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

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 result 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 more accurate approximation for the volatility swap strike than the at-the-money implied volatility.