IDEAS home Printed from https://ideas.repec.org/p/cfi/fseres/cf256.html
   My bibliography  Save this paper

An Asymptotic Expansion with Push-Down of Malliavin Weights

Author

Listed:
  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Toshihiro Yamada

    (Mitsubishi UFJ Trust Investment Technology Institute Co.,Ltd. (MTEC))

Abstract

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2011. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CARF F-Series CARF-F-256, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    • Handle: RePEc:cfi:fseres:cf256
    as

    Download full text from publisher

    File URL: https://www.carf.e.u-tokyo.ac.jp/old/pdf/workingpaper/fseries/281.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method (Published in "Journal of Japan Statistical Society", Vol.35-2, 171-203, 2005. )," CARF F-Series CARF-F-030, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    3. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
    4. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2009. "Computation in an Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-621, CIRJE, Faculty of Economics, University of Tokyo.
    5. Akihiko Takahashi & Nakahiro Yoshida, 2004. "An Asymptotic Expansion Scheme for Optimal Investment Problems," Statistical Inference for Stochastic Processes, Springer, vol. 7(2), pages 153-188, May.
    6. Maria Siopacha & Josef Teichmann, 2010. "Weak and strong Taylor methods for numerical solutions of stochastic differential equations," Quantitative Finance, Taylor & Francis Journals, vol. 11(4), pages 517-528.
    7. Fabio Antonelli & Sergio Scarlatti, 2009. "Pricing options under stochastic volatility: a power series approach," Finance and Stochastics, Springer, vol. 13(2), pages 269-303, April.
    8. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    9. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2009. "Computation in an Asymptotic Expansion Method," CARF F-Series CARF-F-149, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    Full references (including those not matched with items on IDEAS)

    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. Akihiko Takahashi & Yoshihiro Yajima, 2011. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CIRJE F-Series CIRJE-F-824, CIRJE, Faculty of Economics, University of Tokyo.
    2. Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CARF F-Series CARF-F-194, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Apr 2011.
    3. Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CIRJE F-Series CIRJE-F-695, CIRJE, Faculty of Economics, University of Tokyo.
    4. Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2010. "Application Of A High-Order Asymptotic Expansion Scheme To Long-Term Currency Options," CARF F-Series CARF-F-225, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Akihiko Takahashi & Toshihiro Yamada, 2013. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -Application to Stochastic Volatility Model-," CARF F-Series CARF-F-324, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2014.
    6. Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2010. "Application of a High-Order Asymptotic Expansion Scheme to Long-Term Currency Options," CIRJE F-Series CIRJE-F-753, CIRJE, Faculty of Economics, University of Tokyo.
    7. 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.
    8. Akihiko Takahashi & Masashi Toda, 2013. "Note on an Extension of an Asymptotic Expansion Scheme," CARF F-Series CARF-F-312, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. 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.
    10. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2012. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-272, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    11. Akihiko Takahashi & Masashi Toda, 2012. "Note on an Extension of an Asymptotic Expansion Scheme," CIRJE F-Series CIRJE-F-860, CIRJE, Faculty of Economics, University of Tokyo.
    12. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    13. Akihiko Takahashi & Yukihiro Tsuzuki, 2013. "A New Improvement Scheme for Approximation Methods of Probability Density Functions," CIRJE F-Series CIRJE-F-874, CIRJE, Faculty of Economics, University of Tokyo.
    14. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-242, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jul 2011.
    15. Akihiko Takahashi & Masashi Toda, 2012. "Note on an Extension of an Asymptotic Expansion Scheme," CARF F-Series CARF-F-286, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Dec 2012.
    16. Akihiko Takahashi & Toshihiro Yamada, 2013. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -- Application to Stochastic Volatility Model --," CIRJE F-Series CIRJE-F-897, CIRJE, Faculty of Economics, University of Tokyo.
    17. Kenichiro Shiraya & Akihiko Takahashi & Akira Yamazaki, 2010. "Pricing Swaptions under the Libor Market Model of Interest Rates with Local-Stochastic Volatility Models," CARF F-Series CARF-F-214, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    18. Kohta Takehara & Akihiko Takahashi & Masashi Toda, 2010. "New Unified Computational Algorithm in a High-Order Asymptotic Expansion Scheme ( Forthcoming in "The Proceedings of KIER-TMU International Workshop on Financial Engineering 2009".)," CARF F-Series CARF-F-212, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    19. Akihiko Takahashi & Toshihiro Yamada, 2014. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -Application to Stochastic Volatility Model- (Revised version of CARF-F-324; Forthcoming in "Mathematics of Operations Research," CARF F-Series CARF-F-347, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2014.
    20. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-787, CIRJE, Faculty of Economics, University of Tokyo.

    More about this item

    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:cfi:fseres:cf256. 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: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/catokjp.html .

    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.