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

Author

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  • Alexander Gairat
  • Vadim Shcherbakov

    (Royal Holloway University of London)

Abstract

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our 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, 2024. "Extreme ATM skew in a local volatility model with discontinuity: joint density approach," Finance and Stochastics, Springer, vol. 28(4), pages 1179-1202, October.
    • Handle: RePEc:spr:finsto:v:28:y:2024:i:4:d:10.1007_s00780-024-00545-1
    DOI: 10.1007/s00780-024-00545-1
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    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)

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    More about this item

    Keywords

    Local volatility model; Skew Brownian motion; Implied volatility; At-the-money skew;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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