On the Skew and Curvature of the Implied and Local Volatilities

Elisa Alòs, David García-Lorite and Makar Pravosud

Applied Mathematical Finance, 2023, vol. 30, issue 1, 47-67

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} $ 1H+3/2 rule (where H denotes the Hurst parameter of the volatility process) for rough volatilities (see F. Bourgey, S. De Marco, P. Friz, and P. Pigato. 2022. “Local Volatility under Rough Volatility.” arXiv:2204.02376v1 [q-fin.MF] https://doi.org/10.48550/arXiv.2204.02376.), that states that the short-time skew slope of the at-the-money implied volatility is $ \frac {1}{H+3/2} $ 1H+3/2 of 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 vice versa. Additionally, this relationship depends on H.

Date: 2023
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DOI: 10.1080/1350486X.2023.2261459

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