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

TL;DR

This paper introduces a Malliavin calculus-based method to analyze the implied volatility surface of exotic options and volatility derivatives, focusing on short-term behavior and skew properties, especially for VIX options.

Contribution

It provides a novel analytical framework to describe the at-the-money implied volatility and skew for volatility derivatives, linking them to the underlying process characteristics.

Findings

Short-time ATMI level and skew depend on the Hurst parameter.

The model class generating positive VIX skew is characterized.

Asymptotic formulas remain accurate for large maturities.

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.