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March  2020, 16(2): 531-551. doi: 10.3934/jimo.2018166

Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching

Ping Chen 1, and  Haixiang Yao 2,,

1. 

Department of Economics, The University of Melbourne, VIC 3010, Australia
2. 

School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China

* Corresponding author: Haixiang Yao. Tel.:+86 2037105360

Received  January 2018 Revised  April 2018 Published  March 2020 Early access  October 2018

Fund Project: This research was supported by the National Natural Science Foundation of China (Nos. 71871071, 71471045, 71721001), the Natural Science Foundation of Guangdong Province of China (Nos. 2018B030311004, 2017A030313399, 2017A030313397), the Innovation Team Project of Guangdong Colleges and Universities (No. 2016WCXTD012), the Innovative School Project in Higher Education of Guangdong Province of China (No. GWTP-GC-2017-03)

The present article investigates a continuous-time mean-variance portfolio selection problem with regime-switching under the constraint of no-shorting. The literature along this line is essentially dominated by the Hamilton-Jacobi-Bellman (HJB) equation approach. However, in the presence of switching regimes, a system of HJB equations rather than a single equation need to be tackled concurrently, which might not be solvable in terms of classical solutions, or even not in the weaker viscosity sense as well. Instead, we first introduce a general result on the sign of geometric Brownian motion with jumps, then derive the efficient portfolio and frontier via the maximum principle approach; in particular, we observe, under a mild technical assumption on the initial conditions, that the no-shorting constraint will consistently be satisfied over the whole finite time horizon. Further numerical illustrations will be provided.

Keywords: No-shorting constraint, regime-switching, mean-variance, efficient frontier, Riccati equation.
Mathematics Subject Classification: Primary: 49L99; Secondary: 60J27.
Citation: Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166
References:
[1]

L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar

[2]

A. Bensoussan, K. J. Sung, S. C. H. Yam and S. P. Yung, A non-zero stochastic differential reinsurance game with mixed regime-switching, working paper, 2011. Google Scholar

[3]

J. N. Bi, J. Y. Guo and L. H. Bai, Optimal multi-asset investment with no-shorting constraint under mean-variance criterion for an insurer, Journal of Systems Science and Complexity, 24 (2011), 291-307.  doi: 10.1007/s11424-011-8014-7.  Google Scholar

[4]

T. Björk, Finite dimensional optimal filters for a class of Itô-processes with jumping parameters, Stochastics, 4 (1980), 167-183.  doi: 10.1080/17442508008833160.  Google Scholar

[5]

E. Çanakoǧlu and S. Özekici, Portfolio selection in stochastic markets with HARA utility functions, European Journal of Operational Research, 201 (2010), 520-536.  doi: 10.1016/j.ejor.2009.03.017.  Google Scholar

[6]

P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[7]

Y. Z. Hu, Multi-dimensional geometric Brownian motions, Onsager-Machlup functions, and applications to mathematical finance, Acta Mathematica Scientia, 20 (2000), 341-358.  doi: 10.1016/S0252-9602(17)30641-0.  Google Scholar

[8]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[9]

X. Li and Z. Q. Xu, Continuous-time Markowitzs model with constraints on wealth and portfolio, Operations Research Letters, 44 (2016), 729-736.  doi: 10.1016/j.orl.2016.09.004.  Google Scholar

[10]

X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shoring constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[11]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.   Google Scholar

[12]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous - time case, Review of Economics and Statistics, 51 (1969), 247-257.   Google Scholar

[13]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[14]

P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246.   Google Scholar

[15]

L. R. Sotomayor and A. Cadenillas, Explicit solutions of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 19 (2009), 251-279.  doi: 10.1111/j.1467-9965.2009.00366.x.  Google Scholar

[16]

J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance: Mathematics and Economics, 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008.  Google Scholar

[17]

H. Wu and H. Chen, Nash equilibrium strategy for a multi-period mean-ariance portfolio selection problem with regime switching, Economic Modelling, 46 (2015), 79-90.   Google Scholar

[18]

G. L. Xu and S. E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition: Ⅱ. Constant market coefficients, The Annals of Applied Probability, 2 (1992), 314-328.  doi: 10.1214/aoap/1177005706.  Google Scholar

[19]

M. Zhang and P. Chen, Mean-variance portfolio selection with regime switching under shorting prohibition, Operations Research Letters, 44 (2016), 658-662.  doi: 10.1016/j.orl.2016.07.008.  Google Scholar

[20]

X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.  Google Scholar

show all references

References:
[1]

L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar

[2]

A. Bensoussan, K. J. Sung, S. C. H. Yam and S. P. Yung, A non-zero stochastic differential reinsurance game with mixed regime-switching, working paper, 2011. Google Scholar

[3]

J. N. Bi, J. Y. Guo and L. H. Bai, Optimal multi-asset investment with no-shorting constraint under mean-variance criterion for an insurer, Journal of Systems Science and Complexity, 24 (2011), 291-307.  doi: 10.1007/s11424-011-8014-7.  Google Scholar

[4]

T. Björk, Finite dimensional optimal filters for a class of Itô-processes with jumping parameters, Stochastics, 4 (1980), 167-183.  doi: 10.1080/17442508008833160.  Google Scholar

[5]

E. Çanakoǧlu and S. Özekici, Portfolio selection in stochastic markets with HARA utility functions, European Journal of Operational Research, 201 (2010), 520-536.  doi: 10.1016/j.ejor.2009.03.017.  Google Scholar

[6]

P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[7]

Y. Z. Hu, Multi-dimensional geometric Brownian motions, Onsager-Machlup functions, and applications to mathematical finance, Acta Mathematica Scientia, 20 (2000), 341-358.  doi: 10.1016/S0252-9602(17)30641-0.  Google Scholar

[8]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[9]

X. Li and Z. Q. Xu, Continuous-time Markowitzs model with constraints on wealth and portfolio, Operations Research Letters, 44 (2016), 729-736.  doi: 10.1016/j.orl.2016.09.004.  Google Scholar

[10]

X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shoring constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[11]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.   Google Scholar

[12]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous - time case, Review of Economics and Statistics, 51 (1969), 247-257.   Google Scholar

[13]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[14]

P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246.   Google Scholar

[15]

L. R. Sotomayor and A. Cadenillas, Explicit solutions of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 19 (2009), 251-279.  doi: 10.1111/j.1467-9965.2009.00366.x.  Google Scholar

[16]

J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance: Mathematics and Economics, 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008.  Google Scholar

[17]

H. Wu and H. Chen, Nash equilibrium strategy for a multi-period mean-ariance portfolio selection problem with regime switching, Economic Modelling, 46 (2015), 79-90.   Google Scholar

[18]

G. L. Xu and S. E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition: Ⅱ. Constant market coefficients, The Annals of Applied Probability, 2 (1992), 314-328.  doi: 10.1214/aoap/1177005706.  Google Scholar

[19]

M. Zhang and P. Chen, Mean-variance portfolio selection with regime switching under shorting prohibition, Operations Research Letters, 44 (2016), 658-662.  doi: 10.1016/j.orl.2016.07.008.  Google Scholar

[20]

X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.  Google Scholar

Figure 1.  The value of the stochastic process $P(t, \alpha(t))[x(t)+(\lambda-z)H(t, \alpha(t))]$
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Figure 2.  A sample path of the efficient portfolio $u^{*} = (u_1, u_2, u_3)'$
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Figure 3.  The process $-[x(t)+(\lambda^{*}-z)H(t, \alpha(t))]$
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Figure 4.  The corresponding efficient frontier
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Figure 5.  The effects of initial market mode $i_0$
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