American Institute of Mathematical Sciences

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

An FBSDE approach to market impact games with stochastic parameters

Samuel Drapeau 1, , Peng Luo 2,, , Alexander Schied 3, and  Dewen Xiong 2,

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.

Keywords: Market impact game, Nash equilibrium, FBSDE.
Mathematics Subject Classification: 93E20, 60H30.
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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[33]

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

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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[33]

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

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