Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints (2403.10441v2)

Published 15 Mar 2024 in q-fin.MF and math.OC

Abstract: We consider both $N$-player and mean-field games of optimal portfolio liquidation in which the players are not allowed to change the direction of trading. Players with an initially short position of stocks are only allowed to buy while players with an initially long position are only allowed to sell the stock. Under suitable conditions on the model parameters we show that the games are equivalent to games of timing where the players need to determine the optimal times of market entry and exit. We identify the equilibrium entry and exit times and prove that equilibrium mean-trading rates can be characterized in terms of the solutions to a highly non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium both in the mean-field and the $N$-player game.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (43)
  1. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics and Games, 8(4):331–358, 2021.
  2. R. Almgren and N. Chriss. Optimal execution of portfolio transactions. Journal of Risk, 3:5–40, 2001.
  3. BSDEs with singular terminal condition and a control problem with constraints. SIAM Journal on Control and Optimization, 52(2):893–913, 2014.
  4. P. Bank and M. Voß. Linear quadratic stochastic control problems with singular stochastic terminal constraint. SIAM Journal on Control and Optimization, 56(2):672–669, 2018.
  5. P. R. Beesack. Comparison theorems and integral inequalities for Volterra integral equations. Proc. Am. Math. Soc., 20:61–66, 1969.
  6. A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints. SIAM Journal on Control and Optimization, 61(1):105–134, 2023.
  7. Mean-field games of optimal stopping: A relaxed solution approach. SIAM Journal on Control and Optimization, 58(4):1795–1821, 2020.
  8. L. Campi and M. Burzoni. Mean field games with absorption and common noise with a model of bank run. Stochastic Processes and Their Applications, 164:206–241, 2023.
  9. L. Campi and M. Fischer. N𝑁Nitalic_N-player games and mean-field games with absorption. Annals of Applied Probability, 28(4):2188–2242, 2018.
  10. N𝑁Nitalic_N-player games and mean-field games with smooth dependence on past absorptions. Annales de l’Institut Henri Poincaré, 57(4):1901–1939, 2021.
  11. P. Cardaliaguet and C. Lehalle. Mean field game of controls and an application to trade crowding. Mathematics and Financial Economics, 12(3):335–363, 2018.
  12. Episodic liquidity crises: Cooperative and predatory trading. Journal of Finance, 62(5):2235–2274, 2007.
  13. Mean field games of timing and models for bank runs. Applied Mathematics & Optimization, 76(1):217–260, 2017.
  14. P. Casgrain and S. Jaimungal. Mean field games with partial information for algorithmic trading. arXiv:1803.04094, 2018.
  15. P. Casgrain and S. Jaimungal. Mean-field games with differing beliefs for algorithmic trading. Mathematical Finance, 30(3):995–1034, 2020.
  16. R. Cesari and H. Zheng. Stochastic maximum principle for optimal liquidation with control-dependent terminal time. Applied Mathematics and Optimization, 85:43(3), 2022.
  17. An FBSDE approach to market impact games with stochastic parameters. Probability, Uncertainty and Quantitative Risk, 6(3):237–260, 2019.
  18. Control and optimal stopping mean field games: a linear programming approach. Electronic Journal of Probability, 26:1–49, 2021.
  19. Optimal trade execution and price manipulation in order books with time-varying liquidity. Mathematical Finance, 24(4):651–695, 2014.
  20. A mean field game of optimal portfolio liquidation. Mathematics of Operations Research, 46(4):1251–1281, 2021.
  21. Mean-field liquidation games with market drop-out. to appear in Mathematical Finance, 2023.
  22. G. Fu and U. Horst. Mean-field leader-follower games with terminal state constraint. SIAM Journal on Control and Optimization, 58(4):2078–2113, 2020.
  23. A mean-field control problem of optimal portfolio liquidation with semimartingale strategies. to appear in Mathematics of Operations Research, 2022.
  24. Portfolio liquidation games with self exciting order flow. Mathematical Finance, 30(4):1020–1065, 2022.
  25. J. Gatheral and A. Schied. Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework. International Journal of Theoretical and Applied Finance, 14(3):353–368, 2011.
  26. P. Graewe and U. Horst. Optimal trade exection with instantaneous price impact and stochastic resilience. SIAM Journal on Control and Optimization, 55(6):3707–3725, 2017.
  27. A maximum principle approach to a deterministic mean field game of control with absorption. SIAM Journal on Control and Optimization, 60(5):3173–3190, 2022.
  28. U. Horst. Stationary equilibria in discounted stochastic games with weakly interacting players. Games and Economic Behavior, 51(1):83–108, 2005.
  29. U. Horst and J. Scheinkman. Equilibria in systems of social interactions. Journal of Economic Theory, 130(1):44–77, 2006.
  30. Portfolio liquidation under factor uncertainty. Annals of Applied Probability, 32(1):80–123, 2022.
  31. Mean-field game strategies for optimal execution. Applied Mathematical Finance, 26:153–185, 2019.
  32. P. Kratz. An explicit solution of a nonlinear-quadratic constrained stochastic control problem with jumps: Optimal liquidation in dark pools with adverse selection. Mathematics of Operations Research, 39(4):1198–1220, 2014.
  33. T. Kruse and A. Popier. Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting. Stochastic Processes and their Applications, 126(9):2554–2592, 2016.
  34. X. Luo and A. Schied. Nash equilibrium for risk-averse investors in a market impact game with transient price impact. Market Microstructure and Liquidity, 5, 2020.
  35. Closed-loop nash competition for liquidity. Mathematical Finance, 33(4):1082–1118, 2023.
  36. E. Neuman and M. Voß. Trading with the crowd. Mathematical Finance, 33(3):548–617, 2023.
  37. M. Nutz. A mean field games of optimal stopping. SIAM Journal on Control and Optimization, 56:1206–1221, 2018.
  38. A. Obizhaeva and J. Wang. Optimal trading strategy and supply/demand dynamics. Journal of Financial Markets, 16(1):1–32, 2013.
  39. A. Popier and C. Zhou. Second order BSDE under monotonicity condition and liquidation problem under uncertainty. Annals of Applied Probability, 29(3), 2019.
  40. High-frequency limit of Nash equilibria in a market impact game with transient price impact. SIAM Journal on Financial Mathematics, 8(1):589–634, 2017.
  41. A. Schied and T. Zhang. A market impact game under transient price impact. Mathematics of Operations Research, 44(1):102–121, 2019.
  42. E. Strehle. Optimal execution in a multiplayer model of transient price impact. Market Microstructure and Liquidity, 3(3-4):1850007, 2018.
  43. G. Teschl. Ordinary Differential Equations and Dynamical Systems. AMS, 2016.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now