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Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation

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  • Wang, J.
  • Forsyth, P.A.

Abstract

We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic linear-quadratic (LQ) problems using the method in Zhou and Li (2000) and Li and Ng (2000). We use a finite difference method with fully implicit timestepping to solve the resulting nonlinear Hamilton-Jacobi-Bellman (HJB) PDE, and present the solutions in terms of an efficient frontier and an optimal asset allocation strategy. The numerical scheme satisfies sufficient conditions to ensure convergence to the viscosity solution of the HJB PDE. We handle various constraints on the optimal policy. Numerical tests indicate that realistic constraints can have a dramatic effect on the optimal policy compared to the unconstrained solution.

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  • Wang, J. & Forsyth, P.A., 2010. "Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation," Journal of Economic Dynamics and Control, Elsevier, vol. 34(2), pages 207-230, February.
    • Handle: RePEc:eee:dyncon:v:34:y:2010:i:2:p:207-230
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    3. Van Staden, Pieter M. & Dang, Duy-Minh & Forsyth, Peter A., 2018. "Time-consistent mean–variance portfolio optimization: A numerical impulse control approach," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 9-28.
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    5. Wang, J. & Forsyth, P.A., 2011. "Continuous time mean variance asset allocation: A time-consistent strategy," European Journal of Operational Research, Elsevier, vol. 209(2), pages 184-201, March.
    6. Guiyuan Ma & Song-Ping Zhu & Boda Kang, 2020. "A Numerical Solution of Optimal Portfolio Selection Problem with General Utility Functions," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 957-981, March.
    7. Jiamian Lin & Xi Li & SingRu (Celine) Hoe & Zhongfeng Yan, 2023. "A Generalized Finite Difference Method for Solving Hamilton–Jacobi–Bellman Equations in Optimal Investment," Mathematics, MDPI, vol. 11(10), pages 1-20, May.
    8. Xavier Warin, 2021. "Deep learning for efficient frontier calculation in finance," Papers 2101.02044, arXiv.org, revised Feb 2022.
    9. Aivaliotis, Georgios & Palczewski, Jan, 2014. "Investment strategies and compensation of a mean–variance optimizing fund manager," European Journal of Operational Research, Elsevier, vol. 234(2), pages 561-570.
    10. Dang, D.M. & Forsyth, P.A., 2016. "Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton–Jacobi–Bellman equation approach," European Journal of Operational Research, Elsevier, vol. 250(3), pages 827-841.
    11. Strub, Moris S. & Li, Duan & Cui, Xiangyu & Gao, Jianjun, 2019. "Discrete-time mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR," Journal of Economic Dynamics and Control, Elsevier, vol. 108(C).
    12. Cong, F. & Oosterlee, C.W., 2016. "On pre-commitment aspects of a time-consistent strategy for a mean-variance investor," Journal of Economic Dynamics and Control, Elsevier, vol. 70(C), pages 178-193.
    13. F. Cong & C. W. Oosterlee, 2017. "On Robust Multi-Period Pre-Commitment And Time-Consistent Mean-Variance Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-26, November.
    14. Chi Kin Lam & Yuhong Xu & Guosheng Yin, 2016. "Dynamic portfolio selection without risk-free assets," Papers 1602.04975, arXiv.org.
    15. Forsyth, P.A. & Kennedy, J.S. & Tse, S.T. & Windcliff, H., 2012. "Optimal trade execution: A mean quadratic variation approach," Journal of Economic Dynamics and Control, Elsevier, vol. 36(12), pages 1971-1991.
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    17. Ballestra, Luca Vincenzo & Pacelli, Graziella, 2013. "Pricing European and American options with two stochastic factors: A highly efficient radial basis function approach," Journal of Economic Dynamics and Control, Elsevier, vol. 37(6), pages 1142-1167.

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