IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1310.3061.html
   My bibliography  Save this paper

Small-maturity asymptotics for the at-the-money implied volatility slope in L\'evy models

Author

Listed:
  • Stefan Gerhold
  • I. Cetin Gulum
  • Arpad Pinter

Abstract

We consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the behavior of the slope for infinite activity exponential L\'evy models including a Brownian component. As auxiliary results, we obtain asymptotic expansions of short maturity at-the-money digital call options, using Mellin transform asymptotics. Finally, we discuss when the at-the-money slope is consistent with the steepness of the smile wings, as given by Lee's moment formula.

Suggested Citation

  • Stefan Gerhold & I. Cetin Gulum & Arpad Pinter, 2013. "Small-maturity asymptotics for the at-the-money implied volatility slope in L\'evy models," Papers 1310.3061, arXiv.org, revised May 2016.
    • Handle: RePEc:arx:papers:1310.3061
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1310.3061
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Figueroa-López, José E. & Houdré, Christian, 2009. "Small-time expansions for the transition distributions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3862-3889, November.
    2. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    3. 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.
    4. 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.
    5. Valdo Durrleman, 2010. "From implied to spot volatilities," Finance and Stochastics, Springer, vol. 14(2), pages 157-177, April.
    6. L. De Leo & V. Vargas & S. Ciliberti & J. -P. Bouchaud, 2012. "We've walked a million miles for one of these smiles," Papers 1203.5703, arXiv.org, revised Apr 2012.
    7. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
    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. Jean-Philippe Aguilar, 2019. "The value of power-related options under spectrally negative L\'evy processes," Papers 1910.07971, arXiv.org, revised Jan 2021.
    2. Jean-Philippe Aguilar, 2021. "The value of power-related options under spectrally negative Lévy processes," Review of Derivatives Research, Springer, vol. 24(2), pages 173-196, July.
    3. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    4. Federico De Olivera & José Fajardo & Ernesto Mordecki, 2018. "Skewed Lévy Models And Implied Volatility Skew," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-16, March.
    5. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps," Finance and Stochastics, Springer, vol. 20(4), pages 973-1020, October.

    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. 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.
    2. 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.
    3. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    4. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps," Finance and Stochastics, Springer, vol. 20(4), pages 973-1020, October.
    5. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
    6. 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.
    7. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    8. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    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. Francesco Caravenna & Jacopo Corbetta, 2015. "The asymptotic smile of a multiscaling stochastic volatility model," Papers 1501.03387, arXiv.org, revised Jul 2017.
    11. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    12. 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.
    13. 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.
    14. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
    15. Cyril Grunspan & Joris van der Hoeven, 2017. "Effective asymptotic analysis for finance," Working Papers hal-01573621, HAL.
    16. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    17. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.
    18. Jos'e E. Figueroa-L'opez & Sveinn 'Olafsson, 2015. "Short-time asymptotics for the implied volatility skew under a stochastic volatility model with L\'evy jumps," Papers 1502.02595, arXiv.org, revised Dec 2015.
    19. Francesco Caravenna & Jacopo Corbetta, 2014. "General smile asymptotics with bounded maturity," Papers 1411.1624, arXiv.org, revised Jul 2016.
    20. 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.

    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:arx:papers:1310.3061. 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.