IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2305.10849.html
   My bibliography  Save this paper

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

Author

Listed:
  • Alexander Gairat
  • Vadim Shcherbakov

Abstract

This paper concerns a local volatility model in which volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold value. The model is known, and a number of results have been obtained for it. In particular, a power law behaviour of the implied volatility skew has been established in the case when the threshold is taken at the money. This result as well as some others have been obtained by techniques based on the Laplace transform. The purpose of this paper is to demonstrate how to obtain similar results by another method. The proposed alternative approach is based on the natural relationship of the model with Skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

Suggested Citation

  • Alexander Gairat & Vadim Shcherbakov, 2023. "Extreme ATM skew in a local volatility model with discontinuity: joint density approach," Papers 2305.10849, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2305.10849
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2305.10849
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    2. Alexander Gairat & Vadim Shcherbakov, 2017. "Density Of Skew Brownian Motion And Its Functionals With Application In Finance," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 1069-1088, October.
    3. Gairat, Alexander & Shcherbakov, Vadim, 2022. "Skew Brownian motion with dry friction: Joint density approach," Statistics & Probability Letters, Elsevier, vol. 187(C).
    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. Guangli Xu & Xingchun Wang, 2021. "On the Transition Density and First Hitting Time Distributions of the Doubly Skewed CIR Process," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 735-752, September.
    2. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    3. Andrey Itkin & Alexander Lipton & Dmitry Muravey, 2021. "Multilayer heat equations and their solutions via oscillating integral transforms," Papers 2112.00949, arXiv.org, revised Dec 2021.
    4. 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.
    5. Li, Dan & Liu, Lixin & Xu, Guangli, 2023. "Psychological barriers and option pricing in a local volatility model," The North American Journal of Economics and Finance, Elsevier, vol. 64(C).
    6. Che Guo & Xingchun Wang, 2022. "Pricing vulnerable options under correlated skew Brownian motions," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(5), pages 852-867, May.
    7. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    8. Akahori, Jirô & Fan, Jie Yen & Imamura, Yuri, 2023. "On the convergence order of a binary tree approximation of symmetrized diffusion processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 263-277.
    9. Manuel L. Esquível & Nadezhda P. Krasii & Pedro P. Mota & Victoria V. Shamraeva, 2023. "Coupled Price–Volume Equity Models with Auto-Induced Regime Switching," Risks, MDPI, vol. 11(11), pages 1-20, November.
    10. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    11. Yizhou Bai & Zhiyu Guo, 2019. "An Empirical Investigation to the “Skew” Phenomenon in Stock Index Markets: Evidence from the Nikkei 225 and Others," Sustainability, MDPI, vol. 11(24), pages 1-17, December.
    12. Pasricha, Puneet & He, Xin-Jiang, 2022. "Skew-Brownian motion and pricing European exchange options," International Review of Financial Analysis, Elsevier, vol. 82(C).
    13. Endre Csáki & Miklós Csörgő & Antónia Földes & Pál Révész, 2019. "Limit Theorems for Local and Occupation Times of Random Walks and Brownian Motion on a Spider," Journal of Theoretical Probability, Springer, vol. 32(1), pages 330-352, March.
    14. Kolb, Aaron M., 2019. "Strategic real options," Journal of Economic Theory, Elsevier, vol. 183(C), pages 344-383.
    15. Gairat, Alexander & Shcherbakov, Vadim, 2022. "Skew Brownian motion with dry friction: Joint density approach," Statistics & Probability Letters, Elsevier, vol. 187(C).
    16. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    17. Lou, Shuwen, 2023. "On transition density functions of skew Brownian motions with two-valued drift," Statistics & Probability Letters, Elsevier, vol. 193(C).
    18. Masaaki Fukasawa, 2020. "Volatility has to be rough," Papers 2002.09215, arXiv.org.

    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:2305.10849. 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.