A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility with Rough Market Price of Volatility Risk

In the option pricing literature, it is well known that:(i) the decrease in the smile amplitude is much slower than standard stochastic volatility models and, (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models and, while:(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 the implied volatility. Namely, we construct a novel fractional volatility model with rough market price of volatility risk and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle