Skip to main content Accessibility help
×
Home
Hostname: page-component-79b67bcb76-vkbph Total loading time: 0.21 Render date: 2021-05-17T05:01:41.260Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

How much market making does a market need?

Published online by Cambridge University Press:  16 November 2018

Vít Peržina
Affiliation:
Charles University
Jan M. Swart
Affiliation:
The Czech Academy of Sciences
Corresponding

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.

Type
Applied Probability Trust Lecture
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Barabási, A.-L. (2005). The origin of bursts and heavy tails in human dynamics. Nature 435, 207211.CrossRefGoogle ScholarPubMed
[2]Bak, P. and Sneppen, K. (1993). Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 71, 40834086.CrossRefGoogle Scholar
[3]Chakraborti, A., Muni Toke, I., Patriarca, M. and Abergel, F. (2011). Econophysics review: II. Agent-based models. Quant. Finance 11, 10131041.CrossRefGoogle Scholar
[4]Cont, R., Stoikov, S. and Talreja, R. (2010). A stochastic model for order book dynamics. Operat. Res. 58, 549563.CrossRefGoogle Scholar
[5]Formentin, M. and Swart, J. M. (2016). The limiting shape of a full mailbox. ALEA Latin Amer. J. Prob. Math. Statist. 13, 11511164.Google Scholar
[6]Gabrielli, A. and Caldarelli, G. (2009). Invasion percolation and the time scaling behavior of a queuing model of human dynamics. J. Statist. Mech. 2009, P02046.CrossRefGoogle Scholar
[7]Kelly, F. and Yudovina, E. (2018). A Markov model of a limit order book: thresholds, recurrence, and trading strategies. Math. Operat. Res. 43, 181203.CrossRefGoogle Scholar
[8]Luckock, H. (2003). A steady-state model of the continuous double auction. Quant. Finance 3, 385404.CrossRefGoogle Scholar
[9]Maslov, S. (2000). Simple model of a limit order-driven market. Physica A 278, 571578.CrossRefGoogle Scholar
[10]Meester, R. and Sarkar, A. (2012). Rigorous self-organised criticality in the modified Bak-Sneppen model. J. Statist. Phys. 149, 964968.CrossRefGoogle Scholar
[11]Plačková, J. (2011). Shluky volatility a dynamika poptávky a nabídky. Masters Thesis, Charles University. (In Czech.)Google Scholar
[12]Scalas, E., Rapallo, F. and Radivojević, T. (2017). Low-traffic limit and first-passage times for a simple model of the continuous double auction. Physica A 485, 6172.CrossRefGoogle Scholar
[13]Slanina, F. (2014). Essentials of Econophysics Modelling. Oxford University Press.Google Scholar
[14]Šmíd, M. (2012). Probabilistic properties of the continuous double auction. Kybernetika 48, 5082.Google Scholar
[15]Stigler, G. J. (1964). Public regulation of the securities markets. J. Business 37, 117142.CrossRefGoogle Scholar
[16]Swart, J. M. (2017). A simple rank-based Markov chain with self-organized criticality. Markov Process. Relat. Fields 23, 87102.Google Scholar
[17]Swart, J. M. (2018). Rigorous results for the Stigler-Luckock model for the evolution of an order book. Ann. Appl. Prob. 28, 14911535.CrossRefGoogle Scholar
[18]Walras, L. (1874). Éléments d'Économie politique pure; ou, Théorie de la Richesse Sociale. Corbaz, Lausanne. (In French.)Google Scholar
[19]Yudovina, E. (2012). A simple model of a limit order book. Preprint. Available at https://arxiv.org/abs/1205.7017v2.Google Scholar
[20]Yudovina, E. (2012). Collaborating queues: large service network and a limit order book. Doctoral thesis. University of Cambridge.Google Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

How much market making does a market need?
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

How much market making does a market need?
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

How much market making does a market need?
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *