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Mean-field equilibrium price formation with exponential utility

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  • Masaaki Fujii
  • Masashi Sekine

Abstract

In this paper, using the mean-field game theory, we study a problem of equilibrium price formation among many investors with exponential utility in the presence of liabilities unspanned by the security prices. The investors are heterogeneous in their initial wealth, risk-averseness parameter, as well as stochastic liability at the terminal time. We characterize the equilibrium risk-premium process of the risky stocks in terms of the solution to a novel mean-field backward stochastic differential equation (BSDE), whose driver has quadratic growth both in the stochastic integrands and in their conditional expectations. We prove the existence of a solution to the mean-field BSDE under several conditions and show that the resultant risk-premium process actually clears the market in the large population limit.

Suggested Citation

  • Masaaki Fujii & Masashi Sekine, 2023. "Mean-field equilibrium price formation with exponential utility," Papers 2304.07108, arXiv.org, revised Oct 2023.
    • Handle: RePEc:arx:papers:2304.07108
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    References listed on IDEAS

    as
    1. Guanxing Fu, 2022. "Mean Field Portfolio Games with Consumption," Papers 2206.05425, arXiv.org, revised Dec 2022.
    2. Masaaki Fujii, 2022. "Equilibrium pricing of securities in the co-presence of cooperative and non-cooperative populations," Papers 2209.12639, arXiv.org, revised Jun 2023.
    3. Hu, Ying & Tang, Shanjian, 2016. "Multi-dimensional backward stochastic differential equations of diagonally quadratic generators," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1066-1086.
    4. Guanxing Fu & Xizhi Su & Chao Zhou, 2020. "Mean Field Exponential Utility Game: A Probabilistic Approach," Papers 2006.07684, arXiv.org, revised Jul 2020.
    5. Masaaki Fujii, 2022. "Equilibrium Pricing of Securities in the Co-Presence of Cooperative and Non-Cooperative Populations," CIRJE F-Series CIRJE-F-1201, CIRJE, Faculty of Economics, University of Tokyo.
    6. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability," Papers 1512.05924, arXiv.org, revised Sep 2017.
    7. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    8. Guanxing Fu & Chao Zhou, 2021. "Mean Field Portfolio Games," Papers 2106.06185, arXiv.org, revised Apr 2022.
    9. Daniel Lacker & Thaleia Zariphopoulou, 2019. "Mean field and n‐agent games for optimal investment under relative performance criteria," Mathematical Finance, Wiley Blackwell, vol. 29(4), pages 1003-1038, October.
    10. Dena Firoozi & Arvind V Shrivats & Sebastian Jaimungal, 2021. "Principal agent mean field games in REC markets," Papers 2112.11963, arXiv.org, revised Jun 2022.
    11. Gilles-Edouard Espinosa & Nizar Touzi, 2015. "Optimal Investment Under Relative Performance Concerns," Mathematical Finance, Wiley Blackwell, vol. 25(2), pages 221-257, April.
    12. David Evangelista & Yuri Saporito & Yuri Thamsten, 2022. "Price formation in financial markets: a game-theoretic perspective," Papers 2202.11416, arXiv.org.
    13. Tevzadze, Revaz, 2008. "Solvability of backward stochastic differential equations with quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 503-515, March.
    14. Masaaki Fujii & Akihiko Takahashi, 2021. "Equilibrium Price Formation with a Major Player and its Mean Field Limit," Papers 2102.10756, arXiv.org, revised Feb 2022.
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    Cited by:

    1. Masaaki Fujii & Masashi Sekine, 2024. "Mean Field Equilibrium Asset Pricing Model with Habit Formation," CIRJE F-Series CIRJE-F-1229, CIRJE, Faculty of Economics, University of Tokyo.
    2. Masaaki Fujii & Masashi Sekine, 2024. "Mean field equilibrium asset pricing model with habit formation (Forthcoming in Asia-Pacific Financial Markets)," CARF F-Series CARF-F-587, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Nov 2024.
    3. David Evangelista & Yuri Thamsten, 2023. "Approximately optimal trade execution strategies under fast mean-reversion," Papers 2307.07024, arXiv.org, revised Aug 2023.

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