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A closed-form solution to the problem of super-replication under transaction costs


  • HuyËn Pham

    () (Equipe d'Analyse et de MathÊmatiques AppliquÊes, UniversitÊ Marne-la-VallÊe, F-77454 Marne-la-VallÊe Cedex 2, France and CREST)

  • Nizar Touzi

    () (CEREMADE, UniversitÊ Paris Dauphine, F-75775 Paris Cedex, France and CREST Manuscript)

  • Jaksa Cvitanic

    () (Department of Statistics, Columbia University, New York, NY 10027, USA)


We study the problem of finding the minimal price needed to dominate European-type contingent claims under proportional transaction costs in a continuous-time diffusion model. The result we prove has already been known in special cases - the minimal super-replicating strategy is the least expensive buy-and-hold strategy. Our contribution consists in showing that this result remains valid for general path-independent claims, and in providing a shorter and more intuitive, financial mathematics-type proof. It is based on a previously known representation of the minimal price as a supremum of the prices in corresponding shadow markets, and on a PDE (viscosity) characterization of that representation.

Suggested Citation

  • HuyËn Pham & Nizar Touzi & Jaksa Cvitanic, 1999. "A closed-form solution to the problem of super-replication under transaction costs," Finance and Stochastics, Springer, vol. 3(1), pages 35-54.
  • Handle: RePEc:spr:finsto:v:3:y:1999:i:1:p:35-54
    Note: received: May 1997; final version received: October 1997

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    Cited by:

    1. C. Atkinson & C. A. Alexandropoulos, 2006. "Pricing a European Basket Option in the Presence of Proportional Transaction Costs," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(3), pages 191-214.
    2. Paolo Guasoni & Mikl'os R'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416,
    3. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
    4. Yan Dolinsky, 2011. "Hedging of Game Options With the Presence of Transaction Costs," Papers 1103.1165,, revised Mar 2012.
    5. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633,, revised Jan 2013.
    6. Tomasz R. Bielecki & Igor Cialenco & Rodrigo Rodriguez, 2012. "No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs," Papers 1205.6254,, revised Jun 2013.
    7. De Jong, Frank, 2008. "Valuation of pension liabilities in incomplete markets," Journal of Pension Economics and Finance, Cambridge University Press, vol. 7(03), pages 277-294, November.
    8. Tzuu-Shuh Chiang & Shang-Yuan Shiu & Shuenn-Jyi Sheu, 2007. "Price systems for markets with transaction costs and control problems for some finance problems," Papers math/0702828,
    9. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the existence of shadow prices," Working Papers hal-00645980, HAL.

    More about this item


    Transaction costs; super-replicating strategies; viscosity solutions;

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing


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