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On using shadow prices in portfolio optimization with transaction costs

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  • J. Kallsen
  • J. Muhle-Karbe
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    Abstract

    In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process. In this paper, we show that this dual approach can actually be used for both deriving a candidate solution and verification in Merton's problem with logarithmic utility and proportional transaction costs. In particular, we determine the shadow price process.

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    File URL: http://arxiv.org/pdf/1010.4989
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1010.4989.

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    Date of creation: Oct 2010
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    Publication status: Published in Annals of Applied Probability 2010, Vol. 20, No. 4, 1341-1358
    Handle: RePEc:arx:papers:1010.4989

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    Web page: http://arxiv.org/

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    Cited by:
    1. Irene Klein & Emmanuel Lepinette & Lavinia Ostafe, 2012. "Large Financial Markets and Asymptotic Arbitrage with Small Transaction Costs," Papers 1211.0443, arXiv.org.
    2. Dmitry B. Rokhlin, 2011. "On the game interpretation of a shadow price process in utility maximization problems under transaction costs," Papers 1112.2406, arXiv.org, revised Dec 2011.
    3. Jan Kallsen & Johannes Muhle-Karbe, 2013. "The General Structure of Optimal Investment and Consumption with Small Transaction Costs," Papers 1303.3148, arXiv.org.
    4. Dmitry Rokhlin, 2013. "On the game interpretation of a shadow price process in utility maximization problems under transaction costs," Finance and Stochastics, Springer, vol. 17(4), pages 819-838, October.
    5. Jan Kallsen & Johannes Muhle-Karbe, 2012. "Option Pricing and Hedging with Small Transaction Costs," Papers 1209.2555, arXiv.org, revised Dec 2012.
    6. Jin Hyuk Choi & Mihai Sirbu & Gordan Zitkovic, 2012. "Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs," Papers 1204.0305, arXiv.org, revised Jun 2012.
    7. Johannes Muhle-Karbe & Ren Liu, 2012. "Portfolio Selection with Small Transaction Costs and Binding Portfolio Constraints," Papers 1205.4588, arXiv.org, revised Jan 2013.
    8. 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.
    9. Christian Bayer & Bezirgen Veliyev, 2014. "Utility Maximization In A Binomial Model With Transaction Costs: A Duality Approach Based On The Shadow Price Process," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(04), pages 1450022-1-1.
    10. 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.
    11. Christoph K\"uhn & Johannes Muhle-Karbe, 2013. "Optimal Liquidity Provision in Limit Order Markets," Papers 1309.5235, arXiv.org, revised May 2014.
    12. Stefan Gerhold & Johannes Muhle-Karbe & Walter Schachermayer, 2010. "The dual optimizer for the growth-optimal portfolio under transaction costs," Papers 1005.5105, arXiv.org, revised Oct 2010.
    13. Jan Kallsen & Johannes Muhle-Karbe, 2011. "Existence of shadow prices in finite probability spaces," Computational Statistics, Springer, vol. 73(2), pages 251-262, April.
    14. Stefan Gerhold & Paolo Guasoni & Johannes Muhle-Karbe & Walter Schachermayer, 2011. "Transaction Costs, Trading Volume, and the Liquidity Premium," Papers 1108.1167, arXiv.org, revised Jan 2013.
    15. S\"oren Christensen & Marc Wittlinger, 2012. "Optimal relaxed portfolio strategies for growth rate maximization problems with transaction costs," Papers 1209.0305, arXiv.org, revised Jun 2013.
    16. Dai, Min & Wang, Hefei & Yang, Zhou, 2012. "Leverage management in a bull–bear switching market," Journal of Economic Dynamics and Control, Elsevier, vol. 36(10), pages 1585-1599.
    17. Jin Hyuk Choi, 2013. "Asymptotic analysis for Merton's problem with transaction costs in power utility case," Papers 1309.3721, arXiv.org, revised Sep 2013.
    18. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2013. "On the existence of shadow prices," Finance and Stochastics, Springer, vol. 17(4), pages 801-818, October.

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