🤖 AI Summary
This study addresses the lack of efficient computational methods for capturing the dependence of the fractional Riccati equation on the Hurst parameter $H$ in the rough Heston model. The authors establish, for the first time, the analyticity of the solution with respect to $H$ and develop a unified Taylor expansion framework valid around any $H_0 \in (-1/2, 1/2]$. Expansion coefficients are obtained recursively by solving linear Volterra equations with fractional-logarithmic kernels. Combined with an affine transformation and Fourier-based techniques, this approach enables accurate approximation of the characteristic function and European call option prices. Numerical experiments demonstrate that low-order expansions around $H_0 = 1/2$ (classical Heston) or $H_0 = 0$ yield highly precise implied volatility surfaces across a broad range of $H$, including the super-rough regime.
📝 Abstract
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 Padé 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.