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

Jiannan Zhang; Ping Chen; Zhuo Jin; Shuanming Li

doi:10.3934/jimo.2019133

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2021, Volume 17, Issue 2: 765-777. Doi: 10.3934/jimo.2019133

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

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

^* Corresponding author: Ping Chen
^* Corresponding author: Ping Chen

Received: March 2019

Revised: July 2019

Early access: October 2019

Published: March 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 90B50, 93E20; Secondary: 91G80.

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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.
[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.
[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.
[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.
[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.
[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.
[7]	M. Dai, H. Jin, S. Kou and Y. Xu, Robo-advising: A dynamic mean-variance analysis, work in progress.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x.
[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.
[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.
[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.
[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.
[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.