SIAM Journal on Financial Mathematics, Volume 12, Issue 2, Page 690-723, January 2021.
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.

中文翻译：

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论波动率掉期行权与零范纳隐含波动率的区别

SIAM Journal on Financial Mathematics，第 12 卷，第 2 期，第 690-723 页，2021
年1 月。在本文中，Malliavin 演算被用于得出波动率掉期行使价与波动率驱动的零瓦纳隐含波动率之间差异的精确公式由分数噪声。据我们所知，我们的结果是第一个推导出零瓦纳隐含波动率和波动率掉期执行之间严格关系的结果。特别是，我们将看到零瓦纳隐含波动率比平价隐含波动率更准确地近似于波动率掉期行使价。