IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1105.3359.html
   My bibliography  Save this paper

Asymptotic Expansion for the Normal Implied Volatility in Local Volatility Models

Author

Listed:
  • Viorel Costeanu
  • Dan Pirjol

Abstract

We study the dynamics of the normal implied volatility in a local volatility model, using a small-time expansion in powers of maturity T. At leading order in this expansion, the asymptotics of the normal implied volatility is similar, up to a different definition of the moneyness, to that of the log-normal volatility. This relation is preserved also to order O(T) in the small-time expansion, and differences with the log-normal case appear first at O(T^2). The results are illustrated on a few examples of local volatility models with analytical local volatility, finding generally good agreement with exact or numerical solutions. We point out that the asymptotic expansion can fail if applied naively for models with nonanalytical local volatility, for example which have discontinuous derivatives. Using perturbation theory methods, we show that the ATM normal implied volatility for such a model contains a term ~ \sqrt{T}, with a coefficient which is proportional with the jump of the derivative.

Suggested Citation

  • Viorel Costeanu & Dan Pirjol, 2011. "Asymptotic Expansion for the Normal Implied Volatility in Local Volatility Models," Papers 1105.3359, arXiv.org.
    • Handle: RePEc:arx:papers:1105.3359
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1105.3359
    File Function: Latest version
    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. Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    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. 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.
    2. V. M. Belyaev, 2023. "Local Volatility in Interest Rate Models," Papers 2301.13595, arXiv.org, revised Mar 2024.

    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. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    3. 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.
    4. Dan Pirjol & Lingjiong Zhu, 2017. "Short Maturity Asian Options for the CEV Model," Papers 1702.03382, arXiv.org.
    5. Stefano De Marco, 2020. "On the harmonic mean representation of the implied volatility," Papers 2007.03585, arXiv.org.
    6. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    7. Masaaki Fukasawa, 2022. "On asymptotically arbitrage-free approximations of the implied volatility," Papers 2201.02752, arXiv.org, revised Jan 2022.
    8. Wen Su, 2021. "Default Distances Based on the CEV-KMV Model," Papers 2107.10226, arXiv.org, revised May 2022.
    9. Aït-Sahalia, Yacine & Li, Chenxu & Li, Chen Xu, 2021. "Closed-form implied volatility surfaces for stochastic volatility models with jumps," Journal of Econometrics, Elsevier, vol. 222(1), pages 364-392.
    10. 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.
    11. Richard Jordan & Charles Tier, 2016. "Asymptotic Approximations For Pricing Derivatives Under Mean-Reverting Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-31, August.
    12. Jos'e E. Figueroa-L'opez & Yankeng Luo & Cheng Ouyang, 2011. "Small-time expansions for local jump-diffusion models with infinite jump activity," Papers 1108.3386, arXiv.org, revised Jul 2014.
    13. Dan Pirjol & Lingjiong Zhu, 2024. "Short-maturity asymptotics for option prices with interest rates effects," Papers 2402.14161, arXiv.org.
    14. 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.
    15. Brace, Alan & Fabbri, Giorgio & Goldys, Benjamin, 2007. "An Hilbert space approach for a class of arbitrage free implied volatilities models," MPRA Paper 6321, University Library of Munich, Germany.
    16. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    17. Christian Keller & Michael C. Tseng, 2023. "Arrow-Debreu Meets Kyle: Price Discovery for Derivatives," Papers 2302.13426, arXiv.org, revised Mar 2024.
    18. Dan Stefanica & Radoš Radoičić, 2016. "A sharp approximation for ATM-forward option prices and implied volatilites," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 1-24, March.
    19. Jan Obloj, 2007. "Fine-tune your smile: Correction to Hagan et al," Papers 0708.0998, arXiv.org, revised Mar 2008.
    20. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.

    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:1105.3359. 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.