On the difference between the volatility swap strike and the zero vanna implied volatility

TL;DR

This paper uses Malliavin calculus to derive exact formulas for the difference between volatility swap strikes and zero vanna implied volatility in fractional noise models, establishing a novel relationship and showing improved approximation over ATMI.

Contribution

It provides the first rigorous derivation of the relationship between zero vanna implied volatility and volatility swap strike in fractional noise models.

Findings

Zero vanna implied volatility better approximates volatility swap strike than ATMI.

Exact formulas derived using Malliavin calculus.

First rigorous relationship established for fractional noise driven volatilities.

Abstract

In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our estimate is the first to derive the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility is a better approximation for the volatility swap strike than the ATMI.