September  2021, 6(3): 237-260. doi: 10.3934/puqr.2021012

An FBSDE approach to market impact games with stochastic parameters

1. 

SAIF/CAFR/CMAR and School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200030, China

2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

3. 

Department of Statistics and Actuarial Science, University of Waterloo, Canada

Email: peng.luo@sjtu.edu.cn (Corresponding author)

Received  February 05, 2021 Accepted  August 20, 2021 Published  September 2021

Fund Project: The authors thank two anonymous referees for their constructive comments and suggestions, which have significantly improved the manuscript. Samuel Drapeau gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 11971310) and “Assessment of Risk and Uncertainty in Finance” (Grant No. AF0710020) from Shanghai Jiao Tong University. Peng Luo gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 12101400). Peng Luo and Alexander Schied gratefully acknowledge the support from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2017-04054). Dewen Xiong gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 11671257).

In this study, we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium.

Citation: Samuel Drapeau, Peng Luo, Alexander Schied, Dewen Xiong. An FBSDE approach to market impact games with stochastic parameters. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 237-260. doi: 10.3934/puqr.2021012
References:
[1]

Almgren, R., Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Mathematical Finance, 2003, 10(1): 1-18. doi: 10.1080/135048602100056.  Google Scholar

[2]

Almgren, R., Optimal trading with stochastic liquidity and volatility, SIAM J. Financial Math., 2012, 3(1): 163-181. doi: 10.1137/090763470.  Google Scholar

[3]

Ankirchner, S., Fromm, A., Kruse, T. and Popier, A., Optimal position targeting via decoupling fields, Annals of Applied Probability, 2020, 30(2): 644-672. Google Scholar

[4]

Ankirchner, S., Jeanblanc, M. and Kruse, T., BSDEs with singular terminal condition and a control problem with constraints, SIAM J. Control Optim., 2014, 52(2): 893-913. doi: 10.1137/130923518.  Google Scholar

[5]

Antonelli, F., Backward-forward stochastic differential equations, Ann. Appl. Probab., 1993, 3(3): 777-793. Google Scholar

[6]

Bismut, J. M., Linear quadratic optimal control with random coeffcients, SIAM J. Control Optim., 1976, 14(3): 419-444. doi: 10.1137/0314028.  Google Scholar

[7]

Cardaliaguet, P. and Lehalle, C. A., Mean field game of controls and an application to trade crowding, Mathematics and Financial Economics, 2018, 12(3): 335-363. doi: 10.1007/s11579-017-0206-z.  Google Scholar

[8]

Carlin, B. I., Lobo, M. S. and Viswanathan, S., Episodic liquidity crises: Cooperative and predatory trading, Journal of Finance, 2007, 62(5): 2235-2274. Google Scholar

[9]

Carmona, R. A. and Yang, J., Predatory trading: A game on volatility and liquidity, Quantitative Finance Preprint, 2011. Google Scholar

[10]

Casgrain, P. and Jaimungal, S., Algorithmic trading with partial information: A mean field game approach, arXiv: 1803.04094, 2018. Google Scholar

[11]

Forsyth, P., Kennedy, J., Tse, T. S. and Windclif, H., Optimal trade execution: A mean-quadratic-variation approach, Journal of Economic Dynamics and Control, 2012, 36(12): 1971-1991. doi: 10.1016/j.jedc.2012.05.007.  Google Scholar

[12]

Gatheral, J., No-dynamic-arbitrage and market impact, Quant. Finance, 2010, 10(7): 749-759. doi: 10.1080/14697680903373692.  Google Scholar

[13]

Gatheral, J. and Schied, A., Dynamical models of market impact and algorithms for order execution, In J.-P. Fouque and J. Langsam, editors, Handbook on Systemic Risk, Cambridge University Press, 2013: 579-602. Google Scholar

[14]

Graewe, P., Horst, U. and Qiu, J., A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions, SIAM J. Control Optim., 2015, 53(2): 690-711. doi: 10.1137/130944084.  Google Scholar

[15]

