American Institute of Mathematical Sciences

Advanced
September  2015, 5(3): 475-488. doi: 10.3934/mcrf.2015.5.475

Continuous-time portfolio selection under ambiguity

Hanqing Jin 1, and  Xun Yu Zhou 1,

1. 

Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom, United Kingdom

Received  August 2014 Revised  November 2014 Published  July 2015

In a financial market, the appreciation rates of stocks are statistically difficult to estimate, and typically only some confidence intervals in which the rates reside can be estimated. In this paper we study continuous-time portfolio selection under ambiguity, in the sense that the appreciation rates are only known to be in a certain convex closed set and the portfolios are allowed to be based on only the historical stocks prices. We formulate the problem in both the expected utility and the mean--variance frameworks, and derive robust portfolios explicitly for both models.
Keywords: Portfolio selection, continuous time, mean--variance, robust portfolio., ambiguity, Sharpe ratio, expected utility.
Mathematics Subject Classification: Primary: 91G10, 93E20; Secondary: 90B5.
Citation: Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control & Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475
References:
[1]

A. Ben-Tal and A. Nemirovski, Robust convex optimization,, Math. Oepr. Res., 23 (1998), 796.  doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

S. Boyd, L. ElGhaoui, E. Feron and B. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, SIAM Studies in Applied Mathematics, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[3]

T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition,, Math. Finance, 15 (2005), 213.  doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[4]

Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time,, Econometrica, 70 (2002), 1403.  doi: 10.1111/1468-0262.00337.  Google Scholar

[5]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances and covariances on optimal portfolio choice,, J. Portfolio Management, (1993), 6.  doi: 10.3905/jpm.1993.409440.  Google Scholar

[6]

I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior,, Journal of Mathematical Economics, 18 (1989), 141.  doi: 10.1016/0304-4068(89)90018-9.  Google Scholar

[7]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems,, Mathematics of Operations Research, 28 (2003), 1.  doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[8]

A. Gundel, Robust utility maximization for complete and incomplete market models,, Finance Stochast, 9 (2005), 151.  doi: 10.1007/s00780-004-0148-1.  Google Scholar

[9]

L. P. Hanse and T. J. Sargent, Robust control and model uncertainty,, The American Economic Review, 91 (2001), 60.   Google Scholar

[10]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Second edition. North-Holland Mathematical Library, (1989).   Google Scholar

[11]

J. E. Ingersoll, Theory of Financial Decision Making,, Rowman and Littlefield, (1987).   Google Scholar

[12]

H. Jin, Z. Q. Xu and X. Y. Zhou, A convex stochastic optimization problem arising from portfolio selection,, Math. Finance, 18 (2008), 171.  doi: 10.1111/j.1467-9965.2007.00327.x.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer-Verlag, (1998).  doi: 10.1007/b98840.  Google Scholar

[14]

R. W. Klein and V. S. Bawa, The effect of estimation risk on optimal portfolio choice,, J. Financial Economics, 3 (1976), 215.  doi: 10.1016/0304-405X(76)90004-0.  Google Scholar

[15]

P. Lakner, Utility maximization with partial information,, Stochastic Processes and their Applications, 56 (1995), 247.  doi: 10.1016/0304-4149(94)00073-3.  Google Scholar

[16]

H. Markowitz, Portfolio selection,, J. Finance, 7 (1952), 77.   Google Scholar

[17]

C. Skiadas, Robust control and recursive utility,, Finance Stochast., 7 (2003), 475.  doi: 10.1007/s007800300100.  Google Scholar

[18]

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

show all references

References:
[1]

A. Ben-Tal and A. Nemirovski, Robust convex optimization,, Math. Oepr. Res., 23 (1998), 796.  doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

S. Boyd, L. ElGhaoui, E. Feron and B. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, SIAM Studies in Applied Mathematics, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[3]

T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition,, Math. Finance, 15 (2005), 213.  doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[4]

Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time,, Econometrica, 70 (2002), 1403.  doi: 10.1111/1468-0262.00337.  Google Scholar

[5]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances and covariances on optimal portfolio choice,, J. Portfolio Management, (1993), 6.  doi: 10.3905/jpm.1993.409440.  Google Scholar

[6]

