On the curvature of the bachelier implied volatility

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• dc.contributor.author Alòs, Elisa
• dc.contributor.author García-Lorite, David
• dc.date.accessioned 2025-05-20T13:19:36Z
• dc.date.available 2025-05-20T13:19:36Z
• dc.date.issued 2025
• dc.date.updated 2025-05-20T13:19:36Z
• dc.description.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 (Formula presented.) and (Formula presented.) 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.
• dc.description.sponsorship This research was funded by Ministerio de Ciencia e Innovación (Spain). Grant number PID2020-118339GB.
• dc.format.mimetype application/pdf
• dc.identifier.citation Alòs E, García-Lorite D. On the curvature of the bachelier implied volatility. Risks. 2025;13(2):27. DOI: 10.3390/risks13020027
• dc.identifier.doi http://dx.doi.org/10.3390/risks13020027
• dc.identifier.issn 2227-9091
• dc.identifier.uri http://hdl.handle.net/10230/70443
• dc.language.iso eng
• dc.publisher MDPI
• dc.relation.ispartof Risks. 2025;13(2):27
• dc.relation.projectID info:eu-repo/grantAgreement/ES/2PE/PID2020-118339GB
• dc.rights © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
• dc.rights.accessRights info:eu-repo/semantics/openAccess
• dc.rights.uri http://creativecommons.org/licenses/by/4.0/
• dc.subject.keyword Malliavin calculus
• dc.subject.keyword Bachelier implied volatility
• dc.subject.keyword Implied volatility curvature
• dc.subject.keyword Fractional Brownian motion
• dc.title On the curvature of the bachelier implied volatility
• dc.type info:eu-repo/semantics/article
• dc.type.version info:eu-repo/semantics/publishedVersion