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Continuous-Time Markowitz’s Mean-Variance Model Under Different Borrowing and Saving Rates

Author

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  • Chonghu Guan

    (Jiaying University)

  • Xiaomin Shi

    (Shandong University of Finance and Economics)

  • Zuo Quan Xu

    (The Hong Kong Polytechnic University)

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 $$C^{3,2}$$ C 3 , 2 smooth. It is also shown that there are a borrowing boundary and a saving boundary 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 turns out to be 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 the 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, 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.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:1:d:10.1007_s10957-023-02259-4
    DOI: 10.1007/s10957-023-02259-4
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    References listed on IDEAS

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    1. Wendell H. Fleming & Thaleia Zariphopoulou, 1991. "An Optimal Investment/Consumption Model with Borrowing," Mathematics of Operations Research, INFORMS, vol. 16(4), pages 802-822, November.
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    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.

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