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No-Arbitrage and Hedging with Liquid American Options

Author

Listed:
  • Erhan Bayraktar

    (Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109)

  • Zhou Zhou

    (School of Mathematics and Statistics, University of Sydney, Camperdown NSW 2006, Australia)

Abstract

Since most of the traded options on individual stocks are of American type, it is of interest to generalize the results obtained in semistatic trading to the case when one is allowed to statically trade American options. However, this problem has proved to be elusive so far because of the asymmetric nature of the positions of holding versus shorting such options. Here, we provide a unified framework and generalize the fundamental theorem of asset pricing (FTAP) and hedging dualities in Bayraktar and Zhou [Bayraktar E, Zhou Z (2016) Arbitrage, hedging and utility maximization using semi-static trading strategies with American options. Ann. Appl. Probab. 26(6):3531–3558.] to the case where the investor can also short American options. Following Bayraktar and Zhou [Bayraktar E, Zhou Z (2016) Arbitrage, hedging, and utility maximization using semistatic trading strategies with American options. Ann. Appl. Probab. 26(6):3531–3558.], we assume that the longed American options are divisible. As for the shorted American options, we show that divisibility plays no role regarding arbitrage property and hedging prices. Then, using the method of enlarging probability spaces proposed in Deng and Tan [Deng S, Tan X (2016) Duality in nondominated discrete-time models for Americain options. ArXiv e-prints.], we convert the shorted American options to European options and establish the FTAP and subhedging and superhedging dualities in the enlarged space both with and without model uncertainty.

Suggested Citation

  • Erhan Bayraktar & Zhou Zhou, 2019. "No-Arbitrage and Hedging with Liquid American Options," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 468-486, May.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:2:p:468-486
    DOI: 10.1287/moor.2018.0932
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    References listed on IDEAS

    as
    1. Erhan Bayraktar & Zhou Zhou, 2015. "Arbitrage, hedging and utility maximization using semi-static trading strategies with American options," Papers 1502.06681, arXiv.org, revised Feb 2016.
    2. Erhan Bayraktar & Yuchong Zhang & Zhou Zhou, 2014. "A Note on the Fundamental Theorem of Asset Pricing under Model Uncertainty," Risks, MDPI, vol. 2(4), pages 1-9, October.
    3. Alexander M. G. Cox & Christoph Hoeggerl, 2013. "Model-independent no-arbitrage conditions on American put options," Papers 1301.5467, arXiv.org.
    4. repec:dau:papers:123456789/5710 is not listed on IDEAS
    5. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    6. Erhan Bayraktar & Zhou Zhou, 2017. "Super-Hedging American Options With Semi-Static Trading Strategies Under Model Uncertainty," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-10, September.
    7. David Hobson & Anthony Neuberger, 2017. "Model uncertainty and the pricing of American options," Finance and Stochastics, Springer, vol. 21(1), pages 285-329, January.
    8. Erhan Bayraktar & Yu-Jui Huang & Zhou Zhou, 2013. "On hedging American options under model uncertainty," Papers 1309.2982, arXiv.org, revised Apr 2015.
    9. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
    10. Anna Aksamit & Shuoqing Deng & Jan Obl'oj & Xiaolu Tan, 2016. "Robust pricing--hedging duality for American options in discrete time financial markets," Papers 1604.05517, arXiv.org, revised Apr 2017.
    11. Alexander M. G. Cox & Christoph Hoeggerl, 2016. "Model-Independent No-Arbitrage Conditions On American Put Options," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 431-458, April.
    12. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    13. B. Acciaio & M. Beiglböck & F. Penkner & W. Schachermayer, 2016. "A Model-Free Version Of The Fundamental Theorem Of Asset Pricing And The Super-Replication Theorem," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 233-251, April.
    14. Mark H. A. Davis & David G. Hobson, 2007. "The Range Of Traded Option Prices," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 1-14, January.
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    Cited by:

    1. Erhan Bayraktar & Matteo Burzoni, 2020. "On the quasi-sure superhedging duality with frictions," Finance and Stochastics, Springer, vol. 24(1), pages 249-275, January.
    2. Anna Aksamit & Ivan Guo & Shidan Liu & Zhou Zhou, 2021. "Superhedging duality for multi-action options under model uncertainty with information delay," Papers 2111.14502, arXiv.org, revised Nov 2023.
    3. Erhan Bayraktar & Jingjie Zhang & Zhou Zhou, 2018. "Time Consistent Stopping For The Mean-Standard Deviation Problem --- The Discrete Time Case," Papers 1802.08358, arXiv.org, revised Apr 2019.
    4. Tongseok Lim, 2023. "Optimal exercise decision of American options under model uncertainty," Papers 2310.14473, arXiv.org, revised Nov 2023.
    5. Erhan Bayraktar & Zhou Zhou, 2017. "Super-Hedging American Options With Semi-Static Trading Strategies Under Model Uncertainty," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-10, September.

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