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Mixing Monte-Carlo and Partial Differential Equations for Pricing Options

Author

Listed:
  • Tobias Lipp

    (LJLL - Laboratoire Jacques-Louis Lions - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Grégoire Loeper

    (Monash University [Clayton])

  • Olivier Pironneau

    (LJLL - Laboratoire Jacques-Louis Lions - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

Abstract

There is a need for very fast option pricers when the financial objects are mod-eled by complex systems of stochastic differential equations. Here the authors investigate option pricers based on mixed Monte-Carlo partial differential solvers for stochastic volatility models such as Heston's. It is found that orders of magnitude in speed are gained on full Monte-Carlo algorithms by solving all equations but one by a Monte-Carlo method, and pricing the underlying asset by a partial differential equation with random coefficients, derived by Itô calculus. This strategy is investigated for vanilla options, barrier options and American options with stochastic volatilities and jumps optionally.

Suggested Citation

  • Tobias Lipp & Grégoire Loeper & Olivier Pironneau, 2013. "Mixing Monte-Carlo and Partial Differential Equations for Pricing Options," Post-Print hal-01558826, HAL.
    • Handle: RePEc:hal:journl:hal-01558826
    DOI: 10.1007/s11401-013-0763-2
    Note: View the original document on HAL open archive server: https://hal.sorbonne-universite.fr/hal-01558826
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    References listed on IDEAS

    as
    1. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    2. Kaushik Amin & Ajay Khanna, 1994. "Convergence Of American Option Values From Discrete‐ To Continuous‐Time Financial Models1," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 289-304, October.
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    4. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893, September.
    5. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
    8. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
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    Cited by:

    1. Andrei Cozma & Christoph Reisinger, 2015. "A mixed Monte Carlo and PDE variance reduction method for foreign exchange options under the Heston-CIR model," Papers 1509.01479, arXiv.org, revised Apr 2016.
    2. Jan Posp'iv{s}il & Vladim'ir v{S}v'igler, 2019. "Isogeometric analysis in option pricing," Papers 1910.00258, arXiv.org.
    3. David Farahany & Kenneth Jackson & Sebastian Jaimungal, 2018. "Mixing LSMC and PDE Methods to Price Bermudan Options," Papers 1803.07216, arXiv.org, revised May 2020.

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