Hamadène, S., Backward–forward sdes and stochastic differential games, Stoch. Process. Appl., 1998, 77(1): 1-15. doi: 10.1016/S0304-4149(98)00038-6.  Google Scholar

[16]

Hamadène, S., Nonzero sum linear–quadratic stochastic differential games and backward–forward equations, Stoch. Anal. Appl., 1999, 17(1): 117-130. doi: 10.1080/07362999908809591.  Google Scholar

[17]

Kazamaki, N., Continuous Exponential Martingale and BMO, Lecture Notes in Mathematics, vol. 1579, Springer-Verlag, Berlin, 1994. Google Scholar

[18]

Lacker, D., On the convergence of closed-loop nash equilibria to the mean field game limit, Ann. Appl. Probab., 2020, 30(4): 1693-1761. Google Scholar

[19]

Luo, X. and Schied, A., Nash equilibrium for risk-averse investors in a market impact game: Finite and infinite time horizons, Market Microstructure and Liquidity, Preprint, 2020. Google Scholar

[20]

Moallemi, C. C., Park, B. and Van Roy, B., Strategic execution in the presence of an uninformed arbitrageur, Journal of Financial Markets, 2012, 15(4): 361-391. doi: 10.1016/j.finmar.2011.11.002.  Google Scholar

[21]

Pardoux, E. and Peng, S. G., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 1990, 14(1): 55-61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[22]

Peng, S. and Wu, Z., Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 1999, 37(3): 825-843. doi: 10.1137/S0363012996313549.  Google Scholar

[23]

Schied, A., A control problem with fuel constraint and Dawson–Watanabe superprocesses, Ann. Appl. Probab., 2013, 23(6): 2472-2499. Google Scholar

[24]

Schied, A., Strehle, E. and Zhang, T., High-frequency limit of Nash equilibria in a market impact game with transient price impact, SIAM J. Financial Math., 2017, 8(1): 589-634. doi: 10.1137/16M107030X.  Google Scholar

[25]

Schied, A. and Zhang, T., A state-constrained differential game arising in optimal portfolio liquidation, Math. Finance, 2017, 27(3): 779-802. doi: 10.1111/mafi.12108.  Google Scholar

[26]

Schied, A. and Zhang, T., A market impact game under transient price impact, Mathematics of Operations Research, 2019, 44(1): 102-121. Google Scholar

[27]

Schöneborn, T., Optimal trade execution for time-inconsistent mean-variance criteria and risk functions, SIAM J. Financial Math., 2015, 6(1): 1044-1067. doi: 10.1137/15M1007537.  Google Scholar

[28]

Schöneborn, T. and Schied, A., Liquidation in the face of adversity: stealth vs. sunshine trading, SSRN Preprint 1007014, 2009. Google Scholar

[29]

Tang, S., General linear quadratic optimal stochastic control problems with random coeffcients: Linear stochastic hamilton systems and backward stochastic riccati equations, SIAM J. Control Optim., 2003, 42(1): 53-75. doi: 10.1137/S0363012901387550.  Google Scholar

[30]

Tse, S. T., Forsyth, P. A., Kennedy, J. S. and Windcliff, H., Comparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies, Appl. Math. Finance, 2013, 20(5): 415-449. doi: 10.1080/1350486X.2012.755817.  Google Scholar

[31]

Yong, J., Linear forward—backward stochastic differential equations, Appl. Math. Optim., 1999, 39(1): 93-119. doi: 10.1007/s002459900100.  Google Scholar

[32]

Yong, J., Linear forward-backward stochastic differential equations with random coeffcients, Probab. Theory Relat. Fields, 2006, 135(1): 53-83. doi: 10.1007/s00440-005-0452-5.  Google Scholar

[33]

Yong, J. and Zhou, X., Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, Berlin, 1999. Google Scholar

show all references

References:
[1]

Almgren, R., Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Mathematical Finance, 2003, 10(1): 1-18. doi: 10.1080/135048602100056.  Google Scholar

[2]

