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A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs

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  • Arash Fahim
  • Wan-Yu Tsai

Abstract

This paper concerns the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional transaction costs. We consider the optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).

Suggested Citation

  • Arash Fahim & Wan-Yu Tsai, 2017. "A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs," Papers 1711.01017, arXiv.org.
    • Handle: RePEc:arx:papers:1711.01017
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    References listed on IDEAS

    as
    1. Li, Geng, 2009. "Transaction costs and consumption," Journal of Economic Dynamics and Control, Elsevier, vol. 33(6), pages 1263-1277, June.
    2. Halil Mete Soner & Nizar Touzi, 2012. "Homogenization and Asymptotics for Small Transaction Costs," Swiss Finance Institute Research Paper Series 12-13, Swiss Finance Institute.
    3. repec:dau:papers:123456789/5524 is not listed on IDEAS
    4. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    5. Dylan Possamai & H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs: the multidimensional case," Papers 1212.6275, arXiv.org, revised Jan 2013.
    6. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989, arXiv.org.
    7. repec:dau:papers:123456789/6373 is not listed on IDEAS
    8. Erhan Bayraktar & Arash Fahim, 2011. "A Stochastic Approximation for Fully Nonlinear Free Boundary Parabolic Problems," Papers 1109.5752, arXiv.org, revised Nov 2013.
    9. Kumar Muthuraman & Sunil Kumar, 2006. "Multidimensional Portfolio Optimization With Proportional Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 301-335, April.
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    11. Albert Altarovici & Max Reppen & H. Mete Soner, 2016. "Optimal Consumption and Investment with Fixed and Proportional Transaction Costs," Papers 1610.03958, arXiv.org.
    12. Magill, Michael J. P. & Constantinides, George M., 1976. "Portfolio selection with transactions costs," Journal of Economic Theory, Elsevier, vol. 13(2), pages 245-263, October.
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    15. Tourin, Agnes & Zariphopoulou, Thaleia, 1994. "Numerical Schemes for Investment Models with Singular Transactions," Computational Economics, Springer;Society for Computational Economics, vol. 7(4), pages 287-307.
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