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A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options

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  • Sergey Smirnov

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory 1/52, 119991 Moscow, Russia
    Financial Engineering and Risk Management Laboratory, National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, 101000 Moscow, Russia)

Abstract

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions.

Suggested Citation

  • Sergey Smirnov, 2019. "A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options," Mathematics, MDPI, vol. 7(12), pages 1-19, December.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1246-:d:298974
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    References listed on IDEAS

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    1. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. "General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
    2. Leon A Petrosyan & Nikolay A Zenkevich, 2016. "Game Theory," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 9824, September.
    3. repec:dau:papers:123456789/10794 is not listed on IDEAS
    4. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    5. Dana, Rose-Anne & Le Van, Cuong & Magnien, Francois, 1999. "On the Different Notions of Arbitrage and Existence of Equilibrium," Journal of Economic Theory, Elsevier, vol. 87(1), pages 169-193, July.
    6. Z. Hucki & V. N. Kolokoltsov, 2007. "Pricing Of Rainbow Options: Game Theoretic Approach," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 215-242.
    7. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    8. Erhan Bayraktar & Zhou Zhou, 2017. "On Arbitrage And Duality Under Model Uncertainty And Portfolio Constraints," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 988-1012, October.
    9. Matteo Burzoni & Marco Frittelli & Marco Maggis, 2016. "Universal arbitrage aggregator in discrete-time markets under uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 1-50, January.
    10. Sanjeet Singh & Nav Bhardwaj & Gagan Deep Sharma & Tuğberk Kaya & Mandeep Mahendru & Burak Erkut, 2019. "Research in market-calibrated option pricing analysis," Qualitative Research in Financial Markets, Emerald Group Publishing Limited, vol. 12(2), pages 159-176, July.
    11. Jan Obloj & Johannes Wiesel, 2018. "A unified Framework for Robust Modelling of Financial Markets in discrete time," Papers 1808.06430, arXiv.org, revised Dec 2019.
    12. Matteo Burzoni & Marco Frittelli & Marco Maggis, 2016. "Universal arbitrage aggregator in discrete-time markets under uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 1-50, January.
    13. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    14. Yaacov Z. Bergman & Bruce D. Grundy & Zvi Wiener, "undated". "General Properties of Option Prices (Revision of 11-95) (Reprint 058)," Rodney L. White Center for Financial Research Working Papers 1-96, Wharton School Rodney L. White Center for Financial Research.
    15. repec:dau:papers:123456789/6228 is not listed on IDEAS
    16. Erhan Bayraktar & Yuchong Zhang, 2016. "Fundamental Theorem of Asset Pricing Under Transaction Costs and Model Uncertainty," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 1039-1054, August.
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    Cited by:

    1. Sergey Smirnov, 2022. "Correction: Smirnov, S. A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options. Mathematics 2019, 7 , 1246," Mathematics, MDPI, vol. 10(23), pages 1-4, November.

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