Almgren, R., Optimal trading with stochastic liquidity and volatility, SIAM J. Financial Math., 2012, 3(1): 163-181. doi: 10.1137/090763470.  Google Scholar

[3]

Ankirchner, S., Fromm, A., Kruse, T. and Popier, A., Optimal position targeting via decoupling fields, Annals of Applied Probability, 2020, 30(2): 644-672. Google Scholar

[4]

Ankirchner, S., Jeanblanc, M. and Kruse, T., BSDEs with singular terminal condition and a control problem with constraints, SIAM J. Control Optim., 2014, 52(2): 893-913. doi: 10.1137/130923518.  Google Scholar

[5]

Antonelli, F., Backward-forward stochastic differential equations, Ann. Appl. Probab., 1993, 3(3): 777-793. Google Scholar

[6]

Bismut, J. M., Linear quadratic optimal control with random coeffcients, SIAM J. Control Optim., 1976, 14(3): 419-444. doi: 10.1137/0314028.  Google Scholar

[7]

Cardaliaguet, P. and Lehalle, C. A., Mean field game of controls and an application to trade crowding, Mathematics and Financial Economics, 2018, 12(3): 335-363. doi: 10.1007/s11579-017-0206-z.  Google Scholar

[8]

Carlin, B. I., Lobo, M. S. and Viswanathan, S., Episodic liquidity crises: Cooperative and predatory trading, Journal of Finance, 2007, 62(5): 2235-2274. Google Scholar

[9]

Carmona, R. A. and Yang, J., Predatory trading: A game on volatility and liquidity, Quantitative Finance Preprint, 2011. Google Scholar

[10]

Casgrain, P. and Jaimungal, S., Algorithmic trading with partial information: A mean field game approach, arXiv: 1803.04094, 2018. Google Scholar

[11]

Forsyth, P., Kennedy, J., Tse, T. S. and Windclif, H., Optimal trade execution: A mean-quadratic-variation approach, Journal of Economic Dynamics and Control, 2012, 36(12): 1971-1991. doi: 10.1016/j.jedc.2012.05.007.  Google Scholar

[12]

Gatheral, J., No-dynamic-arbitrage and market impact, Quant. Finance, 2010, 10(7): 749-759. doi: 10.1080/14697680903373692.  Google Scholar

[13]

Gatheral, J. and Schied, A., Dynamical models of market impact and algorithms for order execution, In J.-P. Fouque and J. Langsam, editors, Handbook on Systemic Risk, Cambridge University Press, 2013: 579-602. Google Scholar

[14]

Graewe, P., Horst, U. and Qiu, J., A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions, SIAM J. Control Optim., 2015, 53(2): 690-711. doi: 10.1137/130944084.  Google Scholar

[15]

Hamadène, S., Backward–forward sdes and stochastic differential games, Stoch. Process. Appl., 1998, 77(1): 1-15. doi: 10.1016/S0304-4149(98)00038-6.  Google Scholar

[16]

Hamadène, S., Nonzero sum linear–quadratic stochastic differential games and backward–forward equations, Stoch. Anal. Appl., 1999, 17(1): 117-130. doi: 10.1080/07362999908809591.  Google Scholar

[17]

Kazamaki, N., Continuous Exponential Martingale and BMO, Lecture Notes in Mathematics, vol. 1579, Springer-Verlag, Berlin, 1994. Google Scholar

[18]

Lacker, D., On the convergence of closed-loop nash equilibria to the mean field game limit, Ann. Appl. Probab., 2020, 30(4): 1693-1761. Google Scholar

[19]

Luo, X. and Schied, A., Nash equilibrium for risk-averse investors in a market impact game: Finite and infinite time horizons, Market Microstructure and Liquidity, Preprint, 2020. Google Scholar

[20]

Moallemi, C. C., Park, B. and Van Roy, B., Strategic execution in the presence of an uninformed arbitrageur, Journal of Financial Markets, 2012, 15(4): 361-391. doi: 10.1016/j.finmar.2011.11.002.  Google Scholar

[21]

