IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1009.4818.html
   My bibliography  Save this paper

Semi-Closed Form Cubature and Applications to Financial Diffusion Models

Author

Listed:
  • Christian Bayer
  • Peter Friz
  • Ronnie Loeffen

Abstract

Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka~[Adv.~Math.~Econ.~6, 69--83, 2004] and Lyons--Victoir~[Proc.~R.~Soc.\\Lond.~Ser.~A 460, 169--198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm~[Appl.~Math.~Fin.~15, 107--121, 2008], which corresponds to a concrete level $5$ cubature method, we study some parametric diffusion models motivated from financial applications, and exhibit structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies, by introducing a (model-dependent) variation of the Ninomiya-Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.

Suggested Citation

  • Christian Bayer & Peter Friz & Ronnie Loeffen, 2010. "Semi-Closed Form Cubature and Applications to Financial Diffusion Models," Papers 1009.4818, arXiv.org.
    • Handle: RePEc:arx:papers:1009.4818
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1009.4818
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Mariko Ninomiya & Syoiti Ninomiya, 2009. "A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method," Finance and Stochastics, Springer, vol. 13(3), pages 415-443, September.
    2. Syoiti Ninomiya & Nicolas Victoir, 2008. "Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 107-121.
    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. Eva Lütkebohmert & Lydienne Matchie, 2014. "Value-At-Risk Computations In Stochastic Volatility Models Using Second-Order Weak Approximation Schemes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 1-26.

    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. Denis Belomestny & Tigran Nagapetyan, 2014. "Multilevel path simulation for weak approximation schemes," Papers 1406.2581, arXiv.org, revised Oct 2014.
    2. Masahiro Nishiba, 2013. "Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 20(2), pages 147-182, May.
    3. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    4. Aur'elien Alfonsi & Edoardo Lombardo, 2022. "High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids," Papers 2209.13334, arXiv.org, revised Apr 2023.
    5. Kazuhiro Yoshikawa, 2015. "An Approximation Scheme for Diffusion Processes Based on an Antisymmetric Calculus over Wiener Space," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(2), pages 185-207, May.
    6. Akiyama, Naho & Yamada, Toshihiro, 2024. "A weak approximation for Bismut’s formula: An algorithmic differentiation method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 386-396.
    7. Riu Naito & Toshihiro Yamada, 2024. "Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance," Computational Economics, Springer;Society for Computational Economics, vol. 64(3), pages 1443-1461, September.
    8. Yusuke Morimoto & Makiko Sasada, 2015. "Algebraic Structure of Vector Fields in Financial Diffusion Models and its Applications," Papers 1510.02013, arXiv.org, revised Dec 2015.
    9. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    10. Shigeto Kusuoka & Mariko Ninomiya & Syoiti Ninomiya, 2012. "Application Of The Kusuoka Approximation To Barrier Options," CARF F-Series CARF-F-277, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    11. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.
    12. Mariko Ninomiya, 2011. "Sde Weak Approximation Library (Sde Wa) (Version 1.0)," CARF F-Series CARF-F-274, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    13. Mariko Ninomiya & Syoiti Ninomiya, 2009. "A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method," Finance and Stochastics, Springer, vol. 13(3), pages 415-443, September.
    14. Christian Bayer & Peter Friz & Ronnie Loeffen, 2012. "Semi-closed form cubature and applications to financial diffusion models," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 769-782, October.
    15. Kenichiro Shiraya & Akihiko Takahashi & Masashi Toda, 2010. "Pricing Barrier and Average Options under Stochastic Volatility Environment," CIRJE F-Series CIRJE-F-745, CIRJE, Faculty of Economics, University of Tokyo.
    16. Susana Alvarez Diez & Samuel Baixauli & Luis Eduardo Girón, 2019. "Valoración de opciones call asiáticas Promedio Aritmético usando Taylor Estocástico 1.5," Working Papers 44, Faculty of Economics and Management, Pontificia Universidad Javeriana Cali.
    17. Hocquet, Antoine & Vogler, Alexander, 2023. "An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 183-217.
    18. Christian Bayer & Peter K. Friz & Paul Gassiat & Joerg Martin & Benjamin Stemper, 2017. "A regularity structure for rough volatility," Papers 1710.07481, arXiv.org.
    19. Al Gerbi, A. & Jourdain, B. & Clément, E., 2018. "Asymptotics for the normalized error of the Ninomiya–Victoir scheme," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 1889-1928.
    20. Rey Clément, 2017. "Convergence in total variation distance of a third order scheme for one-dimensional diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 1-12, March.

    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:1009.4818. 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.