  We solve a family of fractional Riccati differential equations with constant
(possibly complex) coefficients. These equations arise, e.g., in fractional
Heston stochastic volatility models, that have received great attention in the
recent financial literature thanks to their ability to reproduce a rough
volatility behavior. We first consider the case of a zero initial value
corresponding to the characteristic function of the log-price. Then we
investigate the case of a general starting value associated to a transform also
involving the volatility process. The solution to the fractional Riccati
equation takes the form of power series, whose convergence domain is typically
finite. This naturally suggests a hybrid numerical algorithm to explicitly
obtain the solution also beyond the convergence domain of the power series
representation. Our numerical tests show that the hybrid algorithm turns out to
be extremely fast and stable. When applied to option pricing, our method
largely outperforms the only available alternative in the literature, based on
the Adams method.
