Low-traffic limit and first-passage times for a simple model of the continuous double auction

TL;DR

This paper analyzes a simplified continuous double auction model, deriving exact price distribution formulas in the low-traffic limit and validating them with first passage time analysis.

Contribution

It introduces a tractable model linking auction dynamics to queueing theory and provides explicit formulas for price distribution in the low-traffic regime.

Findings

Exact price distribution in the low-traffic limit

Approximate validity when order ratio is below 1/2

First passage time analysis confirms model predictions

Abstract

We consider a simplified model of the continuous double auction where prices are integers varying from $1$ to $N$ with limit orders and market orders, but quantity per order limited to a single share. For this model, the order process is equivalent to two $M/M/1$ queues. We study the behaviour of the auction in the low-traffic limit where limit orders are immediately transformed into market orders. In this limit, the distribution of prices can be computed exactly and gives a reasonable approximation of the price distribution when the ratio between the rate of order arrivals and the rate of order executions is below $1/2$. This is further confirmed by the analysis of the first passage time in $1$ or $N$.