On the Curvature of the Bachelier Implied Volatility

Elisa Alòs () and David García-Lorite
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Elisa Alòs: Departament d’Economia i Empresa, Universitat Pompeu Fabra and Barcelona SE, c/Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain
David García-Lorite: CaixaBank, Quantitative Analyst Team, Plaza de Castilla, 3, 28046 Madrid, Spain

Risks, 2025, vol. 13, issue 2, 1-19

Abstract: Our aim in this paper is to analytically compute the at-the-money second derivative of the Bachelier implied volatility curve as a function of the strike price for correlated stochastic volatility models. We also obtain an expression for the short-term limit of this second derivative in terms of the first and second Malliavin derivatives of the volatility process and the correlation parameter. Our analysis does not need the volatility to be Markovian and can be applied to the case of fractional volatility models, both with H < 1 / 2 and H > 1 / 2 . More precisely, we start our analysis with an adequate decomposition formula of the curvature as the curvature in the uncorrelated case (where the Brownian motions describing asset price and volatility dynamics are uncorrelated) plus a term due to the correlation. Then, we compute the curvature in the uncorrelated case via Malliavin calculus. Finally, we add the corresponding correlation correction and we take limits as the time to maturity tends to zero. The presented results can be an interesting tool in financial modeling and in the computation of the corresponding Greeks. Moreover, they allow us to obtain general formulas that can be applied to a wide class of models. Finally, they provide us with a precise interpretation of the impact of the Hurst parameter H on this curvature.

Keywords: Malliavin calculus; Bachelier implied volatility; implied volatility curvature; fractional Brownian motion (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2025
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