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Existence of shadow prices in finite probability spaces


  • Jan Kallsen


  • Johannes Muhle-Karbe



A shadow price is a process $${\widetilde{S}}$$ lying within the bid/ask prices $${\underline{S},\overline{S}}$$ of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process $${\widetilde{S}}$$ leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price. Copyright Springer-Verlag 2011

Suggested Citation

  • 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.
  • Handle: RePEc:spr:mathme:v:73:y:2011:i:2:p:251-262
    DOI: 10.1007/s00186-011-0345-6

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    References listed on IDEAS

    1. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
    2. 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.
    3. Lamberton, Damien & Pham, Huyên & Schweizer, Martin, 1998. "Local risk-minimization under transaction costs," SFB 373 Discussion Papers 1998,18, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    4. 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.
    5. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    6. Paolo Guasoni & Mikl'os R'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416,
    7. 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.
    8. repec:dau:papers:123456789/5630 is not listed on IDEAS
    9. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989,
    10. 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.
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    Cited by:

    1. Christoph Czichowsky & Walter Schachermayer, 2015. "Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion," Papers 1505.02416,, revised Aug 2016.
    2. Jörn Sass & Manfred Schäl, 2014. "Numeraire portfolios and utility-based price systems under proportional transaction costs," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 37(2), pages 195-234, October.
    3. 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.
    4. Christoph Czichowsky & R'emi Peyre & Walter Schachermayer & Junjian Yang, 2016. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Papers 1608.01415,
    5. Lingqi Gu & Yiqing Lin & Junjian Yang, 2016. "On the existence of shadow prices for optimal investment with random endowment," Papers 1602.01109,, revised Feb 2017.
    6. 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.
    7. Lingqi Gu & Yiqing Lin & Junjian Yang, 2017. "Utility maximization problem under transaction costs: optimal dual processes and stability," Papers 1710.04363,
    8. Yiqing Lin & Junjian Yang, 2016. "Utility maximization problem with random endowment and transaction costs: when wealth may become negative," Papers 1604.08224,, revised Sep 2016.
    9. repec:spr:finsto:v:22:y:2018:i:1:d:10.1007_s00780-017-0351-5 is not listed on IDEAS

    More about this item


    Transactions costs; Portfolio optimization; Shadow price; 91B28; 91B16;

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