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March  2021, 17(2): 765-777. doi: 10.3934/jimo.2019133

Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model

Jiannan Zhang , Ping Chen , , Zhuo Jin and  Shuanming Li

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC, 3010, Australia

* Corresponding author: Ping Chen

Received  March 2019 Revised  July 2019 Published  March 2021 Early access  October 2019

This paper investigates a continuous-time mean-variance portfolio selection problem based on a log-return model. The financial market is composed of one risk-free asset and multiple risky assets whose prices are modelled by geometric Brownian motions. We derive a sufficient condition for open-loop equilibrium strategies via forward backward stochastic differential equations (FBSDEs). An equilibrium strategy is derived by solving the system. To illustrate our result, we consider a special case where the interest rate process is described by the Vasicek model. In this case, we also derive the closed-loop equilibrium strategy through the dynamic programming approach.

Keywords: Mean-variance, log-return, time-consistent, open-loop equilibrium strategy, closed-loop equilibrium strategy.
Mathematics Subject Classification: Primary: 90B50, 93E20; Secondary: 91G80.
Citation: Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133
References:
[1]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.  Google Scholar

[2]

A. Bensoussan, K. C. Wong, S. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM J. Financial Math., 5 (2014), 153-190.  doi: 10.1137/130914139.  Google Scholar

[3]

T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[4]

T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance Stoch., 18 (2014), 545-592.  doi: 10.1007/s00780-014-0234-y.  Google Scholar

[5]

T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar

[6]

M. C. Chiu and D. Li, Asset and liability management under a continuous-time mean-variance optimization framework, Insurance Math. Econom., 39 (2006), 330-355.  doi: 10.1016/j.insmatheco.2006.03.006.  Google Scholar

[7]

M. Dai, H. Jin, S. Kou and Y. Xu, Robo-advising: A dynamic mean-variance analysis, work in progress. Google Scholar

[8]

C. Fu, A. Lari-Lavassani and X. Li, Dynamic mean-variance portfolio selection with borrowing constraint, European J. Oper. Res., 200 (2010), 312-319.  doi: 10.1016/j.ejor.2009.01.005.  Google Scholar

[9]

C. Gollier and R. J. Zeckhauser, Horizon length and portfolio risk, J. Risk and Uncertainty, 24 (2002), 195-212.  doi: 10.1023/A:1015697417916.  Google Scholar

[10]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[11]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279.  doi: 10.1137/15M1019040.  Google Scholar

[12]

F. E. Kydland and E. C. Prescott, Rules rather than discretion: The inconsistency of optimal plans, Journal of Political Economy, 85 (2010), 473-491.  doi: 10.1086/260580.  Google Scholar

[13]

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

[14]

A. E. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Math. Oper. Res., 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337.  Google Scholar

[15]

H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[16]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, Palgrave, London, 1973, 128–143. doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[17]

J. Wei and T. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011.  Google Scholar

[18]

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 Math. Econom., 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008.  Google Scholar

[19]

S. Xie, Z. Li and S. Wang, Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach, Insurance Math. Econom., 42 (2008), 943-953.  doi: 10.1016/j.insmatheco.2007.10.014.  Google Scholar

[20]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

show all references

References:
[1]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.  Google Scholar

[2]

A. Bensoussan, K. C. Wong, S. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM J. Financial Math., 5 (2014), 153-190.  doi: 10.1137/130914139.  Google Scholar

[3]

T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[4]

T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance Stoch., 18 (2014), 545-592.  doi: 10.1007/s00780-014-0234-y.  Google Scholar

[5]

T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar

[6]

M. C. Chiu and D. Li, Asset and liability management under a continuous-time mean-variance optimization framework, Insurance Math. Econom., 39 (2006), 330-355.  doi: 10.1016/j.insmatheco.2006.03.006.  Google Scholar

[7]

M. Dai, H. Jin, S. Kou and Y. Xu, Robo-advising: A dynamic mean-variance analysis, work in progress. Google Scholar

[8]

C. Fu, A. Lari-Lavassani and X. Li, Dynamic mean-variance portfolio selection with borrowing constraint, European J. Oper. Res., 200 (2010), 312-319.  doi: 10.1016/j.ejor.2009.01.005.  Google Scholar

[9]

C. Gollier and R. J. Zeckhauser, Horizon length and portfolio risk, J. Risk and Uncertainty, 24 (2002), 195-212.  doi: 10.1023/A:1015697417916.  Google Scholar

[10]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[11]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279.  doi: 10.1137/15M1019040.  Google Scholar

[12]

F. E. Kydland and E. C. Prescott, Rules rather than discretion: The inconsistency of optimal plans, Journal of Political Economy, 85 (2010), 473-491.  doi: 10.1086/260580.  Google Scholar

[13]

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

[14]

A. E. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Math. Oper. Res., 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337.  Google Scholar

[15]

H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[16]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, Palgrave, London, 1973, 128–143. doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[17]

J. Wei and T. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011.  Google Scholar

[18]

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 Math. Econom., 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008.  Google Scholar

[19]

S. Xie, Z. Li and S. Wang, Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach, Insurance Math. Econom., 42 (2008), 943-953.  doi: 10.1016/j.insmatheco.2007.10.014.  Google Scholar

[20]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

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