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March  2020, 25(3): 1015-1041. doi: 10.3934/dcdsb.2019206

## A dynamic model of the limit order book

 1 Department of Mathematics, Penn State University, McAllister Building, University Park, PA 16802, USA 2 Institut de Mathématiques de Jussieu - Paris Rive Gauche, CNRS, Sorbonne Université, Case 247, 4 Place Jussieu, 75252 Paris, France

Received  April 2018 Revised  April 2019 Published  March 2020 Early access  September 2019

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.

Citation: Alberto Bressan, Marco Mazzola, Hongxu Wei. A dynamic model of the limit order book. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1015-1041. doi: 10.3934/dcdsb.2019206
##### References:
 [1] K. Back and S. Baruch, Information in securities markets: Kyle meets Glosten and Milgrom, Econometrica, 72 (2004), 433-465.  doi: 10.1111/j.1468-0262.2004.00497.x. [2] K. Back and S. Baruch, Strategic liquidity provision in limit order markets, Econometrica, 81 (2013), 363-392.  doi: 10.3982/ECTA10018. [3] P. Bank and D. Kramkov, A model for a large investor trading at market indifference prices. Ⅰ: Single-period case, Finance Stoch., 19 (2015), 449-472.  doi: 10.1007/s00780-015-0258-y. [4] P. Bank and D. Kramkov, A model for a large investor trading at market indifference prices. Ⅱ: Continuous-time case, Ann. Appl. Probab., 25 (2015), 2708-2742.  doi: 10.1214/14-AAP1059. [5] A. Bressan and G. Facchi, A bidding game in a continuum limit order book, SIAM J. Control Optim., 51 (2013), 3459-3485.  doi: 10.1137/120896359. [6] A. Bressan and G. Facchi, Discrete bidding strategies for a random incoming order, SIAM J. Financial Math., 5 (2014), 50-70.  doi: 10.1137/130917685. [7] A. Bressan and D. Wei, A bidding game with heterogeneous players, J. Optim. Theory Appl., 163 (2014), 1018-1048.  doi: 10.1007/s10957-014-0551-5. [8] A. Bressan and H. Wei, Dynamic stability of the Nash equilibrium for a bidding game, Analysis & Applications, 14 (2016), 591-614.  doi: 10.1142/S0219530515500098. [9] U. Cetin, R. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance Stoch., 8 (2004), 311-341.  doi: 10.1007/s00780-004-0123-x. [10] R. Cont and A. Larrard, Price dynamics in a Markovian limit order book market, SIAM J. Financial Math., 4 (2013), 1-25.  doi: 10.1137/110856605. [11] R. Cont, S. Stoikov and R. Talreja, A stochastic model for order book dynamics, Operations Research, 58 (2010), 549-563.  doi: 10.1287/opre.1090.0780. [12] R. Gayduk and S. Nadtochiy, Liquidity effects of trading frequency, Math. Finance, 28 (2018), 839-876.  doi: 10.1111/mafi.12157. [13] R. Gayduk and S. Nadtochiy, Endogenous formation of limit order book: The effects of trading frequency, SIAM J. Control Optim., 56 (2018), 1577-1619.  doi: 10.1137/16M1078045. [14] M. D. Gould, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn and S. D. Howison, Limit order books, Quantitative Finance, 13 (2013), 1709-1742.  doi: 10.1080/14697688.2013.803148. [15] F. Kelly and E. Yudovina, A Markov model of the limit order book: thresholds, recurrence, and trading strategies, Journal Math. of Operations Research, 43 (2018), 181-203.  doi: 10.1287/moor.2017.0857. [16] A. Lachapelle, J. M. Lasry, C. A. Lehalle and P. L. Lions, Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis, Math. Financ. Econ., 10 (2016), 223-262.  doi: 10.1007/s11579-015-0157-1. [17] C. Parlour and D. J. Seppi, Limit order markets: A survey, in Proceedings of the Handbook of Financial Intermediation and Banking (eds. A. Thakor and A. Boot), Elsevier, (2008), 63–96. doi: 10.1016/B978-044451558-2.50007-6. [18] I. Rosu, A dynamic model of the limit order book, Review of Financial Studies, 22 (2009), 4601-4641. [19] T. W. Yang and L. Zhu, A reduced-form model for level-1 limit order books, Market Microstructure and Liquidity, 2 (2016), 1650008. doi: 10.1142/S2382626616500088.

