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Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

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  • Michael B. Giles
  • Christoph Reisinger

Abstract

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.

Suggested Citation

  • Michael B. Giles & Christoph Reisinger, 2012. "Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance," Papers 1204.1442, arXiv.org.
    • Handle: RePEc:arx:papers:1204.1442
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    References listed on IDEAS

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    1. Buckwar, Evelyn & Sickenberger, Thorsten, 2011. "A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(6), pages 1110-1127.
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    5. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Kamm, Kevin & Pagliarani, Stefano & Pascucci, Andrea, 2023. "Numerical solution of kinetic SPDEs via stochastic Magnus expansion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 189-208.
    2. Ben Hambly & Nikolaos Kolliopoulos, 2020. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Finance and Stochastics, Springer, vol. 24(3), pages 757-794, July.
    3. Marco Di Francesco & Kevin Kamm, 2022. "CDO calibration via Magnus Expansion and Deep Learning," Papers 2212.12318, arXiv.org.
    4. Ben Hambly & Nikolaos Kolliopoulos, 2018. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Papers 1811.08808, arXiv.org, revised Feb 2020.
    5. 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.
    6. Ben Hambly & Nikolaos Kolliopoulos, 2019. "Stochastic PDEs for large portfolios with general mean-reverting volatility processes," Papers 1906.05898, arXiv.org, revised Mar 2024.
    7. A. K. Bahl & O. Baltzer & A. Rau-Chaplin & B. Varghese & A. Whiteway, 2013. "Achieving Speedup in Aggregate Risk Analysis using Multiple GPUs," Papers 1308.2572, arXiv.org.
    8. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.

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