IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1408.0916.html
   My bibliography  Save this paper

A system of quadratic BSDEs arising in a price impact model

Author

Listed:
  • Dmitry Kramkov
  • Sergio Pulido

Abstract

We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand. We characterize these prices in terms of a system of BSDEs with quadratic growth. We show that this system admits a unique solution for every bounded demand if and only if the market maker's risk-aversion is sufficiently small. The uniqueness is established in the natural class of solutions, without any additional norm restrictions. To the best of our knowledge, this is the first study that proves such (global) uniqueness result for a system of fully coupled quadratic BSDEs.

Suggested Citation

  • Dmitry Kramkov & Sergio Pulido, 2014. "A system of quadratic BSDEs arising in a price impact model," Papers 1408.0916, arXiv.org, revised May 2016.
    • Handle: RePEc:arx:papers:1408.0916
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1408.0916
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. repec:bla:jfinan:v:43:y:1988:i:3:p:617-37 is not listed on IDEAS
    2. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    3. Grossman, S.J. & Miller, M.H., 1988. "Liquidity And Market Structure," Papers 88, Princeton, Department of Economics - Financial Research Center.
    4. Nicolae Garleanu & Lasse Heje Pedersen & Allen M. Poteshman, 2009. "Demand-Based Option Pricing," The Review of Financial Studies, Society for Financial Studies, vol. 22(10), pages 4259-4299, October.
    5. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    6. Frei, Christoph, 2014. "Splitting multidimensional BSDEs and finding local equilibria," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2654-2671.
    7. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zongxia Liang & Jianming Xia & Fengyi Yuan, 2023. "Dynamic portfolio selection for nonlinear law-dependent preferences," Papers 2311.06745, arXiv.org, revised Nov 2023.
    2. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamaï, 2021. "Equilibrium asset pricing with transaction costs," Finance and Stochastics, Springer, vol. 25(2), pages 231-275, April.
    3. Dmitry Kramkov & Sergio Pulido, 2019. "Density of the set of probability measures with the martingale representation property," Post-Print hal-01598651, HAL.
    4. Ying Hu & Gechun Liang & Shanjian Tang, 2018. "Systems of ergodic BSDEs arising in regime switching forward performance processes," Papers 1807.01816, arXiv.org, revised Jun 2020.
    5. Luis Escauriaza & Daniel C. Schwarz & Hao Xing, 2020. "Radner equilibrium and systems of quadratic BSDEs with discontinuous generators," Papers 2008.03500, arXiv.org, revised May 2021.
    6. Dmitry Kramkov & Sergio Pulido, 2014. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Papers 1410.6144, arXiv.org, revised Aug 2016.
    7. Peter Bank & Ibrahim Ekren & Johannes Muhle-Karbe, 2018. "Liquidity in Competitive Dealer Markets," Papers 1807.08278, arXiv.org, revised Mar 2021.
    8. Jana Bielagk & Arnaud Lionnet & Goncalo Dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Papers 1511.04218, arXiv.org, revised Feb 2017.
    9. Peter Bank & Ibrahim Ekren & Johannes Muhle‐Karbe, 2021. "Liquidity in competitive dealer markets," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 827-856, July.
    10. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Working Papers hal-01181147, HAL.
    11. Kim Weston, 2022. "Existence of an equilibrium with limited participation," Papers 2206.12399, arXiv.org.
    12. Jarrow, Robert & Li, Siguang, 2021. "Endogenous liquidity risk and dealer market structure," The Quarterly Review of Economics and Finance, Elsevier, vol. 81(C), pages 449-453.
    13. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Working Papers hal-01598651, HAL.
    14. 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.
    15. Guanxing Fu & Xizhi Su & Chao Zhou, 2020. "Mean Field Exponential Utility Game: A Probabilistic Approach," Papers 2006.07684, arXiv.org, revised Jul 2020.
    16. Ying Hu & Xiaomin Shi & Zuo Quan Xu, 2022. "Optimal consumption-investment with coupled constraints on consumption and investment strategies in a regime switching market with random coefficients," Papers 2211.05291, arXiv.org.
    17. Michail Anthropelos & Scott Robertson & Konstantinos Spiliopoulos, 2021. "Optimal investment, derivative demand, and arbitrage under price impact," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 3-35, January.
    18. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Papers 1709.07329, arXiv.org, revised Jul 2019.
    19. Jana Bielagk & Arnaud Lionnet & Gonçalo dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Working Papers hal-01245812, HAL.
    20. Hernández, Camilo, 2023. "On quadratic multidimensional type-I BSVIEs, infinite families of BSDEs and their applications," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 249-298.
    21. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    22. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamai, 2019. "Equilibrium Asset Pricing with Transaction Costs," Papers 1901.10989, arXiv.org, revised Sep 2020.
    23. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Post-Print hal-01181147, HAL.
    24. Constantinos Kardaras & Hao Xing & Gordan v{Z}itkovi'c, 2015. "Incomplete stochastic equilibria for dynamic monetary utility," Papers 1505.07224, arXiv.org, revised Feb 2017.
    25. Xing, Hao & Žitković, Gordan, 2018. "A class of globally solvable Markovian quadratic BSDE systems and applications," LSE Research Online Documents on Economics 73440, London School of Economics and Political Science, LSE Library.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. repec:hal:wpaper:hal-01147411 is not listed on IDEAS
    2. Dmitry Kramkov & Sergio Pulido, 2016. "A system of quadratic BSDEs arising in a price impact model," Post-Print hal-01147411, HAL.
    3. Dmitry Kramkov & Sergio Pulido, 2014. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Papers 1410.6144, arXiv.org, revised Aug 2016.
    4. Jana Bielagk & Arnaud Lionnet & Gonçalo dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Working Papers hal-01245812, HAL.
    5. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Post-Print hal-01181147, HAL.
    6. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Working Papers hal-01181147, HAL.
    7. Jana Bielagk & Arnaud Lionnet & Goncalo Dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Papers 1511.04218, arXiv.org, revised Feb 2017.
    8. Guanxing Fu & Xizhi Su & Chao Zhou, 2020. "Mean Field Exponential Utility Game: A Probabilistic Approach," Papers 2006.07684, arXiv.org, revised Jul 2020.
    9. Hiroaki Hata & Shuenn-Jyi Sheu & Li-Hsien Sun, 2019. "Expected exponential utility maximization of insurers with a general diffusion factor model : The complete market case," Papers 1903.08957, arXiv.org.
    10. Kim, Kon-Gun & Kim, Mun-Chol & O, Hun, 2022. "Local existence and uniqueness of solutions to quadratic BSDEs with weak monotonicity and general growth generators," Statistics & Probability Letters, Elsevier, vol. 186(C).
    11. Andrew Papanicolaou, 2018. "Backward SDEs for Control with Partial Information," Papers 1807.08222, arXiv.org.
    12. Kupper, Michael & Luo, Peng & Tangpi, Ludovic, 2019. "Multidimensional Markovian FBSDEs with super-quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 902-923.
    13. Masashi Sekine, 2024. "Mean field equilibrium asset pricing model under partial observation: An exponential quadratic Gaussian approach," Papers 2410.01352, arXiv.org, revised Apr 2025.
    14. Christoph Belak & An Chen & Carla Mereu & Robert Stelzer, 2014. "Optimal investment with time-varying stochastic endowments," Papers 1406.6245, arXiv.org, revised Feb 2022.
    15. Adam Zaremba & Jacob Koby Shemer, 2018. "Price-Based Investment Strategies," Springer Books, Springer, number 978-3-319-91530-2, July.
    16. Guanxing Fu, 2023. "Mean field portfolio games with consumption," Mathematics and Financial Economics, Springer, volume 17, number 4, February.
    17. Hu, Ying & Lin, Yiqing & Soumana Hima, Abdoulaye, 2018. "Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3724-3750.
    18. Zixin Feng & Dejian Tian, 2021. "Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints," Papers 2111.09032, arXiv.org, revised May 2023.
    19. Péter Kondor & Dimitri Vayanos, 2019. "Liquidity Risk and the Dynamics of Arbitrage Capital," Journal of Finance, American Finance Association, vol. 74(3), pages 1139-1173, June.
    20. Dimitri Vayanos & Jiang Wang, 2012. "Market Liquidity -- Theory and Empirical Evidence," NBER Working Papers 18251, National Bureau of Economic Research, Inc.
    21. David M. Kreps & Walter Schachermayer, 2020. "Convergence of optimal expected utility for a sequence of discrete‐time markets," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1205-1228, October.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1408.0916. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.