### Abstract

We consider an equilibrium model of the Limit Order Book in a stock market, where a large number of competing agents post “buy” or “sell” orders. For the “one-shot” game, it is shown that the two sides of the LOB are determined by the distribution of the random size of the incoming order, and by the maximum price accepted by external buyers (or the minimum price accepted by external sellers). We then consider an iterated game, where more agents come to the market, posting both market orders and limit orders. Equilibrium strategies are found by backward induction, in terms of a value function which depends on the current sizes of the two portions of the LOB. The existence of a unique Nash equilibrium is proved under a natural assumption, namely: the probability that the external order is so large that it wipes out the entire LOB should be sufficiently small.

Original language | English (US) |
---|---|

Pages (from-to) | 1015-1041 |

Number of pages | 27 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 25 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2020 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems - Series B*,

*25*(3), 1015-1041. https://doi.org/10.3934/dcdsb.2019206

}

*Discrete and Continuous Dynamical Systems - Series B*, vol. 25, no. 3, pp. 1015-1041. https://doi.org/10.3934/dcdsb.2019206

**A dynamic model of the limit order book.** / Bressan, Alberto; Mazzola, Marco; Wei, Hongxu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A dynamic model of the limit order book

AU - Bressan, Alberto

AU - Mazzola, Marco

AU - Wei, Hongxu

PY - 2020/1/1

Y1 - 2020/1/1

N2 - We consider an equilibrium model of the Limit Order Book in a stock market, where a large number of competing agents post “buy” or “sell” orders. For the “one-shot” game, it is shown that the two sides of the LOB are determined by the distribution of the random size of the incoming order, and by the maximum price accepted by external buyers (or the minimum price accepted by external sellers). We then consider an iterated game, where more agents come to the market, posting both market orders and limit orders. Equilibrium strategies are found by backward induction, in terms of a value function which depends on the current sizes of the two portions of the LOB. The existence of a unique Nash equilibrium is proved under a natural assumption, namely: the probability that the external order is so large that it wipes out the entire LOB should be sufficiently small.

AB - We consider an equilibrium model of the Limit Order Book in a stock market, where a large number of competing agents post “buy” or “sell” orders. For the “one-shot” game, it is shown that the two sides of the LOB are determined by the distribution of the random size of the incoming order, and by the maximum price accepted by external buyers (or the minimum price accepted by external sellers). We then consider an iterated game, where more agents come to the market, posting both market orders and limit orders. Equilibrium strategies are found by backward induction, in terms of a value function which depends on the current sizes of the two portions of the LOB. The existence of a unique Nash equilibrium is proved under a natural assumption, namely: the probability that the external order is so large that it wipes out the entire LOB should be sufficiently small.

UR - http://www.scopus.com/inward/record.url?scp=85076441748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85076441748&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2019206

DO - 10.3934/dcdsb.2019206

M3 - Article

AN - SCOPUS:85076441748

VL - 25

SP - 1015

EP - 1041

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 3

ER -