  We consider the fractional Heston model originally proposed by Comte, Coutin
and Renault. Inspired by recent ground-breaking work on rough volatility, which
showed that models with volatility driven by fractional Brownian motion with
short memory allows for better calibration of the volatility surface and more
robust estimation of time series of historical volatility, we provide a
characterisation of the short- and long-maturity asymptotics of the implied
volatility smile. Our analysis reveals that the short-memory property precisely
provides a jump-type behaviour of the smile for short maturities, thereby
fixing the well-known standard inability of classical stochastic volatility
models to fit the short-end of the volatility smile.

Mathematical Finance