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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.
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    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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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Economic Dynamics and Control.

    Volume (Year): 34 (2010)
    Issue (Month): 2 (February)
    Pages: 207-230

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    Handle: RePEc:eee:dyncon:v:34:y:2010:i:2:p:207-230

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    Web page: http://www.elsevier.com/locate/jedc

    Related research

    Keywords: Optimal control Mean variance tradeoff HJB equation Viscosity solution;

    References

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    1. Munk, Claus, 2000. "Optimal consumption/investment policies with undiversifiable income risk and liquidity constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 24(9), pages 1315-1343, August.
    2. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
    3. Chellathurai, Thamayanthi & Draviam, Thangaraj, 2007. "Dynamic portfolio selection with fixed and/or proportional transaction costs using non-singular stochastic optimal control theory," Journal of Economic Dynamics and Control, Elsevier, vol. 31(7), pages 2168-2195, July.
    4. Chiu, Mei Choi & Li, Duan, 2006. "Asset and liability management under a continuous-time mean-variance optimization framework," Insurance: Mathematics and Economics, Elsevier, vol. 39(3), pages 330-355, December.
    5. A. C. Belanger & P. A. Forsyth & G. Labahn, 2009. "Valuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 451-496.
    6. Cairns, Andrew J.G. & Blake, David & Dowd, Kevin, 2006. "Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans," Journal of Economic Dynamics and Control, Elsevier, vol. 30(5), pages 843-877, May.
    7. Markus LEIPPOLD & Fabio TROJANI & Paolo VANINI, 2002. "A Geometric Approach to Multiperiod Mean Variance Optimization of Assets and Liabilities," FAME Research Paper Series rp48, International Center for Financial Asset Management and Engineering.
    8. Gerrard, Russell & Haberman, Steven & Vigna, Elena, 2004. "Optimal investment choices post-retirement in a defined contribution pension scheme," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 321-342, October.
    9. Wang, Zengwu & Xia, Jianming & Zhang, Lihong, 2007. "Optimal investment for an insurer: The martingale approach," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 322-334, March.
    10. Nguyen, Pascal & Portait, Roland, 2002. "Dynamic asset allocation with mean variance preferences and a solvency constraint," Journal of Economic Dynamics and Control, Elsevier, vol. 26(1), pages 11-32, January.
    11. Duan Li & Wan-Lung Ng, 2000. "Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation," Mathematical Finance, Wiley Blackwell, vol. 10(3), pages 387-406.
    12. Damgaard, Anders, 2006. "Computation of reservation prices of options with proportional transaction costs," Journal of Economic Dynamics and Control, Elsevier, vol. 30(3), pages 415-444, March.
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    Citations

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    Cited by:
    1. 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.
    2. 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.
    3. 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.
    4. 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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