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The super-replication theorem under proportional transaction costs revisited

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  • Walter Schachermayer

Abstract

We consider a financial market with one riskless and one risky asset. The super-replication theorem states that there is no duality gap in the problem of super-replicating a contingent claim under transaction costs and the associated dual problem. We give two versions of this theorem. The first theorem relates a num\'eraire-based admissibility condition in the primal problem to the notion of a local martingale in the dual problem. The second theorem relates a num\'eraire -free admissibility condition in the primal problem to the notion of a uniformly integrable martingale in the dual problem.

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  • Walter Schachermayer, 2014. "The super-replication theorem under proportional transaction costs revisited," Papers 1405.1266, arXiv.org.
    • Handle: RePEc:arx:papers:1405.1266
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    References listed on IDEAS

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    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. Luciano Campi & Walter Schachermayer, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Finance and Stochastics, Springer, vol. 10(4), pages 579-596, December.
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    Citations

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    Cited by:

    1. Irene Klein & Emmanuel Lépinette & Lavinia Perez-Ostafe, 2014. "Asymptotic arbitrage with small transaction costs," Finance and Stochastics, Springer, vol. 18(4), pages 917-939, October.
    2. Christoph Kuhn, 2023. "The fundamental theorem of asset pricing with and without transaction costs," Papers 2307.00571, arXiv.org.
    3. Lingqi Gu & Yiqing Lin & Junjian Yang, 2016. "A note on utility maximization with transaction costs and random endoment: num\'eraire-based model and convex duality," Papers 1602.01070, arXiv.org, revised Feb 2016.
    4. Christoph Czichowsky & Raphael Huwyler, 2022. "Robust utility maximisation under proportional transaction costs for c\`adl\`ag price processes," Papers 2211.00532, arXiv.org, revised May 2023.
    5. Yan Dolinsky & H. Mete Soner, 2017. "Convex Duality with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 448-471, May.
    6. Francesca Biagini & Thomas Reitsam, 2021. "A dynamic version of the super-replication theorem under proportional transaction costs," Papers 2107.02628, arXiv.org.
    7. Matteo Burzoni, 2015. "Arbitrage and Hedging in model-independent markets with frictions," Papers 1512.01488, arXiv.org, revised Aug 2016.
    8. Peter Bank & Yan Dolinsky, 2018. "Continuous-time Duality for Super-replication with Transient Price Impact," Papers 1808.09807, arXiv.org, revised May 2019.
    9. Francesca Biagini & Thomas Reitsam, 2019. "Asset Price Bubbles in market models with proportional transaction costs," Papers 1911.10149, arXiv.org, revised Dec 2020.
    10. Erhan Bayraktar & Xiang Yu, 2015. "Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices," Papers 1504.00310, arXiv.org, revised Aug 2018.
    11. 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.
    12. 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.
    13. 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.
    14. Francesca Biagini & Lukas Gonon & Thomas Reitsam, 2021. "Neural network approximation for superhedging prices," Papers 2107.14113, arXiv.org.
    15. Yan Dolinsky & H. Mete Soner, 2015. "Convex duality with transaction costs," Papers 1502.01735, arXiv.org, revised Oct 2015.
    16. Christoph Czichowsky & Walter Schachermayer, 2014. "Duality Theory for Portfolio Optimisation under Transaction Costs," Papers 1408.5989, arXiv.org.

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