  It has been recently shown that rough volatility models, where the volatility
is driven by a fractional Brownian motion with small Hurst parameter, provide
very relevant dynamics in order to reproduce the behavior of both historical
and implied volatilities. However, due to the non-Markovian nature of the
fractional Brownian motion, they raise new issues when it comes to derivatives
pricing. Using an original link between nearly unstable Hawkes processes and
fractional volatility models, we compute the characteristic function of the
log-price in rough Heston models. In the classical Heston model, the
characteristic function is expressed in terms of the solution of a Riccati
equation. Here we show that rough Heston models exhibit quite a similar
structure, the Riccati equation being replaced by a fractional Riccati
equation.
