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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

K. Back and S. Baruch, Strategic liquidity provision in limit order markets, Econometrica, 81 (2013), 363-392.  doi: 10.3982/ECTA10018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

U. CetinR. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance Stoch., 8 (2004), 311-341.  doi: 10.1007/s00780-004-0123-x.  Google Scholar

[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.  Google Scholar

[11]

R. ContS. Stoikov and R. Talreja, A stochastic model for order book dynamics, Operations Research, 58 (2010), 549-563.  doi: 10.1287/opre.1090.0780.  Google Scholar

[12]

R. Gayduk and S. Nadtochiy, Liquidity effects of trading frequency, Math. Finance, 28 (2018), 839-876.  doi: 10.1111/mafi.12157.  Google Scholar

[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.  Google Scholar

[14]

M. D. GouldM. A. PorterS. WilliamsM. McDonaldD. J. Fenn and S. D. Howison, Limit order books, Quantitative Finance, 13 (2013), 1709-1742.  doi: 10.1080/14697688.2013.803148.  Google Scholar

[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.  Google Scholar

[16]

A. LachapelleJ. M. LasryC. 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.  Google Scholar

[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.  Google Scholar

[18]

I. Rosu, A dynamic model of the limit order book, Review of Financial Studies, 22 (2009), 4601-4641.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

K. Back and S. Baruch, Strategic liquidity provision in limit order markets, Econometrica, 81 (2013), 363-392.  doi: 10.3982/ECTA10018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

U. CetinR. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance Stoch., 8 (2004), 311-341.  doi: 10.1007/s00780-004-0123-x.  Google Scholar

[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.  Google Scholar

[11]

R. ContS. Stoikov and R. Talreja, A stochastic model for order book dynamics, Operations Research, 58 (2010), 549-563.  doi: 10.1287/opre.1090.0780.  Google Scholar

[12]

R. Gayduk and S. Nadtochiy, Liquidity effects of trading frequency, Math. Finance, 28 (2018), 839-876.  doi: 10.1111/mafi.12157.  Google Scholar

[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.  Google Scholar

[14]

M. D. GouldM. A. PorterS. WilliamsM. McDonaldD. J. Fenn and S. D. Howison, Limit order books, Quantitative Finance, 13 (2013), 1709-1742.  doi: 10.1080/14697688.2013.803148.  Google Scholar

[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.  Google Scholar

[16]

A. LachapelleJ. M. LasryC. 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.  Google Scholar

[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.  Google Scholar

[18]

I. Rosu, A dynamic model of the limit order book, Review of Financial Studies, 22 (2009), 4601-4641.   Google Scholar

[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.  Google Scholar

Figure 1.  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
Figure 2.  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 $
Figure 3.  A plot of the functions $ U(p) $ in (11) and (21), with data as in (39)
Figure 4.  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
[1]

Jin Ma, Xinyang Wang, Jianfeng Zhang. Dynamic equilibrium limit order book model and optimal execution problem. Mathematical Control & Related Fields, 2015, 5 (3) : 557-583. doi: 10.3934/mcrf.2015.5.557

[2]

Wai-Ki Ching, Tang Li, Sin-Man Choi, Issic K. C. Leung. A tandem queueing system with applications to pricing strategy. Journal of Industrial & Management Optimization, 2009, 5 (1) : 103-114. doi: 10.3934/jimo.2009.5.103

[3]

Mitali Sarkar, Young Hae Lee. Optimum pricing strategy for complementary products with reservation price in a supply chain model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1553-1586. doi: 10.3934/jimo.2017007

[4]

Xue-Yan Wu, Zhi-Ping Fan, Bing-Bing Cao. Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-29. doi: 10.3934/jimo.2019040

[5]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1

[6]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

[7]

Tak Kuen Siu, Howell Tong, Hailiang Yang. Option pricing under threshold autoregressive models by threshold Esscher transform. Journal of Industrial & Management Optimization, 2006, 2 (2) : 177-197. doi: 10.3934/jimo.2006.2.177

[8]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237

[9]

Po-Chung Yang, Hui-Ming Wee, Shen-Lian Chung, Yong-Yan Huang. Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand. Journal of Industrial & Management Optimization, 2013, 9 (4) : 769-787. doi: 10.3934/jimo.2013.9.769

[10]

Ali Naimi Sadigh, S. Kamal Chaharsooghi, Majid Sheikhmohammady. A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain. Journal of Industrial & Management Optimization, 2016, 12 (1) : 337-355. doi: 10.3934/jimo.2016.12.337

[11]

Jianxiong Zhang, Zhenyu Bai, Wansheng Tang. Optimal pricing policy for deteriorating items with preservation technology investment. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1261-1277. doi: 10.3934/jimo.2014.10.1261

[12]

Guodong Yi, Xiaohong Chen, Chunqiao Tan. Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1579-1597. doi: 10.3934/jimo.2018112

[13]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1

[14]

María Suárez-Taboada, Carlos Vázquez. Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3503-3523. doi: 10.3934/dcdsb.2018254

[15]

Baojun Bian, Nan Wu, Harry Zheng. Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1401-1420. doi: 10.3934/dcdsb.2016002

[16]

Ruopeng Wang, Jinting Wang, Chang Sun. Optimal pricing and inventory management for a loss averse firm when facing strategic customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1521-1544. doi: 10.3934/jimo.2018019

[17]

Zhijie Sasha Dong, Wei Chen, Qing Zhao, Jingquan Li. Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2018175

[18]

María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004

[19]

Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2019008

[20]

Bing-Bing Cao, Zhi-Ping Fan, Tian-Hui You. The optimal pricing and ordering policy for temperature sensitive products considering the effects of temperature on demand. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1153-1184. doi: 10.3934/jimo.2018090

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (21)
  • HTML views (122)
  • Cited by (0)

Other articles
by authors

[Back to Top]