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We solve a family of fractional Riccati differential equations with constant(possibly complex) coefficients. These equations arise, e.g., in fractionalHeston stochastic volatility models, that have received great attention in therecent financial literature thanks to their ability to reproduce a roughvolatility behavior. We first consider the case of a zero initial valuecorresponding to the characteristic function of the log-price. Then weinvestigate the case of a general starting value associated to a transform alsoinvolving the volatility process. The solution to the fractional Riccatiequation takes the form of power series, whose convergence domain is typicallyfinite. This naturally suggests a hybrid numerical algorithm to explicitlyobtain the solution also beyond the convergence domain of the power seriesrepresentation. Our numerical tests show that the hybrid algorithm turns out tobe extremely fast and stable. When applied to option pricing, our methodlargely outperforms the only available alternative in the literature, based onthe Adams method.

Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)