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Iterative construction of the optimal Bermudan stopping time

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  • Anastasia Kolodko
  • John Schoenmakers

Abstract

We present an iterative procedure for computing the optimal Bermudan stopping time, hence the Bermudan Snell envelope. The method produces an increasing sequence of approximations of the Snell envelope from below, which coincide with the Snell envelope after finitely many steps. Then, by duality, the method induces a convergent sequence of upper bounds as well. In a Markovian setting the presented procedure allows to calculate approximative solutions with only a few nestings of conditional expectations and is therefore tailor-made for a plain Monte Carlo implementation. The method may be considered generic for all discrete optimal stopping problems. The power of the procedure is demonstrated for Bermudan swaptions in a full factor LIBOR market model. Copyright Springer-Verlag Berlin/Heidelberg 2006

Suggested Citation

  • Anastasia Kolodko & John Schoenmakers, 2006. "Iterative construction of the optimal Bermudan stopping time," Finance and Stochastics, Springer, vol. 10(1), pages 27-49, January.
    • Handle: RePEc:spr:finsto:v:10:y:2006:i:1:p:27-49
    DOI: 10.1007/s00780-005-0168-5
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    Citations

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    Cited by:

    1. John Schoenmakers, 2012. "A pure martingale dual for multiple stopping," Finance and Stochastics, Springer, vol. 16(2), pages 319-334, April.
    2. Denis Belomestny & Christian Bender & John Schoenmakers, 2009. "True Upper Bounds For Bermudan Products Via Non‐Nested Monte Carlo," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 53-71, January.
    3. Joerg Kampen & Anastasia Kolodko & John Schoenmakers, 2008. "Monte Carlo Greeks for financial products via approximative transition densities," Papers 0807.1213, arXiv.org.
    4. Dan Andrei Iancu & Nikolaos Trichakis & Do Young Yoon, 2021. "Monitoring with Limited Information," Management Science, INFORMS, vol. 67(7), pages 4233-4251, July.
    5. Soren Christensen & Albrecht Irle & Julian Peter Lemburg, 2021. "Flexible forward improvement iteration for infinite time horizon Markovian optimal stopping problems," Papers 2111.13443, arXiv.org.
    6. Beveridge, Christopher & Joshi, Mark & Tang, Robert, 2013. "Practical policy iteration: Generic methods for obtaining rapid and tight bounds for Bermudan exotic derivatives using Monte Carlo simulation," Journal of Economic Dynamics and Control, Elsevier, vol. 37(7), pages 1342-1361.
    7. Wei, Wei & Zhu, Dan, 2022. "Generic improvements to least squares monte carlo methods with applications to optimal stopping problems," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1132-1144.
    8. Denis Belomestny & Grigori Milstein & Vladimir Spokoiny, 2009. "Regression methods in pricing American and Bermudan options using consumption processes," Quantitative Finance, Taylor & Francis Journals, vol. 9(3), pages 315-327.
    9. Christoph Reisinger & Rasmus Wissmann, 2012. "Numerical Valuation of Derivatives in High-Dimensional Settings via PDE Expansions," Papers 1209.1909, arXiv.org, revised Oct 2013.
    10. Denis Belomestny & G. Milstein & John Schoenmakers, 2010. "Sensitivities for Bermudan options by regression methods," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 33(2), pages 117-138, November.
    11. Christian Bayer & Denis Belomestny & Paul Hager & Paolo Pigato & John Schoenmakers, 2020. "Randomized optimal stopping algorithms and their convergence analysis," Papers 2002.00816, arXiv.org.
    12. Ivan Guo & Nicolas Langren'e & Jiahao Wu, 2023. "Simultaneous upper and lower bounds of American option prices with hedging via neural networks," Papers 2302.12439, arXiv.org, revised Apr 2024.
    13. Denis Belomestny & John Schoenmakers & Fabian Dickmann, 2013. "Multilevel dual approach for pricing American style derivatives," Finance and Stochastics, Springer, vol. 17(4), pages 717-742, October.
    14. Christian Bender & Christian Gaertner & Nikolaus Schweizer, 2016. "Pathwise Iteration for Backward SDEs," Papers 1605.07500, arXiv.org, revised Jun 2016.
    15. Roberto Baviera & Lorenzo Giada, 2013. "A perturbative approach to Bermudan options pricing with applications," Quantitative Finance, Taylor & Francis Journals, vol. 13(2), pages 255-263, January.
    16. John Schoenmakers & Junbo Huang & Jianing Zhang, 2011. "Optimal dual martingales, their analysis and application to new algorithms for Bermudan products," Papers 1111.6038, arXiv.org, revised Feb 2012.
    17. Christian Bender & Anastasia Kolodko & John Schoenmakers, 2008. "Enhanced policy iteration for American options via scenario selection," Quantitative Finance, Taylor & Francis Journals, vol. 8(2), pages 135-146.
    18. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen & Timo Welti, 2019. "Solving high-dimensional optimal stopping problems using deep learning," Papers 1908.01602, arXiv.org, revised Aug 2021.
    19. Joshi, Mark & Tang, Robert, 2014. "Effective sub-simulation-free upper bounds for the Monte Carlo pricing of callable derivatives and various improvements to existing methodologies," Journal of Economic Dynamics and Control, Elsevier, vol. 40(C), pages 25-45.
    20. Denis Belomestny & John Schoenmakers, 2021. "From optimal martingales to randomized dual optimal stopping," Papers 2102.01533, arXiv.org.

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