IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1302.4676.html
   My bibliography  Save this paper

Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

Author

Listed:
  • Michael B. Giles
  • Kristian Debrabant
  • Andreas Ro{ss}ler

Abstract

The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

Suggested Citation

  • Michael B. Giles & Kristian Debrabant & Andreas Ro{ss}ler, 2013. "Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation," Papers 1302.4676, arXiv.org, revised Jun 2019.
    • Handle: RePEc:arx:papers:1302.4676
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1302.4676
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michael Giles & Desmond Higham & Xuerong Mao, 2009. "Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff," Finance and Stochastics, Springer, vol. 13(3), pages 403-413, September.
    2. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    3. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    4. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing," Papers 2209.00821, arXiv.org.
    2. Michael B. Giles & Abdul-Lateef Haji-Ali, 2022. "Multilevel Path Branching for Digital Options," Papers 2209.03017, arXiv.org, revised Jun 2024.
    3. Kahalé, Nabil, 2020. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," European Journal of Operational Research, Elsevier, vol. 287(2), pages 739-748.
    4. Mouna Ben Derouich & Ahmed Kebaier, 2022. "Interpolated Drift Implicit Euler MLMC Method for Barrier Option Pricing and application to CIR and CEV Models," Papers 2210.00779, arXiv.org, revised Sep 2024.
    5. Michael B. Giles & Abdul-Lateef Haji-Ali, 2019. "Sub-sampling and other considerations for efficient risk estimation in large portfolios," Papers 1912.05484, arXiv.org, revised Apr 2022.
    6. Nabil Kahale, 2018. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," Papers 1805.09427, arXiv.org, revised Sep 2018.
    7. Devang Sinha & Siddhartha P. Chakrabarty, 2024. "Multilevel Monte Carlo in Sample Average Approximation: Convergence, Complexity and Application," Papers 2407.18504, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.
    2. Christian Bayer & Chiheb Ben Hammouda & Raul Tempone, 2020. "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities," Papers 2003.05708, arXiv.org, revised Oct 2023.
    3. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Post-Print hal-01214840, HAL.
    4. Hideyuki Tanaka & Toshihiro Yamada, 2013. "Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method (Forthcoming in "International Journal of Theoretical and Applied Finance")," CARF F-Series CARF-F-333, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Ahmed Kebaier & J'er^ome Lelong, 2015. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Papers 1510.03590, arXiv.org, revised Jul 2017.
    6. Hideyuki Tanaka & Toshihiro Yamada, 2012. "Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method," Papers 1210.0670, arXiv.org, revised Nov 2013.
    7. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Post-Print hal-02430430, HAL.
    8. Hoel Håkon & von Schwerin Erik & Szepessy Anders & Tempone Raúl, 2014. "Implementation and analysis of an adaptive multilevel Monte Carlo algorithm," Monte Carlo Methods and Applications, De Gruyter, vol. 20(1), pages 1-41, March.
    9. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Working Papers hal-02430430, HAL.
    10. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c & Ger'onimo Uribe Bravo, 2018. "Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes," Papers 1810.11039, arXiv.org, revised Jun 2021.
    11. Jorge Gonz'alez C'azares & Aleksandar Mijatovi'c, 2020. "Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation," Papers 2011.06618, arXiv.org, revised Mar 2021.
    12. Michael B. Giles & Abdul-Lateef Haji-Ali & Jonathan Spence, 2023. "Efficient Risk Estimation for the Credit Valuation Adjustment," Papers 2301.05886, arXiv.org, revised May 2024.
    13. Daphné Giorgi & Vincent Lemaire & Gilles Pagès, 2020. "Weak Error for Nested Multilevel Monte Carlo," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1325-1348, September.
    14. Andrei Cozma & Christoph Reisinger, 2017. "Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models," Papers 1706.07375, arXiv.org, revised Oct 2018.
    15. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    16. Andrei Cozma & Christoph Reisinger, 2017. "Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process," Papers 1704.07321, arXiv.org, revised Oct 2018.
    17. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
    18. Bourgey Florian & De Marco Stefano & Gobet Emmanuel & Zhou Alexandre, 2020. "Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations," Monte Carlo Methods and Applications, De Gruyter, vol. 26(2), pages 131-161, June.
    19. Ben Alaya Mohamed & Kebaier Ahmed, 2014. "Multilevel Monte Carlo for Asian options and limit theorems," Monte Carlo Methods and Applications, De Gruyter, vol. 20(3), pages 181-194, September.
    20. Dirk Becherer & Plamen Turkedjiev, 2014. "Multilevel approximation of backward stochastic differential equations," Papers 1412.3140, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1302.4676. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.