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

Dynamic equilibrium limit order book model and optimal execution problem

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089

2. 

Institutional Equity Division, Morgan Stanley, New York, NY 10036, United States, United States

Received  January 2014 Revised  October 2014 Published  July 2015

In this paper we propose a dynamic model of Limit Order Book (LOB). The main feature of our model is that the shape of the LOB is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density of the LOB is random, nonlinear, and time inhomogeneous. Consequently, the liquidity cost can be defined dynamically in a natural way.
    We next study an optimal execution problem in our model. We verify that the value function satisfies the Dynamic Programming Principle, and is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation which is in the form of an integro-partial-differential quasi-variational inequality. We also prove the existence and analyze the structure of the optimal strategy via a verification theorem argument, assuming that the PDE has a classical solution.
Citation: 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
References:
[1]

A. Alfonsi, A. Fruth and A. Schied, Constrained portfolio liquidation in a limit order book model,, Advances in mathematics of finance, 83 (2008), 9.  doi: 10.4064/bc83-0-1.  Google Scholar

[2]

A. Alfonsi, A. Fruth and A. Schied, Optimal execution strategies in limit order books with general shape functions,, Quant. Finance, 10 (2010), 143.  doi: 10.1080/14697680802595700.  Google Scholar

[3]

A. Alfonsi and A. Schied, Optimal execution and absence of price manipulations in limit order book models,, SIAM J. Financial Math., 1 (2010), 490.  doi: 10.1137/090762786.  Google Scholar

[4]

A. Alfonsi, A. Schied and A. Slynko, Order book resilience, price manipulation, and the positive portfolio problem,, SIAM J. Finan. Math., 3 (2012), 511.  doi: 10.1137/110822098.  Google Scholar

[5]

M. Avellaneda and S. Stoikov, High-frequency trading in a limit order book,, Quantitative Finance, 8 (2008), 217.  doi: 10.1080/14697680701381228.  Google Scholar

[6]

E. Bayraktar and M. Ludkovski, Optimal trade execution in illiquid markets,, Mathematical Finance, 21 (2011), 681.  doi: 10.1111/j.1467-9965.2010.00446.x.  Google Scholar

[7]

T. Bielecki, M. Jeanblanc and M. Rutkowski, Hedging of defaultable claims,, Paris-Princeton Lectures on Mathematical Finance 2003, 1847 (2004), 1.  doi: 10.1007/978-3-540-44468-8_1.  Google Scholar

[8]

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

[9]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (NS), 27 (1992), 1.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I,, Annals of Probability, ().   Google Scholar

[11]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer-Verlag, (2006).   Google Scholar

[12]

T. Foucault, O. Kadan and E. Kandel, Limit Order Book as a Market for Liquidity,, Review of Financial Studies, 18 (2005), 1171.   Google Scholar

[13]

J. Gatheral, A. Schied and A. Slynko, Transient linear price impact and Fredholm integral equations,, Mathematical Finance, 22 (2012), 445.  doi: 10.1111/j.1467-9965.2011.00478.x.  Google Scholar

[14]

P. Handa and R. A. Schwartz, Limit order trading,, Journal of Finance, 51 (1996), 1835.   Google Scholar

[15]

B. Hollifield, R. A. Miller and P. Sandas, Empirical analysis of limit order markets,, Review of Economic Studies, 71 (2004), 1027.  doi: 10.1111/0034-6527.00313.  Google Scholar

[16]

A. Obizhaeva and J. Wang, Optimal trading strategy and supply/demand dynamics,, Journal of Financial Markets, 16 (2013), 1.   Google Scholar

[17]

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

[18]

S. Predoiu, G. Shaikhet and S. E. Shreve, Optimal execution in a general one-sided limit-order book,, SIAM J. Finan. Math., 2 (2011), 183.  doi: 10.1137/10078534X.  Google Scholar

[19]

M. Soner, N. Touzi and J. Zhang, Dual formulation of the second order target problems,, Annals of Applied Probability, 23 (2013), 308.  doi: 10.1214/12-AAP844.  Google Scholar

show all references

References:
[1]

A. Alfonsi, A. Fruth and A. Schied, Constrained portfolio liquidation in a limit order book model,, Advances in mathematics of finance, 83 (2008), 9.  doi: 10.4064/bc83-0-1.  Google Scholar

[2]

A. Alfonsi, A. Fruth and A. Schied, Optimal execution strategies in limit order books with general shape functions,, Quant. Finance, 10 (2010), 143.  doi: 10.1080/14697680802595700.  Google Scholar

[3]

A. Alfonsi and A. Schied, Optimal execution and absence of price manipulations in limit order book models,, SIAM J. Financial Math., 1 (2010), 490.  doi: 10.1137/090762786.  Google Scholar

[4]

A. Alfonsi, A. Schied and A. Slynko, Order book resilience, price manipulation, and the positive portfolio problem,, SIAM J. Finan. Math., 3 (2012), 511.  doi: 10.1137/110822098.  Google Scholar

[5]

M. Avellaneda and S. Stoikov, High-frequency trading in a limit order book,, Quantitative Finance, 8 (2008), 217.  doi: 10.1080/14697680701381228.  Google Scholar

[6]

E. Bayraktar and M. Ludkovski, Optimal trade execution in illiquid markets,, Mathematical Finance, 21 (2011), 681.  doi: 10.1111/j.1467-9965.2010.00446.x.  Google Scholar

[7]

T. Bielecki, M. Jeanblanc and M. Rutkowski, Hedging of defaultable claims,, Paris-Princeton Lectures on Mathematical Finance 2003, 1847 (2004), 1.  doi: 10.1007/978-3-540-44468-8_1.  Google Scholar

[8]

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

[9]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (NS), 27 (1992), 1.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I,, Annals of Probability, ().   Google Scholar

[11]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer-Verlag, (2006).   Google Scholar

[12]

T. Foucault, O. Kadan and E. Kandel, Limit Order Book as a Market for Liquidity,, Review of Financial Studies, 18 (2005), 1171.   Google Scholar

[13]

J. Gatheral, A. Schied and A. Slynko, Transient linear price impact and Fredholm integral equations,, Mathematical Finance, 22 (2012), 445.  doi: 10.1111/j.1467-9965.2011.00478.x.  Google Scholar

[14]

P. Handa and R. A. Schwartz, Limit order trading,, Journal of Finance, 51 (1996), 1835.   Google Scholar

[15]

B. Hollifield, R. A. Miller and P. Sandas, Empirical analysis of limit order markets,, Review of Economic Studies, 71 (2004), 1027.  doi: 10.1111/0034-6527.00313.  Google Scholar

[16]

A. Obizhaeva and J. Wang, Optimal trading strategy and supply/demand dynamics,, Journal of Financial Markets, 16 (2013), 1.   Google Scholar

[17]

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

[18]

S. Predoiu, G. Shaikhet and S. E. Shreve, Optimal execution in a general one-sided limit-order book,, SIAM J. Finan. Math., 2 (2011), 183.  doi: 10.1137/10078534X.  Google Scholar

[19]

M. Soner, N. Touzi and J. Zhang, Dual formulation of the second order target problems,, Annals of Applied Probability, 23 (2013), 308.  doi: 10.1214/12-AAP844.  Google Scholar

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