Pardoux, E. and Peng, S. G., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 1990, 14(1): 55-61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[22]

Peng, S. and Wu, Z., Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 1999, 37(3): 825-843. doi: 10.1137/S0363012996313549.  Google Scholar

[23]

Schied, A., A control problem with fuel constraint and Dawson–Watanabe superprocesses, Ann. Appl. Probab., 2013, 23(6): 2472-2499. Google Scholar

[24]

Schied, A., Strehle, E. and Zhang, T., High-frequency limit of Nash equilibria in a market impact game with transient price impact, SIAM J. Financial Math., 2017, 8(1): 589-634. doi: 10.1137/16M107030X.  Google Scholar

[25]

Schied, A. and Zhang, T., A state-constrained differential game arising in optimal portfolio liquidation, Math. Finance, 2017, 27(3): 779-802. doi: 10.1111/mafi.12108.  Google Scholar

[26]

Schied, A. and Zhang, T., A market impact game under transient price impact, Mathematics of Operations Research, 2019, 44(1): 102-121. Google Scholar

[27]

Schöneborn, T., Optimal trade execution for time-inconsistent mean-variance criteria and risk functions, SIAM J. Financial Math., 2015, 6(1): 1044-1067. doi: 10.1137/15M1007537.  Google Scholar

[28]

Schöneborn, T. and Schied, A., Liquidation in the face of adversity: stealth vs. sunshine trading, SSRN Preprint 1007014, 2009. Google Scholar

[29]

Tang, S., General linear quadratic optimal stochastic control problems with random coeffcients: Linear stochastic hamilton systems and backward stochastic riccati equations, SIAM J. Control Optim., 2003, 42(1): 53-75. doi: 10.1137/S0363012901387550.  Google Scholar

[30]

Tse, S. T., Forsyth, P. A., Kennedy, J. S. and Windcliff, H., Comparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies, Appl. Math. Finance, 2013, 20(5): 415-449. doi: 10.1080/1350486X.2012.755817.  Google Scholar

[31]

Yong, J., Linear forward—backward stochastic differential equations, Appl. Math. Optim., 1999, 39(1): 93-119. doi: 10.1007/s002459900100.  Google Scholar

[32]

Yong, J., Linear forward-backward stochastic differential equations with random coeffcients, Probab. Theory Relat. Fields, 2006, 135(1): 53-83. doi: 10.1007/s00440-005-0452-5.  Google Scholar

[33]

Yong, J. and Zhou, X., Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, Berlin, 1999. Google Scholar

Figure 1.  Plot of the three agents’ inventory for different values of price impact
Figure 2.  Plot of the three agents’ inventory for different values of slippage effect
Figure 3.  Plot of the three agents’ inventory for the different start values of the first agent’s inventory with two arbitrageurs
[1]

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

[2]

Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics & Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009

[3]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[4]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091

[5]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51

[6]

Marta Faias, Emma Moreno-García, Myrna Wooders. A strategic market game approach for the private provision of public goods. Journal of Dynamics & Games, 2014, 1 (2) : 283-298. doi: 10.3934/jdg.2014.1.283

[7]

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

[8]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153

[9]

Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060

[10]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[11]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123

[12]

Qiang Yan, Mingqiao Luan, Yu Lin, Fangyu Ye. Equilibrium strategies in a supply chain with capital constrained suppliers: The impact of external financing. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3027-3047. doi: 10.3934/jimo.2020106

[13]

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 & Management Optimization, 2021  doi: 10.3934/jimo.2021157

[14]

Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics & Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015

[15]

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

[16]

Moez Kallel, Maher Moakher, Anis Theljani. The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting. Inverse Problems & Imaging, 2015, 9 (3) : 853-874. doi: 10.3934/ipi.2015.9.853

[17]

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

[18]

Narges Torabi Golsefid, Maziar Salahi. Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021032

[19]

Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial & Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843

[20]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010

 Impact Factor: 

Metrics

  • PDF downloads (44)
  • HTML views (157)
  • Cited by (0)

[Back to Top]