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

Chonghu Guan (), Xiaomin Shi () and Zuo Quan Xu ()
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Chonghu Guan: Jiaying University
Xiaomin Shi: Shandong University of Finance and Economics
Zuo Quan Xu: The Hong Kong Polytechnic University

Journal of Optimization Theory and Applications, 2023, vol. 199, issue 1, No 6, 167-208

Abstract: 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.

Keywords: Markowitz’s mean-variance portfolio selection; Fully nonlinear PDE; Free boundary; Dual transformation; Different borrowing and saving rates; 35R35; 35Q93; 91G10; 91G30; 93E20 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02259-4

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