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How much market making does a market need?

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes

Published online by Cambridge University Press:  16 November 2018

Vít Peržina  and
Jan M. Swart
Vít Peržina*
Affiliation:
Charles University
Jan M. Swart*
Affiliation:
The Czech Academy of Sciences
*
* Postal address: Matematicko-fyzikální fakulta, Charles University, Ke Karlovu 3, 121 16 Praha 2, Czech Republic. Email address: perzina@gmail.com
** Postal address: The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic. Email address: swart@utia.cas.cz
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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 (respectively, sell) limit orders that arrive above (respectively, below) the current ask (respectively, bid) price are converted into market orders. There is no cancellation of limit orders. This model has been independently reinvented by several authors, including Stigler (1964), and Luckock (2003), who calculated 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 that introducing market makers reduces the spread, which in the original model was unrealistically large. In particular, we 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 to the Walrasian equilibrium price.

Keywords

Continuous double auctionlimit order bookStigler‒Luckock modelrank-based Markov chain

MSC classification

Primary: 82C27: Dynamic critical phenomena
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C26: Dynamic and nonequilibrium phase transitions (general) 60K25: Queueing theory
Type
Applied Probability Trust Lecture
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 667 - 681
Copyright
Copyright © Applied Probability Trust 2018 

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