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Mixing LSMC and PDE Methods to Price Bermudan Options

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  • David Farahany
  • Kenneth Jackson
  • Sebastian Jaimungal

Abstract

We develop a mixed least squares Monte Carlo-partial differential equation (LSMC-PDE) method for pricing Bermudan style options on assets whose volatility is stochastic. The algorithm is formulated for an arbitrary number of assets and volatility processes and we prove the algorithm converges almost surely for a class of models. We also discuss two methods to improve the algorithm's computational complexity. Our numerical examples focus on the single ($2d$) and multi-dimensional ($4d$) Heston models and we compare our hybrid algorithm with classical LSMC approaches. In each case, we find that the hybrid algorithm outperforms standard LSMC in terms of estimating prices and optimal exercise boundaries.

Suggested Citation

  • David Farahany & Kenneth Jackson & Sebastian Jaimungal, 2018. "Mixing LSMC and PDE Methods to Price Bermudan Options," Papers 1803.07216, arXiv.org, revised May 2020.
    • Handle: RePEc:arx:papers:1803.07216
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Duy-Minh Dang & Kenneth R. Jackson & Scott Sues, 2017. "A dimension and variance reduction Monte-Carlo method for option pricing under jump-diffusion models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(3), pages 175-215, May.
    3. Duy-Minh Dang & Kenneth R. Jackson & Mohammadreza Mohammadi, 2015. "Dimension and variance reduction for Monte Carlo methods for high-dimensional models in finance," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(6), pages 522-552, December.
    4. Farid AitSahlia & Manisha Goswami & Suchandan Guha, 2010. "American option pricing under stochastic volatility: an empirical evaluation," Computational Management Science, Springer, vol. 7(2), pages 189-206, April.
    5. repec:dau:papers:123456789/4273 is not listed on IDEAS
    6. Farid AitSahlia & Manisha Goswami & Suchandan Guha, 2010. "American option pricing under stochastic volatility: an efficient numerical approach," Computational Management Science, Springer, vol. 7(2), pages 171-187, April.
    7. Philip Protter & Emmanuelle Clément & Damien Lamberton, 2002. "An analysis of a least squares regression method for American option pricing," Finance and Stochastics, Springer, vol. 6(4), pages 449-471.
    8. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    9. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, February.
    10. Tobias Lipp & Grégoire Loeper & Olivier Pironneau, 2013. "Mixing Monte-Carlo and Partial Differential Equations for Pricing Options," Post-Print hal-01558826, HAL.
    11. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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    Cited by:

    1. Kaustav Das & Ivan Guo & Gr'egoire Loeper, 2021. "On Stochastic Partial Differential Equations and their applications to Derivative Pricing through a conditional Feynman-Kac formula," Papers 2106.14870, arXiv.org, revised Nov 2023.

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