Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated “1/2 skew rule” linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated “harmonic mean formula” of [Berestycki et al. (2002). Quantitative Finance, 2, 61–69], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant $1/(H+3/2)$ (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.

中文翻译：

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粗暴波动下的局部波动

近年来，已经获得了粗略波动率模型产生的隐含波动率的几个渐近结果（特别是在小期限制度中），这提供了对粗略波动率模型引起的波动率表面形状的更好理解，支持了其校准能力SP500 选项数据。粗糙波动率模型还通过所谓的随机波动率的马尔可夫投影生成局部波动率表面。我们通过研究由一类粗糙随机波动率模型（包括粗糙 Bergomi 模型）生成的局部波动率表面的渐近行为来补充隐含波动率的现有结果。尤其，我们观察到，著名的“1/2 偏斜规则”将隐含波动率的短期平价偏斜与局部波动率的短期平价偏斜联系起来，这是著名的“谐波”的结果。 [Berestycki 等人的平均公式”。（2002）。Quantitative Finance, 2, 61–69 ]，被一条新规则所取代：平值隐含波动率和局部波动率偏斜的比率趋于恒定$1/（H+3/2）$（与常数 1/2 相反），其中H是基础瞬时波动过程的规律性指数。