show all references

##### References:
 [1] K. Back and S. Baruch, Information in securities markets: Kyle meets Glosten and Milgrom, Econometrica, 72 (2004), 433-465.  doi: 10.1111/j.1468-0262.2004.00497.x. [2] K. Back and S. Baruch, Strategic liquidity provision in limit order markets, Econometrica, 81 (2013), 363-392.  doi: 10.3982/ECTA10018. [3] P. Bank and D. Kramkov, A model for a large investor trading at market indifference prices. Ⅰ: Single-period case, Finance Stoch., 19 (2015), 449-472.  doi: 10.1007/s00780-015-0258-y. [4] P. Bank and D. Kramkov, A model for a large investor trading at market indifference prices. Ⅱ: Continuous-time case, Ann. Appl. Probab., 25 (2015), 2708-2742.  doi: 10.1214/14-AAP1059. [5] A. Bressan and G. Facchi, A bidding game in a continuum limit order book, SIAM J. Control Optim., 51 (2013), 3459-3485.  doi: 10.1137/120896359. [6] A. Bressan and G. Facchi, Discrete bidding strategies for a random incoming order, SIAM J. Financial Math., 5 (2014), 50-70.  doi: 10.1137/130917685. [7] A. Bressan and D. Wei, A bidding game with heterogeneous players, J. Optim. Theory Appl., 163 (2014), 1018-1048.  doi: 10.1007/s10957-014-0551-5. [8] A. Bressan and H. Wei, Dynamic stability of the Nash equilibrium for a bidding game, Analysis & Applications, 14 (2016), 591-614.  doi: 10.1142/S0219530515500098. [9] U. Cetin, R. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance Stoch., 8 (2004), 311-341.  doi: 10.1007/s00780-004-0123-x. [10] R. Cont and A. Larrard, Price dynamics in a Markovian limit order book market, SIAM J. Financial Math., 4 (2013), 1-25.  doi: 10.1137/110856605. [11] R. Cont, S. Stoikov and R. Talreja, A stochastic model for order book dynamics, Operations Research, 58 (2010), 549-563.  doi: 10.1287/opre.1090.0780. [12] R. Gayduk and S. Nadtochiy, Liquidity effects of trading frequency, Math. Finance, 28 (2018), 839-876.  doi: 10.1111/mafi.12157. [13] R. Gayduk and S. Nadtochiy, Endogenous formation of limit order book: The effects of trading frequency, SIAM J. Control Optim., 56 (2018), 1577-1619.  doi: 10.1137/16M1078045. [14] M. D. Gould, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn and S. D. Howison, Limit order books, Quantitative Finance, 13 (2013), 1709-1742.  doi: 10.1080/14697688.2013.803148. [15] F. Kelly and E. Yudovina, A Markov model of the limit order book: thresholds, recurrence, and trading strategies, Journal Math. of Operations Research, 43 (2018), 181-203.  doi: 10.1287/moor.2017.0857. [16] A. Lachapelle, J. M. Lasry, C. A. Lehalle and P. L. Lions, Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis, Math. Financ. Econ., 10 (2016), 223-262.  doi: 10.1007/s11579-015-0157-1. [17] C. Parlour and D. J. Seppi, Limit order markets: A survey, in Proceedings of the Handbook of Financial Intermediation and Banking (eds. A. Thakor and A. Boot), Elsevier, (2008), 63–96. doi: 10.1016/B978-044451558-2.50007-6. [18] I. Rosu, A dynamic model of the limit order book, Review of Financial Studies, 22 (2009), 4601-4641. [19] T. W. Yang and L. Zhu, A reduced-form model for level-1 limit order books, Market Microstructure and Liquidity, 2 (2016), 1650008. doi: 10.1142/S2382626616500088.
Left: a distribution function for the random variable $X$, describing the size of the external order. Right: a possible shape of the limit order book. If the external order is a buy order with size $X>0$, all the stocks in the shaded region on the right (with area $= X$), will be sold. If the external order is a sell order for an amount $Y>0$ of stocks, all the buy orders in the shaded region on the left (with area $= Y$), will be executed
A plot of the density function $\phi$, with data as in (39). In this case, solving (36)–(38) we find $p_A = 10.0831$, $p_B = 9.6097$, $\bar p = 9.8464$
A plot of the functions $U(p)$ in (11) and (21), with data as in (39)
The ask price $p_A$ is found by solving the Cauchy problem (107), (15), and finding the price at which $U = 0$. To estimate the rate at which $p_A$ changes with the boundary data $\overline p$, it is convenient to invert the role of the variables $U, p$, thus obtaining the linear ODE (109) for $p = p(U)$. The figure shows how $p_A$ changes when the value of $\overline p$ is increased
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