• Previous Article
    A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior
  • DCDS-B Home
  • This Issue
  • Next Article
    Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form
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. CetinR. 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. ContS. 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. 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.

[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. 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.

[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. CetinR. 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. ContS. 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. 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.

[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. 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.

[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.

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 and Related Fields, 2015, 5 (3) : 557-583. doi: 10.3934/mcrf.2015.5.557

[2]

Ali Naimi-Sadigh, S. Kamal Chaharsooghi, Marzieh Mozafari. Optimal pricing and advertising decisions with suppliers' oligopoly competition: Stakelberg-Nash game structures. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1423-1450. doi: 10.3934/jimo.2020028

[3]

Haijiao Li, Kuan Yang, Guoqing Zhang. Optimal pricing strategy in a dual-channel supply chain: A two-period game analysis. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022072

[4]

Yu Chen, Zixian Cui, Shihan Di, Peibiao Zhao. Capital asset pricing model under distribution uncertainty. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021113

[5]

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

[6]

Jaimie W. Lien, Vladimir V. Mazalov, Jie Zheng. Pricing equilibrium of transportation systems with behavioral commuters. Journal of Dynamics and Games, 2020, 7 (4) : 335-350. doi: 10.3934/jdg.2020026

[7]

Zhenkai Lou, Fujun Hou, Xuming Lou. Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3099-3111. doi: 10.3934/jimo.2020109

[8]

Yu-Chung Tsao, Hanifa-Astofa Fauziah, Thuy-Linh Vu, Nur Aini Masruroh. Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021152

[9]

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

[10]

Xiujing Dang, Yang Xu, Gongbing Bi, Lei Qin. Pricing strategy and product quality design with platform-investment. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2021224

[11]

Jing Zhang, Jianquan Lu, Jinde Cao, Wei Huang, Jianhua Guo, Yun Wei. Traffic congestion pricing via network congestion game approach. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1553-1567. doi: 10.3934/dcdss.2020378

[12]

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

[13]

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 and Management Optimization, 2020, 16 (4) : 1999-2027. doi: 10.3934/jimo.2019040

[14]

Shaokun Tao, Xianjin Du, Suresh P. Sethi, Xiuli He, Yu Li. Equilibrium decisions on pricing and innovation that impact reference price dynamics. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021157

[15]

Patrick Beißner, Emanuela Rosazza Gianin. The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 23-52. doi: 10.3934/puqr.2021002

[16]

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

[17]

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

[18]

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

[19]

Tomasz R. Bielecki, Igor Cialenco, Marek Rutkowski. Arbitrage-free pricing of derivatives in nonlinear market models. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 2-. doi: 10.1186/s41546-018-0027-x

[20]

Marianito R. Rodrigo, Rogemar S. Mamon. Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2685-2702. doi: 10.3934/jimo.2020089

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (310)
  • HTML views (290)
  • Cited by (0)

Other articles
by authors

[Back to Top]