IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1205.4643.html
   My bibliography  Save this paper

Transaction Costs, Shadow Prices, and Duality in Discrete Time

Author

Listed:
  • Christoph Czichowsky
  • Johannes Muhle-Karbe
  • Walter Schachermayer

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.

Suggested Citation

  • Christoph Czichowsky & Johannes Muhle-Karbe & Walter Schachermayer, 2012. "Transaction Costs, Shadow Prices, and Duality in Discrete Time," Papers 1205.4643, arXiv.org, revised Jan 2014.
    • Handle: RePEc:arx:papers:1205.4643
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1205.4643
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
    3. Griselda Deelstra & Huyên Pham & Nizar Touzi, 2001. "Dual formulation of the utility maximisation problem under transaction costs," ULB Institutional Repository 2013/7596, ULB -- Universite Libre de Bruxelles.
    4. repec:dau:papers:123456789/5630 is not listed on IDEAS
    5. 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.
    6. 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.
    7. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
    8. Luciano Campi & Mark Owen, 2011. "Multivariate utility maximization with proportional transaction costs," Finance and Stochastics, Springer, vol. 15(3), pages 461-499, September.
    9. Jakša Cvitanić & Ioannis Karatzas, 1996. "Hedging And Portfolio Optimization Under Transaction Costs: A Martingale Approach12," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 133-165, April.
    10. (**), 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.
    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. 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.
    13. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the existence of shadow prices," Working Papers hal-00645980, HAL.
    14. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633, arXiv.org, revised Jan 2013.
    15. repec:crs:wpaper:9513 is not listed on IDEAS
    16. Giuseppe Benedetti & Luciano Campi, 2011. "Multivariate utility maximization with proportional transaction costs and random endowment," Working Papers hal-00586377, HAL.
    17. repec:dau:papers:123456789/2318 is not listed on IDEAS
    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. Christoph Czichowsky & Rémi Peyre & Walter Schachermayer & Junjian Yang, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Post-Print hal-02373296, HAL.
    2. Christoph Czichowsky & Rémi Peyre & Walter Schachermayer & Junjian Yang, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Finance and Stochastics, Springer, vol. 22(1), pages 161-180, January.
    3. Christoph Kühn & Alexander Molitor, 2022. "Semimartingale price systems in models with transaction costs beyond efficient friction," Finance and Stochastics, Springer, vol. 26(4), pages 927-982, October.
    4. Teemu Pennanen & Ari-Pekka Perkkio, 2016. "Convex duality in optimal investment and contingent claim valuation in illiquid markets," Papers 1603.02867, arXiv.org.
    5. Lingqi Gu & Yiqing Lin & Junjian Yang, 2017. "Utility maximization problem under transaction costs: optimal dual processes and stability," Papers 1710.04363, arXiv.org.
    6. Falkowski, Adrian & Słomiński, Leszek, 2022. "SDEs with two reflecting barriers driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 164-186.
    7. Christoph Czichowsky & Walter Schachermayer, 2015. "Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion," Papers 1505.02416, arXiv.org, revised Aug 2016.
    8. Yiqing Lin & Junjian Yang, 2016. "Utility maximization problem with random endowment and transaction costs: when wealth may become negative," Papers 1604.08224, arXiv.org, revised Sep 2016.
    9. Christoph Czichowsky & R'emi Peyre & Walter Schachermayer & Junjian Yang, 2016. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Papers 1608.01415, arXiv.org.
    10. Lingqi Gu & Yiqing Lin & Junjian Yang, 2016. "On the existence of shadow prices for optimal investment with random endowment," Papers 1602.01109, arXiv.org, revised Feb 2017.
    11. Christoph Czichowsky & Walter Schachermayer & Junjian Yang, 2014. "Shadow prices for continuous processes," Papers 1408.6065, arXiv.org, revised May 2015.
