IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v19y2016i04ns0219024916500254.html
   My bibliography  Save this article

High-Dimensional Portfolio Optimization With Transaction Costs

Author

Listed:
  • MARK BROADIE

    () (Graduate School of Business, Columbia University, New York, NY 10027, USA)

  • WEIWEI SHEN

    (Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA)

Abstract

This paper studies Merton’s portfolio optimization problem with proportional transaction costs in a discrete-time finite horizon. Facing short-sale and borrowing constraints, investors have access to a risk-free asset and multiple risky assets whose returns follow a multivariate geometric Brownian motion. Lower and upper bounds for optimal solutions up to the problem with 20 risky assets and 40 investment periods are computed. Three lower bounds are proposed: the value function optimization (VF), the hyper-sphere and the hyper-cube policy parameterizations (HS and HC). VF attacks the conundrums in traditional value function iteration for high-dimensional dynamic programs with continuous decision and state spaces. HS and HC respectively approximate the geometry of the trading policy in the high-dimensional state space by two surfaces. To evaluate lower bounds, two new upper bounds are provided via a duality method based on a new auxiliary problem (OMG and OMG2). Compared with existing methods across various suites of parameters, new methods lucidly show superiority. The three lower bound methods always achieve higher utilities, HS and HC cut run times by a factor of 100, and OMG and OMG2 mostly provide tighter upper bounds. In addition, how the no-trading region characterizing the optimal policy deforms when short-sale and borrowing constraints bind is investigated.

Suggested Citation

  • 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.
  • Handle: RePEc:wsi:ijtafx:v:19:y:2016:i:04:n:s0219024916500254
    DOI: 10.1142/S0219024916500254
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024916500254
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, vol. 50(9), pages 1222-1234, September.
    2. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    3. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1267-1321, June.
    4. Michael W. Brandt & Amit Goyal & Pedro Santa-Clara & Jonathan R. Stroud, 2005. "A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability," Review of Financial Studies, Society for Financial Studies, vol. 18(3), pages 831-873.
    5. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, January.
    6. David B. Brown & James E. Smith, 2013. "Optimal Sequential Exploration: Bandits, Clairvoyants, and Wildcats," Operations Research, INFORMS, vol. 61(3), pages 644-665, June.
    7. Kenneth L. Judd & Lilia Maliar & Serguei Maliar, 2011. "Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models," Quantitative Economics, Econometric Society, vol. 2(2), pages 173-210, July.
    8. C. Atkinson & S. Mokkhavesa, 2004. "Multi-asset portfolio optimization with transaction cost," Applied Mathematical Finance, Taylor & Francis Journals, vol. 11(2), pages 95-123.
    9. Anthony W. Lynch & Pierluigi Balduzzi, 2000. "Predictability and Transaction Costs: The Impact on Rebalancing Rules and Behavior," Journal of Finance, American Finance Association, vol. 55(5), pages 2285-2309, October.
    10. Andrew J. Morton & Stanley R. Pliska, 1995. "Optimal Portfolio Management With Fixed Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 5(4), pages 337-356.
    11. Gaspar, Jess & L. Judd, Kenneth, 1997. "Solving Large-Scale Rational-Expectations Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(01), pages 45-75, January.
    12. Mark Broadie & Yiping Du & Ciamac C. Moallemi, 2015. "Risk Estimation via Regression," Operations Research, INFORMS, vol. 63(5), pages 1077-1097, October.
    13. Jules Binsbergen & Michael Brandt, 2007. "Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?," Computational Economics, Springer;Society for Computational Economics, vol. 29(3), pages 355-367, May.
    14. Balduzzi, Pierluigi & Lynch, Anthony W., 1999. "Transaction costs and predictability: some utility cost calculations," Journal of Financial Economics, Elsevier, vol. 52(1), pages 47-78, April.
    15. 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.
    16. Lynch, Anthony W. & Tan, Sinan, 2010. "Multiple Risky Assets, Transaction Costs, and Return Predictability: Allocation Rules and Implications for U.S. Investors," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 45(04), pages 1015-1053, August.
    17. John Y. Campbell & João Cocco & Francisco Gomes & Pascal J. Maenhout & Luis M. Viceira, 2001. "Stock Market Mean Reversion and the Optimal Equity Allocation of a Long-Lived Investor," Review of Finance, European Finance Association, vol. 5(3), pages 269-292.
    18. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    19. 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.
    20. S. Gerhold & J. Muhle-Karbe & W. Schachermayer, 2013. "The dual optimizer for the growth-optimal portfolio under transaction costs," Finance and Stochastics, Springer, vol. 17(2), pages 325-354, April.
    21. Dumas, Bernard & Luciano, Elisa, 1991. " An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, vol. 46(2), pages 577-595, June.
    22. 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.
    23. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989, arXiv.org.
    24. Kumar Muthuraman & Sunil Kumar, 2006. "Multidimensional Portfolio Optimization With Proportional Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 301-335.
    25. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    26. Hong Liu, 2004. "Optimal Consumption and Investment with Transaction Costs and Multiple Risky Assets," Journal of Finance, American Finance Association, vol. 59(1), pages 289-338, February.
    27. Martin B. Haugh & Leonid Kogan, 2004. "Pricing American Options: A Duality Approach," Operations Research, INFORMS, vol. 52(2), pages 258-270, April.
    28. Karel Janeček & Steven Shreve, 2004. "Asymptotic analysis for optimal investment and consumption with transaction costs," Finance and Stochastics, Springer, vol. 8(2), pages 181-206, May.
    29. Stefan Gerhold & Paolo Guasoni & Johannes Muhle-Karbe & Walter Schachermayer, 2014. "Transaction costs, trading volume, and the liquidity premium," Finance and Stochastics, Springer, vol. 18(1), pages 1-37, January.
    30. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286.
    31. Guoming Lai & François Margot & Nicola Secomandi, 2010. "An Approximate Dynamic Programming Approach to Benchmark Practice-Based Heuristics for Natural Gas Storage Valuation," Operations Research, INFORMS, vol. 58(3), pages 564-582, June.
    32. Hong Liu & Mark Loewenstein, 2002. "Optimal Portfolio Selection with Transaction Costs and Finite Horizons," Review of Financial Studies, Society for Financial Studies, vol. 15(3), pages 805-835.
    33. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    34. Lorenzo Garlappi & Georgios Skoulakis, 2010. "Solving Consumption and Portfolio Choice Problems: The State Variable Decomposition Method," Review of Financial Studies, Society for Financial Studies, vol. 23(9), pages 3346-3400.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2018. "An improved Least Squares Monte Carlo method for portfolio optimization with high dimensional control," Papers 1803.11467, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wsi:ijtafx:v:19:y:2016:i:04:n:s0219024916500254. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Tai Tone Lim). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.