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Stochastic maximum principle for systems driven by local martingales with spatial parameters
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 |
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.
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. |



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