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On a multi-asset version of the Kusuoka limit theorem of option superreplication under transaction costs

Author

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  • Julien Grépat

    (Université Bourgogne Franche-Comté)

  • Yuri Kabanov

    (Lomonosov Moscow State University and Steklov Mathematical Institute of the Russian Academy of Sciences)

Abstract

We consider, using the geometric description, a sequence of models of multi-asset financial markets with proportional transaction costs vanishing in the limit. We assume that the price processes are He-type multinomial approximations of a process whose components are correlated geometric Brownian motions. For a given vector-valued contingent claim, defined as a continuous function of the price trajectories, we consider for each model the hedging set, that is, the set of all vector-valued initial endowments permitting to superreplicate the contingent claim by the final position of a self-financing portfolio. We calculate the limit of the hedging sets in the closed topology, obtaining in this way a set-valued version of the Kusuoka limit theorem.

Suggested Citation

  • Julien Grépat & Yuri Kabanov, 2021. "On a multi-asset version of the Kusuoka limit theorem of option superreplication under transaction costs," Finance and Stochastics, Springer, vol. 25(1), pages 167-187, January.
    • Handle: RePEc:spr:finsto:v:25:y:2021:i:1:d:10.1007_s00780-020-00441-4
    DOI: 10.1007/s00780-020-00441-4
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    References listed on IDEAS

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    1. Friedrich Hubalek & Walter Schachermayer, 1998. "When Does Convergence of Asset Price Processes Imply Convergence of Option Prices?," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 385-403, October.
    2. Peter Bank & Yan Dolinsky & Ari-Pekka Perkkiö, 2017. "The scaling limit of superreplication prices with small transaction costs in the multivariate case," Finance and Stochastics, Springer, vol. 21(2), pages 487-508, April.
    3. Yuri Kabanov, 2009. "Markets with Transaction Costs. Mathematical Theory," Post-Print hal-00488168, HAL.
    4. He, Hua, 1990. "Convergence from Discrete- to Continuous-Time Contingent Claims Prices," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 523-546.
    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. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, June.
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    Cited by:

    1. Çağın Ararat & Zachary Feinstein, 2021. "Set-valued risk measures as backward stochastic difference inclusions and equations," Finance and Stochastics, Springer, vol. 25(1), pages 43-76, January.

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    More about this item

    Keywords

    Hedging; Multinomial approximation; Transaction costs; Kusuoka theorem; Superreplication;
    All these keywords.

    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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