This paper is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call option. It is well known that the difference between these two quantities converges to zero as the time to maturity decreases. In this paper, we make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of convergence is different in the correlated and uncorrelated cases, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter \(H\). Moreover, in the case \(H\geq 1/2\), we develop a model-free approximation formula for the volatility swap in terms of the ATMI and its skew.

中文翻译：

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从短期波动掉期中估计赫斯特参数：Malliavin演算方法

本文致力于研究波动掉期的公平行使价与欧洲看涨期权的平价隐含波动率（ATMI）之间的区别。众所周知，随着成熟时间的缩短，这两个数量之间的差收敛为零。在本文中，我们利用Malliavin演算方法来得出此差异的精确表达式。这种表示使我们能够确定，在相关和不相关的情况下，收敛的顺序是不同的，并且它取决于波动过程的Malliavin衍生物的行为。特别地，我们看到对于由分数布朗运动驱动的波动率，此顺序取决于相应的赫斯特参数 \（H \）。而且，在\（H \ geq 1/2 \）的情况下，我们根据ATMI及其偏度为波动率互换开发了无模型的近似公式。