Expanding the rough Heston model in $H$

We study the dependence of the fractional Riccati equation in the rough Heston model on the Hurst parameter $H$. For each expansion point $H_0\in(-1/2,1/2]$, we derive a Taylor expansion of the Riccati solution in $H$, whose coefficients are characterized recursively as solutions of linear Volterra equations with fractional-logarithmic kernels. We prove local uniform convergence of the resulting Taylor series and, in particular, analyticity of the fractional Riccati solution in the Hurst parameter. Through the affine transform formula, this yields approximations of the rough Heston characteristic function and Fourier prices. Numerically, once a reference solution at $H_0$ is available, the expansion coefficients can be computed recursively and evaluated for many nearby values of $H$. We implement the method around $H_0=1/2$, using the classical Heston solution, and around $H_0=0$, using a Pad\'e approximation. Experiments for European call options indicate that low expansion orders already provide accurate implied volatilities across a wide range of Hurst parameters, including the hyper-rough regime.