I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior,, Journal of Mathematical Economics, 18 (1989), 141.  doi: 10.1016/0304-4068(89)90018-9.  Google Scholar

[7]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems,, Mathematics of Operations Research, 28 (2003), 1.  doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[8]

A. Gundel, Robust utility maximization for complete and incomplete market models,, Finance Stochast, 9 (2005), 151.  doi: 10.1007/s00780-004-0148-1.  Google Scholar

[9]

L. P. Hanse and T. J. Sargent, Robust control and model uncertainty,, The American Economic Review, 91 (2001), 60.   Google Scholar

[10]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Second edition. North-Holland Mathematical Library, (1989).   Google Scholar

[11]

J. E. Ingersoll, Theory of Financial Decision Making,, Rowman and Littlefield, (1987).   Google Scholar

[12]

H. Jin, Z. Q. Xu and X. Y. Zhou, A convex stochastic optimization problem arising from portfolio selection,, Math. Finance, 18 (2008), 171.  doi: 10.1111/j.1467-9965.2007.00327.x.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer-Verlag, (1998).  doi: 10.1007/b98840.  Google Scholar

[14]

R. W. Klein and V. S. Bawa, The effect of estimation risk on optimal portfolio choice,, J. Financial Economics, 3 (1976), 215.  doi: 10.1016/0304-405X(76)90004-0.  Google Scholar

[15]

P. Lakner, Utility maximization with partial information,, Stochastic Processes and their Applications, 56 (1995), 247.  doi: 10.1016/0304-4149(94)00073-3.  Google Scholar

[16]

H. Markowitz, Portfolio selection,, J. Finance, 7 (1952), 77.   Google Scholar

[17]

C. Skiadas, Robust control and recursive utility,, Finance Stochast., 7 (2003), 475.  doi: 10.1007/s007800300100.  Google Scholar

[18]

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

[1]

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
[2]

Zhilin Kang, Xingyi Li, Zhongfei Li. Mean-CVaR portfolio selection model with ambiguity in distribution and attitude. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019094
[3]

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
[4]

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
[5]

Zhen Wang, Sanyang Liu. Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial & Management Optimization, 2013, 9 (3) : 643-656. doi: 10.3934/jimo.2013.9.643
[6]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2018189
[7]

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, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019133
[8]

Ke Ruan, Masao Fukushima. Robust portfolio selection with a combined WCVaR and factor model. Journal of Industrial & Management Optimization, 2012, 8 (2) : 343-362. doi: 10.3934/jimo.2012.8.343
[9]

Yufei Sun, Ee Ling Grace Aw, Bin Li, Kok Lay Teo, Jie Sun. CVaR-based robust models for portfolio selection. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-11. doi: 10.3934/jimo.2019032
[10]

Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045
[11]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019130
[12]

Peng Zhang. Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection. Journal of Industrial & Management Optimization, 2019, 15 (2) : 537-564. doi: 10.3934/jimo.2018056
[13]

Zhifeng Dai, Huan Zhu, Fenghua Wen. Two nonparametric approaches to mean absolute deviation portfolio selection model. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019054
[14]

Haixiang Yao, Zhongfei Li, Xun Li, Yan Zeng. Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1273-1290. doi: 10.3934/jimo.2016072
[15]

Ioannis Baltas, Anastasios Xepapadeas, Athanasios N. Yannacopoulos. Robust portfolio decisions for financial institutions. Journal of Dynamics & Games, 2018, 5 (2) : 61-94. doi: 10.3934/jdg.2018006
[16]

Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487
[17]

Peng Zhang. Multiperiod mean semi-absolute deviation interval portfolio selection with entropy constraints. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1169-1187. doi: 10.3934/jimo.2016067
[18]

Xueting Cui, Xiaoling Sun, Dan Sha. An empirical study on discrete optimization models for portfolio selection. Journal of Industrial & Management Optimization, 2009, 5 (1) : 33-46. doi: 10.3934/jimo.2009.5.33
[19]

Li Xue, Hao Di. Uncertain portfolio selection with mental accounts and background risk. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1809-1830. doi: 10.3934/jimo.2018124
[20]

Chao Zhang, Jingjing Wang, Naihua Xiu. Robust and sparse portfolio model for index tracking. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1001-1015. doi: 10.3934/jimo.2018082

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences