IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2201.00914.html
   My bibliography  Save this paper

Continuous-time Markowitz's mean-variance model under different borrowing and saving rates

Author

Listed:
  • Chonghu Guan
  • Xiaomin Shi
  • Zuo Quan Xu

Abstract

We study Markowitz's mean-variance portfolio selection problem in a continuous-time Black-Scholes market with different borrowing and saving rates. The associated Hamilton-Jacobi-Bellman equation is fully nonlinear. Using a delicate partial differential equation and verification argument, the value function is proven to be $C^{3,2}$ smooth. It is also shown that there are a borrowing boundary and a saving barrier which divide the entire trading area into a borrowing-money region, an all-in-stock region, and a saving-money region in ascending order. The optimal trading strategy is a mixture of continuous-time strategy (as suggested by most continuous-time models) and discontinuous-time strategy (as suggested by models with transaction costs): one should put all her wealth in the stock in the middle all-in-stock region, and continuously trade it in the other two regions in a feedback form of wealth and time. It is never optimal to short sale the stock. Numerical examples are also presented to verify the theoretical results and to give more financial insights beyond them.

Suggested Citation

  • Chonghu Guan & Xiaomin Shi & Zuo Quan Xu, 2022. "Continuous-time Markowitz's mean-variance model under different borrowing and saving rates," Papers 2201.00914, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2201.00914
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2201.00914
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Fu, Chenpeng & Lari-Lavassani, Ali & Li, Xun, 2010. "Dynamic mean-variance portfolio selection with borrowing constraint," European Journal of Operational Research, Elsevier, vol. 200(1), pages 312-319, January.
    2. Kohlmann, Michael & Tang, Shanjian, 2002. "Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging," Stochastic Processes and their Applications, Elsevier, vol. 97(2), pages 255-288, February.
    3. Huyên Pham, 2000. "On quadratic hedging in continuous time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(2), pages 315-339, April.
    4. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    5. Jie Xiong & Zuo Quan Xu & Jiayu Zheng, 2021. "Mean–variance portfolio selection under partial information with drift uncertainty," Quantitative Finance, Taylor & Francis Journals, vol. 21(9), pages 1461-1473, September.
    6. Jianming Xia, 2005. "Mean–Variance Portfolio Choice: Quadratic Partial Hedging," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 533-538, July.
    7. Tomasz R. Bielecki & Hanqing Jin & Stanley R. Pliska & Xun Yu Zhou, 2005. "Continuous‐Time Mean‐Variance Portfolio Selection With Bankruptcy Prohibition," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 213-244, April.
    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 & Keyu Zhang, 2023. "Equilibrium stochastic control with implicitly defined objective functions," Papers 2312.15173, arXiv.org, revised Dec 2023.

    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. Chonghu Guan & Xiaomin Shi & Zuo Quan Xu, 2023. "Continuous-Time Markowitz’s Mean-Variance Model Under Different Borrowing and Saving Rates," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 167-208, October.
    2. Yuanyuan Zhang & Xiang Li & Sini Guo, 2018. "Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature," Fuzzy Optimization and Decision Making, Springer, vol. 17(2), pages 125-158, June.
    3. Shuzhen Yang, 2019. "A varying terminal time mean-variance model," Papers 1909.13102, arXiv.org, revised Jan 2020.
    4. Min Dai & Zuo Quan Xu & Xun Yu Zhou, 2009. "Continuous-Time Markowitz's Model with Transaction Costs," Papers 0906.0678, arXiv.org.
    5. Yao, Haixiang & Lai, Yongzeng & Li, Yong, 2013. "Continuous-time mean–variance asset–liability management with endogenous liabilities," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 6-17.
    6. Shuzhen Yang, 2020. "Discrete time multi-period mean-variance model: Bellman type strategy and Empirical analysis," Papers 2011.10966, arXiv.org.
    7. Johannes Muhle-Karbe & Xiaofei Shi & Chen Yang, 2020. "An Equilibrium Model for the Cross-Section of Liquidity Premia," Papers 2011.13625, arXiv.org.
    8. Zhang, Miao & Chen, Ping & Yao, Haixiang, 2017. "Mean-variance portfolio selection with only risky assets under regime switching," Economic Modelling, Elsevier, vol. 62(C), pages 35-42.
    9. Cong, F. & Oosterlee, C.W., 2016. "Multi-period mean–variance portfolio optimization based on Monte-Carlo simulation," Journal of Economic Dynamics and Control, Elsevier, vol. 64(C), pages 23-38.
    10. Alev{s} v{C}ern'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy Models and the Time Step Equivalent of Jumps," Papers 1309.7833, arXiv.org, revised Jul 2017.
    11. Aivaliotis, Georgios & Palczewski, Jan, 2014. "Investment strategies and compensation of a mean–variance optimizing fund manager," European Journal of Operational Research, Elsevier, vol. 234(2), pages 561-570.
    12. Lin Chen & Xun Yu Zhou, 2022. "Naive Markowitz Policies," Papers 2212.07516, arXiv.org.
    13. F. Cong & C. W. Oosterlee, 2017. "On Robust Multi-Period Pre-Commitment And Time-Consistent Mean-Variance Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-26, November.
    14. Ying Hu & Xiaomin Shi & Zuo Quan Xu, 2022. "Non-homogeneous stochastic LQ control with regime switching and random coefficients," Papers 2201.01433, arXiv.org, revised Jul 2023.
    15. Tomas Björk & Agatha Murgoci & Xun Yu Zhou, 2014. "Mean–Variance Portfolio Optimization With State-Dependent Risk Aversion," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 1-24, January.
    16. Cong, F. & Oosterlee, C.W., 2016. "On pre-commitment aspects of a time-consistent strategy for a mean-variance investor," Journal of Economic Dynamics and Control, Elsevier, vol. 70(C), pages 178-193.
    17. Wong, K.C. & Yam, S.C.P. & Zeng, J., 2019. "Mean-risk portfolio management with bankruptcy prohibition," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 153-172.
    18. De Gennaro Aquino, Luca & Sornette, Didier & Strub, Moris S., 2023. "Portfolio selection with exploration of new investment assets," European Journal of Operational Research, Elsevier, vol. 310(2), pages 773-792.
    19. Chi Kin Lam & Yuhong Xu & Guosheng Yin, 2016. "Dynamic portfolio selection without risk-free assets," Papers 1602.04975, arXiv.org.
    20. Yao, Haixiang & Zeng, Yan & Chen, Shumin, 2013. "Multi-period mean–variance asset–liability management with uncontrolled cash flow and uncertain time-horizon," Economic Modelling, Elsevier, vol. 30(C), pages 492-500.

    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:2201.00914. 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.