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Optimally stopping a Brownian bridge with an unknown pinning time: A Bayesian approach

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  • Glover, Kristoffer

Abstract

We consider the problem of optimally stopping a Brownian bridge with an unknown pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we assume the stopper has a general continuous prior and is allowed to update their belief about the value of the pinning time through sequential observations of the process. Uncertainty in the pinning time influences both the conditional dynamics of the process and the expected (random) horizon of the optimal stopping problem. We analyse certain gamma and beta distributed priors in detail. Remarkably, the optimal stopping problem in the gamma case becomes time homogeneous and is completely solvable in closed form. Moreover, in the beta case we find that the optimal stopping boundary takes on a square-root form, similar to the classical solution with a known pinning time.

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  • 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.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:919-937
    DOI: 10.1016/j.spa.2020.03.007
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    1. Xin Guo & Robert A. Jarrow & Adrien de Larrard, 2014. "The economic default time and the arcsine law," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(03), pages 1-18.
    2. 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.
    3. Jun Liu, 2004. "Losing Money on Arbitrage: Optimal Dynamic Portfolio Choice in Markets with Arbitrage Opportunities," Review of Financial Studies, Society for Financial Studies, vol. 17(3), pages 611-641.
    4. Brennan, Michael J & Schwartz, Eduardo S, 1990. "Arbitrage in Stock Index Futures," The Journal of Business, University of Chicago Press, vol. 63(1), pages 7-31, January.
    5. Álvaro Cartea & Sebastian Jaimungal & Damir Kinzebulatov, 2016. "Algorithmic Trading With Learning," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-30, June.
    6. Kyle, Albert S, 1985. "Continuous Auctions and Insider Trading," Econometrica, Econometric Society, vol. 53(6), pages 1315-1335, November.
    7. Jean-Paul Décamps & Thomas Mariotti & Stéphane Villeneuve, 2005. "Investment Timing Under Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 472-500, May.
    8. Back, Kerry, 1992. "Insider Trading in Continuous Time," Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 387-409.
    9. Goran Peskir, 2005. "A Change-of-Variable Formula with Local Time on Curves," Journal of Theoretical Probability, Springer, vol. 18(3), pages 499-535, July.
    10. Pavel V. Gapeev, 2012. "Pricing Of Perpetual American Options In A Model With Partial Information," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(01), pages 1-21.
    11. Marco Avellaneda & Michael Lipkin, 2003. "A market-induced mechanism for stock pinning," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 417-425.
    12. Erik Ekström & Bing Lu, 2011. "Optimal Selling of an Asset under Incomplete Information," International Journal of Stochastic Analysis, Hindawi, vol. 2011, pages 1-17, December.
    13. Pavel V. Gapeev, 2012. "Pricing Of Perpetual American Options In A Model With Partial Information," World Scientific Book Chapters, in: Matheus R Grasselli & Lane P Hughston (ed.), Finance at Fields, chapter 14, pages 327-347, World Scientific Publishing Co. Pte. Ltd..
    14. 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.
    15. Bing Lu, 2013. "Optimal Selling of an Asset with Jumps Under Incomplete Information," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(6), pages 599-610, December.
    16. Marc Jeannin & Giulia Iori & David Samuel, 2008. "Modeling stock pinning," Quantitative Finance, Taylor & Francis Journals, vol. 8(8), pages 823-831.
    17. 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.
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    3. 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.

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