论文标题
关于隐含波动率的谐波平均值表示
On the harmonic mean representation of the implied volatility
论文作者
论文摘要
众所周知，在短期成熟度限制中，隐含的波动率接近了局部波动性相对于日志的谐波平均值，请参见[Berestycki等人，渐近和局部波动率模型的校准，定量融资，2，2002]。本文致力于无互补模型的结果：实际上，无套套图隐含的波动率是任何固定成熟度的正函数的谐波平均值。我们研究了后一个功能，该功能与福川的可逆地图紧密相关，$ f_ {1/2} $ [Fukasawa，隐含波动性微笑的正常转换，数学金融，22，2012]及其与局部波动性表面的保留。事实证明，log-strike变换$ z = f_ {1/2}（k）$定义了一个新的坐标系，其中短期的隐含波动率接近局部波动率的算术算术（与谐波）。作为例证，我们考虑了SSVI参数化的情况：在这种情况下，我们从实现方差的选项中获得了波动率交换的明确公式。
It is well know that, in the short maturity limit, the implied volatility approaches the integral harmonic mean of the local volatility with respect to log-strike, see [Berestycki et al., Asymptotics and calibration of local volatility models, Quantitative Finance, 2, 2002]. This 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 $f_{1/2}$ [Fukasawa, The normalizing transformation of the implied volatility smile, Mathematical Finance, 22, 2012], 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. As an illustration, we consider the case of the SSVI parameterization: in this setting, we obtain an explicit formula for the volatility swap from options on realized variance.
