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Simulation‐Based Portfolio Optimization For Large Portfolios With Transaction Costs

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  • Kumar Muthuraman
  • Haining Zha

Abstract

We consider a portfolio optimization problem where the investor's objective is to maximize the long‐term expected growth rate, in the presence of proportional transaction costs. This problem belongs to the class of stochastic control problems with singular controls, which are usually solved by computing solutions to related partial differential equations called the free‐boundary Hamilton–Jacobi–Bellman (HJB) equations. The dimensionality of the HJB equals the number of stocks in the portfolio. The runtime of existing solution methods grow super‐exponentially with dimension, making them unsuitable to compute optimal solutions to portfolio optimization problems with even four stocks. In this work we first present a boundary update procedure that converts the free boundary problem into a sequence of fixed boundary problems. Then by combining simulation with the boundary update procedure, we provide a computational scheme whose runtime, as shown by the numerical tests, scales polynomially in dimension. The results are compared and corroborated against existing methods that scale super‐exponentially in dimension. The method presented herein enables the first ever computational solution to free‐boundary problems in dimensions greater than three.

Suggested Citation

  • Kumar Muthuraman & Haining Zha, 2008. "Simulation‐Based Portfolio Optimization For Large Portfolios With Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 115-134, January.
    • Handle: RePEc:bla:mathfi:v:18:y:2008:i:1:p:115-134
    DOI: 10.1111/j.1467-9965.2007.00324.x
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    References listed on IDEAS

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    Cited by:

    1. Mark Broadie & Weiwei Shen, 2016. "High-Dimensional Portfolio Optimization With Transaction Costs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-49, June.
    2. Yongyang Cai & Kenneth L. Judd & Rong Xu, 2013. "Numerical Solution of Dynamic Portfolio Optimization with Transaction Costs," NBER Working Papers 18709, National Bureau of Economic Research, Inc.
    3. Johannes Ruf & Kangjianan Xie, 2019. "The impact of proportional transaction costs on systematically generated portfolios," Papers 1904.08925, arXiv.org.
    4. Ruf, Johannes & Xie, Kangjianan, 2020. "Impact of proportional transaction costs on systematically generated portfolios," LSE Research Online Documents on Economics 104696, London School of Economics and Political Science, LSE Library.
    5. David B. Brown & James E. Smith, 2011. "Dynamic Portfolio Optimization with Transaction Costs: Heuristics and Dual Bounds," Management Science, INFORMS, vol. 57(10), pages 1752-1770, October.
    6. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2016. "Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach," Papers 1610.07694, arXiv.org, revised Jun 2019.
    7. Nabeel Butt, 2019. "On Discrete Probability Approximations for Transaction Cost Problems," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 365-389, September.
    8. J J Glen, 2011. "Mean-variance portfolio rebalancing with transaction costs and funding changes," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(4), pages 667-676, April.

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