SIAM Journal on Financial Mathematics, Volume 12, Issue 2, Page 551-565, January 2021.
It is well known that, in the short-maturity limit, the implied volatility approaches the integral harmonic mean of the local volatility with respect to log-strike; see [H. Berestycki, Busca, and Florent, Quant. Finance, 2 (2002), pp. 61--69]. This short paper is dedicated to a complementary model-free result: An arbitrage-free implied volatility in fact is the harmonic mean of a positive function for any fixed maturity. We investigate the latter function, which is tightly linked to Fukasawa's invertible map $f_{1/2}$ [M. Fukasawa, Math. Finance, 22 (2012), pp. 753--762], and its relation with the local volatility surface. It turns out that the log-strike transformation $z = f_{1/2}(k)$ defines a new coordinate system in which the short-dated implied volatility approaches the arithmetic (as opposed to harmonic) mean of the local volatility.

中文翻译：

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隐含波动率的调和平均表示

SIAM 金融数学杂志，第 12 卷，第 2 期，第 551-565 页，2021 年 1 月。
众所周知，在短期期限内，隐含波动率接近于对数行权的局部波动率的积分调和平均值；见 [H. Berestycki、Busca 和 Florent、Quant。金融，2 (2002)，第 61--69 页]。这篇简短的论文致力于补充无模型结果：无套利的隐含波动率实际上是任何固定期限的正函数的调和平均值。我们研究后一个函数，它与 Fukasawa 的可逆映射 $f_{1/2}$ [M. 深泽，数学。Finance, 22 (2012), pp. 753--762]，及其与局部波动率表面的关系。事实证明，对数罢工变换 $z = f_{1/2}(k)$ 定义了一个新的坐标系，其中短期隐含波动率接近局部波动率的算术（而不是调和）平均值。