Rigorous results for the Stigler-Luckock model for the evolution of an order book

TL;DR

This paper extends the classical Stigler-Luckock order book model by incorporating market orders, providing conditions for the model's positive recurrence and analyzing how these relate to the original model's critical prices.

Contribution

It introduces market orders into Luckock's model and characterizes the conditions for positive recurrence, linking them to the original model's critical prices.

Findings

Necessary and sufficient conditions for positive recurrence with market orders

Relation between recurrence conditions and Luckock's critical prices

Extension of the model's analytical framework to include market orders

Abstract

In 1964, G.J. Stigler introduced a stochastic model for the evolution of an order book on a stock market. This model was independently rediscovered and generalized by H. Luckock in 2003. In his formulation, traders place buy and sell limit orders of unit size according to independent Poisson processes with possibly different intensities. Newly arriving buy (sell) orders are either immediately matched to the best available matching sell (buy) order or stay in the order book until a matching order arrives. Assuming stationarity, Luckock showed that the distribution functions of the best buy and sell order in the order book solve a differential equation, from which he was able to calculate the position of two prices $J^{\rm c}_-<J^{\rm c}_+$ such that buy orders below $J^{\rm c}_-$ and sell orders above $J^{\rm c}_+$ stay in the order book forever while all other orders are eventually matched. We extend Luckock's model by adding market orders, i.e., with a certain rate traders arrive at the market that take the best available buy or sell offer in the order book, if there is one, and do nothing otherwise. We give necessary and sufficient conditions for such an extended model to be positive recurrent and show how these conditions are related to the prices $J^{\rm c}_-$ and $J^{\rm c}_+$ of Luckock.