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Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning

Author

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  • Bernardo D’Auria

    (Department of Statistics, University Carlos III of Madrid, Avenida de la Universidad 30, 28911 Leganés (Madrid), Spain
    UC3M-BS Institute of Financial Big Data, University Carlos III of Madrid, Calle Madrid 135, 28903 Getafe (Madrid), Spain
    These authors contributed equally to this work.)

  • Eduardo García-Portugués

    (Department of Statistics, University Carlos III of Madrid, Avenida de la Universidad 30, 28911 Leganés (Madrid), Spain
    UC3M-BS Institute of Financial Big Data, University Carlos III of Madrid, Calle Madrid 135, 28903 Getafe (Madrid), Spain
    These authors contributed equally to this work.)

  • Abel Guada

    (Department of Statistics, University Carlos III of Madrid, Avenida de la Universidad 30, 28911 Leganés (Madrid), Spain
    These authors contributed equally to this work.)

Abstract

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge. This model may be used under special market conditions, in particular we focus on what in the literature is known as the “pinning effect”, that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary. Finally, we compare our method with the optimal exercise time based on a geometric Brownian motion by using real data exhibiting pinning.

Suggested Citation

  • Bernardo D’Auria & Eduardo García-Portugués & Abel Guada, 2020. "Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning," Mathematics, MDPI, vol. 8(7), pages 1-27, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1159-:d:384715
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    References listed on IDEAS

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    1. Tiziano De Angelis & Alessandro Milazzo, 2019. "Optimal stopping for the exponential of a Brownian bridge," Papers 1904.00075, arXiv.org, revised Nov 2019.
    2. Jérôme Detemple & Weidong Tian, 2002. "The Valuation of American Options for a Class of Diffusion Processes," Management Science, INFORMS, vol. 48(7), pages 917-937, July.
    3. William M. Boyce, 1970. "Stopping Rules for Selling Bonds," Bell Journal of Economics, The RAND Corporation, vol. 1(1), pages 27-53, Spring.
    4. Tim Leung & Jiao Li & Xin Li, 2018. "Optimal Timing to Trade along a Randomized Brownian Bridge," IJFS, MDPI, vol. 6(3), pages 1-23, August.
    5. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    6. Marco Avellaneda & Michael Lipkin, 2003. "A market-induced mechanism for stock pinning," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 417-425.
    7. Xiaoyan Ni, Sophie & Pearson, Neil D. & Poteshman, Allen M., 2005. "Stock price clustering on option expiration dates," Journal of Financial Economics, Elsevier, vol. 78(1), pages 49-87, October.
    8. Ekström, Erik & Vaicenavicius, Juozas, 2020. "Optimal stopping of a Brownian bridge with an unknown pinning point," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 806-823.
    9. Marc Jeannin & Giulia Iori & David Samuel, 2008. "Modeling stock pinning," Quantitative Finance, Taylor & Francis Journals, vol. 8(8), pages 823-831.
    10. Glover, Kristoffer, 2022. "Optimally stopping a Brownian bridge with an unknown pinning time: A Bayesian approach," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 919-937.
    11. Baurdoux, Erik J. & Chen, Nan & Surya, Budhi & Yamazak, Kazutoshi, 2015. "Optimal double stopping of a Brownian bridge," LSE Research Online Documents on Economics 61618, London School of Economics and Political Science, LSE Library.
    12. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
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    Cited by:

    1. Abel Azze & Bernardo D'Auria & Eduardo Garc'ia-Portugu'es, 2022. "Optimal stopping of Gauss-Markov bridges," Papers 2211.05835, arXiv.org, revised Jul 2024.

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