American Institute of Mathematical Sciences

Advanced
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  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, 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

[1]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[2]

Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039
[3]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95
[4]

Zhiping Chen, Jia Liu, Gang Li. Time consistent policy of multi-period mean-variance problem in stochastic markets. Journal of Industrial & Management Optimization, 2016, 12 (1) : 229-249. doi: 10.3934/jimo.2016.12.229
[5]

Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267
[6]

Robert S. Anderssen, Martin Kružík. Modelling of wheat-flour dough mixing as an open-loop hysteretic process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 283-293. doi: 10.3934/dcdsb.2013.18.283
[7]

Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355
[8]

Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058
[9]

Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi, Andrea De Gaetano. Robust closed-loop control of plasma glycemia: A discrete-delay model approach. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 455-468. doi: 10.3934/dcdsb.2009.12.455
[10]

Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
[11]

Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
[12]

Masoud Mohammadzadeh, Alireza Arshadi Khamseh, Mohammad Mohammadi. A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1041-1064. doi: 10.3934/jimo.2016061
[13]

Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931
[14]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018166
[15]

Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2018180
[16]

Yan Zeng, Zhongfei Li, Jingjun Liu. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. Journal of Industrial & Management Optimization, 2010, 6 (3) : 483-496. doi: 10.3934/jimo.2010.6.483
[17]

Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial & Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022
[18]

Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2019008
[19]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521
[20]

Haixiang Yao, Zhongfei Li, Yongzeng Lai. Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate. Journal of Industrial & Management Optimization, 2016, 12 (1) : 187-209. doi: 10.3934/jimo.2016.12.187

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (38)
  • HTML views (33)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences