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

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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 show 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 has a higher rate of convergence than the at-the-money (ATM) implied volatility for zero correlation, and for non-zero correlation when the Hurst parameter H > 1/2.