    12. Jin Hyuk Choi & Tae Ung Gang, 2021. "Optimal investment in illiquid market with search frictions and transaction costs," Papers 2101.09936, arXiv.org, revised Aug 2021.
    13. Jörn Sass & Martin Smaga, 2014. "FTAP in finite discrete time with transaction costs by utility maximization," Finance and Stochastics, Springer, vol. 18(4), pages 805-823, October.
    14. Christoph Czichowsky & Walter Schachermayer, 2014. "Duality Theory for Portfolio Optimisation under Transaction Costs," Papers 1408.5989, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yiqing Lin & Junjian Yang, 2016. "Utility maximization problem with random endowment and transaction costs: when wealth may become negative," Papers 1604.08224, arXiv.org, revised Sep 2016.
    2. Christoph Czichowsky & Walter Schachermayer, 2014. "Duality Theory for Portfolio Optimisation under Transaction Costs," Papers 1408.5989, arXiv.org.
    3. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633, arXiv.org, revised Jan 2013.
    4. Czichowsky, Christoph & Schachermayer, Walter, 2016. "Duality theory for portfolio optimisation under transaction costs," LSE Research Online Documents on Economics 63362, London School of Economics and Political Science, LSE Library.
    5. 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.
    6. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the existence of shadow prices," Working Papers hal-00645980, HAL.
    7. Lingqi Gu & Yiqing Lin & Junjian Yang, 2017. "Utility maximization problem under transaction costs: optimal dual processes and stability," Papers 1710.04363, arXiv.org.
    8. Czichowsky, Christoph & Schachermayer, Walter & Yang, Junjian, 2017. "Shadow prices for continuous processes," LSE Research Online Documents on Economics 63370, London School of Economics and Political Science, LSE Library.
    9. Czichowsky, Christoph & Schachermayer, Walter, 2017. "Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion," LSE Research Online Documents on Economics 67689, London School of Economics and Political Science, LSE Library.
    10. Christoph Czichowsky & Walter Schachermayer & Junjian Yang, 2014. "Shadow prices for continuous processes," Papers 1408.6065, arXiv.org, revised May 2015.
    11. Christoph Czichowsky & Walter Schachermayer, 2015. "Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion," Papers 1505.02416, arXiv.org, revised Aug 2016.
    12. Christoph Czichowsky & Rémi Peyre & Walter Schachermayer & Junjian Yang, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Post-Print hal-02373296, HAL.
    13. Czichowsky, Christoph Johannes & Peyre, Rémi & Schachermayer, Walter & Yang, Junjian, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," LSE Research Online Documents on Economics 85230, London School of Economics and Political Science, LSE Library.
    14. Christoph Czichowsky & Rémi Peyre & Walter Schachermayer & Junjian Yang, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Finance and Stochastics, Springer, vol. 22(1), pages 161-180, January.
    15. Christoph Czichowsky & R'emi Peyre & Walter Schachermayer & Junjian Yang, 2016. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Papers 1608.01415, arXiv.org.
    16. Jörn Sass & Martin Smaga, 2014. "FTAP in finite discrete time with transaction costs by utility maximization," Finance and Stochastics, Springer, vol. 18(4), pages 805-823, October.
    17. Jan Kallsen & Johannes Muhle-Karbe, 2011. "Existence of shadow prices in finite probability spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 251-262, April.
    18. Roux, Alet, 2011. "The fundamental theorem of asset pricing in the presence of bid-ask and interest rate spreads," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 159-163, March.
    19. Matteo Burzoni, 2015. "Arbitrage and Hedging in model-independent markets with frictions," Papers 1512.01488, arXiv.org, revised Aug 2016.
    20. Jan Kallsen & Johannes Muhle-Karbe, 2009. "Existence of Shadow Prices in Finite Probability Spaces," Papers 0911.4801, arXiv.org, revised Nov 2010.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:1205.4643. See general information about how to correct material in RePEc.

    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 bibliographic 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.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.