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

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  • 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, 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.

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  • 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 2024.
    • Handle: RePEc:arx:papers:2305.10849
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    References listed on IDEAS

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    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).
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