standard stochastic volatility models and, (ii) the term structure of the at-the-money volatility skew is approximated by a …:(i) has been explained by using fractional volatility model with Hurst index H>1/2, (ii) is proved to be satisfied by a {\it … rough} volatility model with H<1/2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in …

In recent years there has been a remarkable growth of volatility options. In particular, VIX options are among the most … actively trading contracts at CBOE. These options exhibit upward sloping volatility skew and the shape of the skew is largely … independent of the volatility level. To take into account these stylized facts, this article introduces a novel two …