IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-01502886.html
   My bibliography  Save this paper

Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options

Author

Listed:
  • Romain Bompis

    (Crédit Agricole - Crédit Agricole)

Abstract

In this work we derive new analytical weak approximations for arithmetic means of geometric Brownian motions using a scalar log-normal Proxy with an averaged volatility. The key features of the approach are to keep the martingale property for the approximations and to provide new integration by parts formulas for geometric Brownian motions. Besides, we also provide tight error bounds using Malliavin calculus, estimates depending on a suitable dispersion measure for the volatilities and on the maturity. As applications we give new price and implied volatility approximation formulas for basket call options. The numerical tests reveal the excellent accuracy of our results and comparison with the other known formulas of the literature show a valuable improvement.

Suggested Citation

  • Romain Bompis, 2017. "Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options," Working Papers hal-01502886, HAL.
    • Handle: RePEc:hal:wpaper:hal-01502886
    Note: View the original document on HAL open archive server: https://hal.science/hal-01502886
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01502886/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. H. Berestycki & J. Busca & I. Florent, 2002. "Asymptotics and calibration of local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 61-69.
    2. Romain Bompis & Emmanuel Gobet, 2012. "Asymptotic and non asymptotic approximations for option valuation," Post-Print hal-00720650, HAL.
    3. E. Benhamou & E. Gobet & M. Miri, 2009. "Smart expansion and fast calibration for jump diffusions," Finance and Stochastics, Springer, vol. 13(4), pages 563-589, September.
    4. Minqiang Li & Jieyun Zhou & Shi-Jie Deng, 2010. "Multi-asset spread option pricing and hedging," Quantitative Finance, Taylor & Francis Journals, vol. 10(3), pages 305-324.
    5. E. Benhamou & E. Gobet & M. Miri, 2010. "Expansion Formulas For European Options In A Local Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(04), pages 603-634.
    6. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    7. Valdo Durrleman, 2010. "From implied to spot volatilities," Finance and Stochastics, Springer, vol. 14(2), pages 157-177, April.
    8. Kenichiro Shiraya & Akihiko Takahashi, 2014. "Pricing Multiasset Cross‐Currency Options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 34(1), pages 1-19, January.
    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. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion formulas for European quanto options in a local volatility FX-LIBOR model," Papers 1801.01205, arXiv.org, revised Apr 2018.
    2. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion Formulas For European Quanto Options In A Local Volatility Fx-Libor Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-43, March.
    3. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    4. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    5. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    6. Cai, Ning & Li, Chenxu & Shi, Chao, 2021. "Pricing discretely monitored barrier options: When Malliavin calculus expansions meet Hilbert transforms," Journal of Economic Dynamics and Control, Elsevier, vol. 127(C).
    7. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    8. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
    9. Emmanuel Gobet & Ali Suleiman, 2013. "New approximations in local volatility models," Post-Print hal-00523369, HAL.
    10. Alev{s} v{C}ern'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy Models and the Time Step Equivalent of Jumps," Papers 1309.7833, arXiv.org, revised Jul 2017.
    11. Pierre Etoré & Emmanuel Gobet, 2012. "Stochastic expansion for the pricing of call options with discrete dividends," Post-Print hal-00507787, HAL.
    12. Stefan Gerhold & Max Kleinert & Piet Porkert & Mykhaylo Shkolnikov, 2012. "Small time central limit theorems for semimartingales with applications," Papers 1208.4282, arXiv.org.
    13. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    14. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    15. Elisa Alòs & Yan Yang, 2014. "A closed-form option pricing approximation formula for a fractional Heston model," Economics Working Papers 1446, Department of Economics and Business, Universitat Pompeu Fabra.
    16. Lingjiong Zhu, 2015. "Options with Extreme Strikes," Risks, MDPI, vol. 3(3), pages 1-16, July.
    17. Amel Bentata & Rama Cont, 2012. "Short-time asymptotics for marginal distributions of semimartingales," Papers 1202.1302, arXiv.org.
    18. Elisa Alòs, 2012. "A decomposition formula for option prices in the Heston model and applications to option pricing approximation," Finance and Stochastics, Springer, vol. 16(3), pages 403-422, July.
    19. Amel Bentata & Rama Cont, 2012. "Short-time asymptotics for marginal distributions of semimartingales," Working Papers hal-00667112, HAL.
    20. Romain Bompis & Emmanuel Gobet, 2012. "Asymptotic and non asymptotic approximations for option valuation," Post-Print hal-00720650, HAL.

    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:hal:wpaper:hal-01502886. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.