论文标题
在粗糙波动下的局部波动率
Local volatility under rough volatility
论文作者
论文摘要
近年来已经获得了由粗糙波动率模型产生的隐含波动率的几个渐近结果（尤其是在小成熟度状态下），从而更好地理解了由粗糙波动率模型引起的波动性表面的形状，并支持其对S＆P500选项数据的校准能力。粗糙的波动率模型还通过随机波动率的所谓马尔可夫投影产生局部波动率。我们通过研究由一类粗糙的随机波动率模型产生的局部波动率表面的渐近行为来补充所隐含波动性的现有结果，并涵盖了粗糙的Bergomi模型。 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, Busca, and Florent, QF 2002], is replaced by a new rule: the ratio of the at-the-money implied and local volatility偏度趋向于常数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。
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, and supporting their calibration power to S&P500 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 the 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, Busca, and Florent, QF 2002], 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.
