, we develop a Markovian model that deals with the volume offered at the best quote of an electronic order book. The volume of the first limit is a stochastic process whose paths are periodically interrupted and reset to a new value, either by a new limit order submitted inside the spread or by a market order that removes the first limit. Using applied probability results on killing and resurrecting Markov processes, we derive the stationary distribution of the volume offered at the best quote. All proposed models are empirically fitted and compared, stressing the importance of the proposed mechanisms.