IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-01573621.html
   My bibliography  Save this paper

Effective asymptotic analysis for finance

Author

Listed:
  • Cyril Grunspan

    (ESILV - École Supérieure d'Ingénierie Léonard de Vinci)

  • Joris van der Hoeven

    (LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

Abstract

It is known that an adaptation of Newton's method allows for the computation of functional inverses of formal power series. We show that it is possible to successfully use a similar algorithm in a fairly general analytical framework. This is well suited for functions that are highly tangent to identity and that can be expanded with respect to asymptotic scales of ‘‘exp-log functions''. We next apply our algorithm to various well-known functions coming from the world of quantitative finance. In particular, we deduce asymptotic expansions for the inverses of the Gaussian and the Black–Scholes functions.

Suggested Citation

  • Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.
    • Handle: RePEc:hal:journl:hal-01573621
    DOI: 10.1142/S0219024920500132
    Note: View the original document on HAL open archive server: https://hal.science/hal-01573621v3
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01573621v3/document
    Download Restriction: no
    File URL: https://libkey.io/10.1142/S0219024920500132?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Michael Roper & Marek Rutkowski, 2009. "On The Relationship Between The Call Price Surface And The Implied Volatility Surface Close To Expiry," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(04), pages 427-441.
    2. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    3. 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.
    4. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    5. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    6. Cyril Grunspan, 2011. "Asymptotic Expansions of the Lognormal Implied Volatility : A Model Free Approach," Papers 1112.1652, arXiv.org.
    7. Mark Craddock & Martino Grasselli, 2016. "Lie Symmetry Methods for Local Volatility Models," Research Paper Series 377, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    9. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
    10. Martin Forde & Antoine Jacquier, 2011. "Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 517-535, April.
    11. Cyril Grunspan, 2011. "A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach," Papers 1112.1782, arXiv.org.
    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. Cyril Grunspan & Joris van der Hoeven, 2017. "Effective asymptotic analysis for finance," Working Papers hal-01573621, HAL.
    2. Cyril Grunspan & Joris Van Der Hoeven, 2020. "Effective Asymptotics Analysis For Finance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(02), pages 1-23, March.
    3. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    4. Francesco Caravenna & Jacopo Corbetta, 2015. "The asymptotic smile of a multiscaling stochastic volatility model," Papers 1501.03387, arXiv.org, revised Jul 2017.
    5. Caravenna, Francesco & Corbetta, Jacopo, 2018. "The asymptotic smile of a multiscaling stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 1034-1071.
    6. Vimal Raval & Antoine Jacquier, 2021. "The Log Moment formula for implied volatility," Papers 2101.08145, arXiv.org.
    7. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Small-time asymptotics for Gaussian self-similar stochastic volatility models," Papers 1505.05256, arXiv.org, revised Mar 2016.
    8. Michael R. Tehranchi, 2020. "A Black–Scholes inequality: applications and generalisations," Finance and Stochastics, Springer, vol. 24(1), pages 1-38, January.
    9. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    10. Choi, Jaehyuk & Wu, Lixin, 2021. "The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-alpha-beta-rho (SABR) model," Journal of Economic Dynamics and Control, Elsevier, vol. 128(C).
    11. Francesco Caravenna & Jacopo Corbetta, 2014. "General smile asymptotics with bounded maturity," Papers 1411.1624, arXiv.org, revised Jul 2016.
    12. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    13. Dan Pirjol & Lingjiong Zhu, 2017. "Short Maturity Asian Options for the CEV Model," Papers 1702.03382, arXiv.org.
    14. Roza Galeeva & Ehud Ronn, 2022. "Oil futures volatility smiles in 2020: Why the bachelier smile is flatter," Review of Derivatives Research, Springer, vol. 25(2), pages 173-187, July.
    15. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    16. Jacquier, Antoine & Roome, Patrick, 2016. "Large-maturity regimes of the Heston forward smile," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1087-1123.
    17. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    18. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    19. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    20. Aleksandar Mijatovi'c & Peter Tankov, 2012. "A new look at short-term implied volatility in asset price models with jumps," Papers 1207.0843, arXiv.org, revised Jul 2012.

    More about this item

    Keywords

    exp-log function; BlackScholes formula A.M.S. subject classification: 68W30; 41A60; 91G80; 16A12; Hardy field; pricing; algorithm; asymptotic expansion; Black-Scholes formula; Asymptotic expansion;
    All these keywords.

    JEL classification:

    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:hal:journl:hal-01573621. 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.