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On the skew and curvature of implied and local volatilities

Author

Listed:
  • Elisa Al`os
  • David Garc'ia-Lorite
  • Makar Pravosud

Abstract

In this paper, we study the relationship between the short-end of the local and the implied volatility surfaces. Our results, based on Malliavin calculus techniques, recover the recent $\frac{1}{H+3/2}$ rule (where $H$ denotes the Hurst parameter of the volatility process) for rough volatilitites (see Bourgey, De Marco, Friz, and Pigato (2022)), that states that the short-time skew slope of the at-the-money implied volatility is $\frac{1}{H+3/2}$ the corresponding slope for local volatilities. Moreover, we see that the at-the-money short-end curvature of the implied volatility can be written in terms of the short-end skew and curvature of the local volatility and viceversa, and that this relationship depends on $H$.

Suggested Citation

  • Elisa Al`os & David Garc'ia-Lorite & Makar Pravosud, 2022. "On the skew and curvature of implied and local volatilities," Papers 2205.11185, arXiv.org, revised Sep 2023.
    • Handle: RePEc:arx:papers:2205.11185
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    References listed on IDEAS

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    1. 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.
    2. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
    3. Roger W. Lee, 2001. "Implied And Local Volatilities Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 45-89.
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    Cited by:

    1. Enrico Dall'Acqua & Riccardo Longoni & Andrea Pallavicini, 2022. "Rough-Heston Local-Volatility Model," Papers 2206.09220, arXiv.org.

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