Title: On smile properties of volatility derivatives and exotic products: understanding the VIX skew,
Abstract: We develop a method to study the implied volatility for exotic options and
volatility derivatives with European payoffs such as VIX options. Our approach,
based on Malliavin calculus techniques, allows us to describe the properties of
the at-the-money implied volatility (ATMI) in terms of the Malliavin
derivatives of the underlying process. More precisely, we study the short-time
behaviour of the ATMI level and skew. As an application, we describe the
short-term behavior of the ATMI of VIX and realized 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. In addition, we find that our ATMI asymptotic formulae perform very
well even for large maturities. Several numerical examples are provided to
support our theoretical results.