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We develop a method to study the implied volatility for exotic options andvolatility derivatives with European payoffs such as VIX options. Our approach,based on Malliavin calculus techniques, allows us to describe the properties ofthe at-the-money implied volatility (ATMI) in terms of the Malliavinderivatives of the underlying process. More precisely, we study the short-timebehaviour of the ATMI level and skew. As an application, we describe theshort-term behavior of the ATMI of VIX and realized variance options in termsof the Hurst parameter of the model, and most importantly we describe the classof volatility processes that generate a positive skew for the VIX impliedvolatility. In addition, we find that our ATMI asymptotic formulae perform verywell even for large maturities. Several numerical examples are provided tosupport our theoretical results.

On smile properties of volatility derivatives and exotic products: understanding the VIX skew