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Transaction Costs, Shadow Prices, and Duality in Discrete Time

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  • Christoph Czichowsky
  • Johannes Muhle-Karbe
  • Walter Schachermayer
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    Abstract

    For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a "shadow price", i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counter-example, we show that shadow prices may fail to exist even in seemingly perfectly benign situations, i.e., for a log-investor trading in an arbitrage-free market with bounded prices and arbitrarily small transaction costs. We also clarify the connection between shadow prices and duality theory. Whereas dual minimizers need not lead to shadow prices in the above "global" sense, we show that they always correspond to a "local" version.

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    File URL: http://arxiv.org/pdf/1205.4643
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1205.4643.

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    Date of creation: May 2012
    Date of revision: Jan 2014
    Handle: RePEc:arx:papers:1205.4643

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    1. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the existence of shadow prices," Working Papers hal-00645980, HAL.
    2. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, vol. 66(1), pages 178-197, June.
    3. 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.
    4. (**), Christophe Stricker & (*), Miklós Rásonyi & Yuri Kabanov, 2002. "No-arbitrage criteria for financial markets with efficient friction," Finance and Stochastics, Springer, vol. 6(3), pages 371-382.
    5. Loewenstein, Mark, 2000. "On optimal portfolio trading strategies for an investor facing transactions costs in a continuous trading market," Journal of Mathematical Economics, Elsevier, vol. 33(2), pages 209-228, March.
    6. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633, arXiv.org, revised Jan 2013.
    7. Kabanov, Yu. M. & Stricker, Ch., 2001. "The Harrison-Pliska arbitrage pricing theorem under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 185-196, April.
    8. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
    9. Luciano Campi & Mark Owen, 2011. "Multivariate utility maximization with proportional transaction costs," Finance and Stochastics, Springer, vol. 15(3), pages 461-499, September.
    10. Kallal, Hedi & Jouini, Elyès, 1995. "Martingales and arbitrage in securities markets with transaction costs," Economics Papers from University Paris Dauphine 123456789/5630, Paris Dauphine University.
    11. Föllmer, Hans & Kramkov, D. O., 1997. "Optional decompositions under constraints," SFB 373 Discussion Papers 1997,31, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    12. repec:fth:inseep:9513 is not listed on IDEAS
    13. repec:wop:humbsf:1997-31 is not listed on IDEAS
    14. Owen, Mark & Campi, Luciano, 2011. "Multivariate Utility Maximization with Proportional Transaction Costs," Economics Papers from University Paris Dauphine 123456789/2318, Paris Dauphine University.
    15. Jaksa Cvitanić & Ioannis Karatzas, 1996. "HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH-super-2," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 133-165.
    16. Giuseppe Benedetti & Luciano Campi, 2011. "Multivariate utility maximization with proportional transaction costs and random endowment," Working Papers hal-00586377, HAL.
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