Alòs, ElisaGarcía-Lorite, DavidMuguruza Gonzalez, Aitor2024-02-192024-02-192022Alòs E, García-Lorite D, Muguruza Gonzalez A. On smile properties of volatility derivatives: understanding the VIX skew. SIAM Journal on Financial Mathematics. 2022;13(1):35 p. DOI: 10.1137/19M12699811945-497Xhttp://hdl.handle.net/10230/59132We develop a method to study the implied volatility of exotic underlyings, with special focus on volatility derivatives such as VIX options. Remarkably, our approach is flexible enough to be applied to any underlying, subject to mild technical conditions. Our method, built upon Malliavin calculus techniques, allows to transform any such underlying into the Black--Scholes model with a particular type of stochastic volatility. This, in turn, allows us to describe the properties of the at-the-money implied volatility (ATMI) in terms of the Malliavin derivatives of the transformed underlying process. Concretely, we study the short-time behavior of the ATMI level and skew. As an application, we describe the short-term behavior of the ATMI of VIX and realize variance options in terms of the Hurst parameter of the model, and most importantly, we describe the class of volatility processes that generate a positive skew for the VIX implied volatility. Several numerical examples are provided to support our theoretical results.application/pdfeng© 2022 by SIAM. Unauthorized reproduction of this article is prohibited. First Published in SIAM Journal on Financial Mathematics in 13,1, published by the Society for Industrial and Applied Mathematics (SIAM).info:eu-repo/semantics/articlehttp://dx.doi.org/10.1137/19M1269981variance optionsVIXimplied volatilityMalliavin calculusstochastic volatility modelsrough volatilityfractional Brownian motioninfo:eu-repo/semantics/openAccess