论文标题
对数修改的粗糙随机波动率模型
Log-modulated rough stochastic volatility models
论文作者
论文摘要
我们提出了一类新的粗糙随机波动率模型，该模型通过调节幂律内核来通过对数项来定义小数布朗运动（FBM），以便即使在消失的赫斯特指数$ h $的限制案例中，内核仍然保持正方形的可集成度。即使在$ h = 0 $的情况下，也是如此的对数编码的分数布朗尼运动（Log-fbm）也是连续的高斯过程。结果，可以在整个范围内分析所得的超高随机波动率模型，而无需进一步归一化。我们获得了$ \ log（1/t）^{ - p} t^{h-1/2} $的偏差渐近学，为$ t \ to 0 $，$ h \ ge 0 $，因此偏差不会以$ h \ to the n.t to $ h \ to 0 $。
We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index $H$. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for $H = 0$. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range $0 \le H < 1/2$ without the need of further normalization. We obtain skew asymptotics of the form $\log(1/T)^{-p} T^{H-1/2}$ as $T\to 0$, $H \ge 0$, so no flattening of the skew occurs as $H \to 0$.
