We solve a family of fraction Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models when computing the characteristic function of the log-spot price, hence in a pricing perspective . 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 coefficients satisfy a convolution equation. We show that this solution has a positive convergence domain , which is typically finite. Our theoretical results naturally suggest a numerical algorithm to explicitly obtain the solution to quadratic ordinary differential equations of fractional type, that turns out to be quite promising in terms of computational performance. In particular, we introduce a hybrid numerical scheme, we test its precision by comparing our results with the (few) benchmark available in the literature, based on the Adams method , and we numerically study the convergence of our procedure .
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 .
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fraction
fractional
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when computing the characteristic function of the log-spot price, hence in a pricing perspective
, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility behavior
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coefficients satisfy a convolution equation. We show that this solution has a positive convergence domain , which
convergence domain
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Our theoretical results naturally suggest a
This naturally suggests a hybrid
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to quadratic ordinary differential equations of fractional type, that
also beyond the convergence domain of the power series representation. Our numerical tests show that the hybrid algorithm
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quite promising in terms of computational performance. In particular, we introduce a hybrid numerical scheme, we test its precision by comparing our results with the (few) benchmark available
extremely fast and stable. When applied to option pricing, our method largely outperforms the only available alternative
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, and we numerically study the convergence of our procedure