How much market making does a market need?

TL;DR

This paper extends a limit order book model by incorporating market makers, demonstrating how their activity reduces the bid-ask spread and identifying the order rate needed to eliminate the spread entirely.

Contribution

It introduces a new model with market makers in a Poisson order flow framework and derives the exact order rate to close the spread, extending previous equilibrium analyses.

Findings

Market makers reduce the bid-ask spread.

A specific order rate can eliminate the spread.

Exceeding this rate leads to a non-Walrasian equilibrium.

Abstract

We consider a simple model for the evolution of a limit order book in which limit orders of unit size arrive according to independent Poisson processes. The frequencies of buy limit orders below a given price level, respectively sell limit orders above a given level are described by fixed demand and supply functions. Buy (resp. sell) limit orders that arrive above (resp. below) the current ask (resp. bid) price are converted into market orders. There is no cancellation of limit orders. This model has independently been reinvented by several authors, including Stigler in 1964 and Luckock in 2003, who was able to calculate the equilibrium distribution of the bid and ask prices. We extend the model by introducing market makers that simultaneously place both a buy and sell limit order at the current bid and ask price. We show how the introduction of market makers reduces the spread, which in the original model is unrealistically large. In particular, we are able to calculate the exact rate at which market makers need to place orders in order to close the spread completely. If this rate is exceeded, we show that the price settles at a random level that in general does not correspond the Walrasian equilibrium price.