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Finite horizon portfolio selection problems with stochastic borrowing constraints
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 |
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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T. Björk, A. Murgoci and X. Y. Zhou,
Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.
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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. |
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M. Dai, H. Jin, S. Kou and Y. Xu, Robo-advising: A dynamic mean-variance analysis, work in progress. |
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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. |
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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. |
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H. Markowitz,
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R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, Palgrave, London, 1973, 128–143.
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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. |
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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. |
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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. |
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. |
| [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. |
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