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Randomized Stopping Times and American Option Pricing with Transaction Costs

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  • Prasad Chalasani
  • Somesh Jha

Abstract

In a general discrete‐time market model with proportional transaction costs, we derive new expectation representations of the range of arbitrage‐free prices of an arbitrary American option. The upper bound of this range is called the upper hedging price, and is the smallest initial wealth needed to construct a self‐financing portfolio whose value dominates the option payoff at all times. A surprising feature of our upper hedging price representation is that it requires the use of randomized stopping times (Baxter and Chacon 1977), just as ordinary stopping times are needed in the absence of transaction costs. We also represent the upper hedging price as the optimum value of a variety of optimization problems. Additionally, we show a two‐player game where at Nash equilibrium the value to both players is the upper hedging price, and one of the players must in general choose a mixture of stopping times. We derive similar representations for the lower hedging price as well. Our results make use of strong duality in linear programming.

Suggested Citation

  • Prasad Chalasani & Somesh Jha, 2001. "Randomized Stopping Times and American Option Pricing with Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 33-77, January.
    • Handle: RePEc:bla:mathfi:v:11:y:2001:i:1:p:33-77
    DOI: 10.1111/1467-9965.00107
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    Citations

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    Cited by:

    1. Bruno Bouchard & Emmanuel Temam, 2005. "On the Hedging of American Options in Discrete Time Markets with Proportional Transaction Costs," Papers math/0502189, arXiv.org.
    2. Alet Roux & Tomasz Zastawniak, 2013. "American options with gradual exercise under proportional transaction costs," Papers 1308.2688, arXiv.org.
    3. Alet Roux, 2015. "Pricing and hedging game options in currency models with proportional transaction costs," Papers 1504.07920, arXiv.org, revised Aug 2015.
    4. Yan Dolinsky & Halil Soner, 2013. "Duality and convergence for binomial markets with friction," Finance and Stochastics, Springer, vol. 17(3), pages 447-475, July.
    5. Alet Roux & Tomasz Zastawniak, 2016. "Game options with gradual exercise and cancellation under proportional transaction costs," Papers 1612.02312, arXiv.org.
    6. Alet Roux, 2016. "Pricing And Hedging Game Options In Currency Models With Proportional Transaction Costs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(07), pages 1-25, November.
    7. Yuri Kifer, 2012. "Dynkin Games and Israeli Options," Papers 1209.1791, arXiv.org.
    8. Denis Belomestny & Volker Krätschmer, 2017. "Optimal Stopping Under Probability Distortions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 806-833, August.
    9. Alet Roux, 2007. "The fundamental theorem of asset pricing under proportional transaction costs," Papers 0710.2758, arXiv.org.
    10. Yuri Kifer, 2012. "Hedging of game options in discrete markets with transaction costs," Papers 1206.4506, arXiv.org.
    11. M. Pınar & A. Camcı, 2012. "An Integer Programming Model for Pricing American Contingent Claims under Transaction Costs," Computational Economics, Springer;Society for Computational Economics, vol. 39(1), pages 1-12, January.
    12. Yan Dolinsky & Halil Mete Soner, 2011. "Duality and Convergence for Binomial Markets with Friction," Papers 1106.2095, arXiv.org.
    13. Bruno Bouchard & Elyès Jouini, 2010. "Transaction Costs in Financial Models," Post-Print halshs-00703138, HAL.
    14. Johannes Gerer & Gregor Dorfleitner, 2018. "Optimal discrete hedging of American options using an integrated approach to options with complex embedded decisions," Review of Derivatives Research, Springer, vol. 21(2), pages 175-199, July.
    15. David A. Goldberg & Yilun Chen, 2018. "Polynomial time algorithm for optimal stopping with fixed accuracy," Papers 1807.02227, arXiv.org, revised May 2024.
    16. Tokarz, Krzysztof & Zastawniak, Tomasz, 2006. "American contingent claims under small proportional transaction costs," Journal of Mathematical Economics, Elsevier, vol. 43(1), pages 65-85, December.
    17. Mustafa Ç. Pinar, 2010. "Buyer's quantile hedge portfolios in discrete-time trading," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 729-738, October.
    18. Yuan-Hung Hsuku, 2007. "Dynamic consumption and asset allocation with derivative securities," Quantitative Finance, Taylor & Francis Journals, vol. 7(2), pages 137-149.
    19. Joao Amaro de Matos & Ana Lacerda, 2004. "Dry markets and superreplication bounds of American derivatives," Nova SBE Working Paper Series wp461, Universidade Nova de Lisboa, Nova School of Business and Economics.

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