Moments in Rough Bergomi and Boundary Attainment in Rough Heston

Abstract

We address two open questions in the rough volatility literature. First, we prove finite positive moments for the rough Bergomi price process, and for a wider class of Gaussian Volterra Bergomi models, in the whole subcritical range under negative correlation. More precisely, if \(ρ\in[-1,0)\), then \(\E[S_T^p]<\infty\) for every \(0<p<p_ρ\), where \(p_{-1}=\infty\) and \(p_ρ=(1-ρ^2)^{-1}\) for \(-1<ρ<0\). For the fractional rough Bergomi kernel, this gives the finite side of the sharp critical moment threshold, complementing the known explosion result above the threshold. Second, we prove that the rough Heston variance process, equivalently the scalar Volterra square-root process with fractional kernel \(K_α(t)=t^{α-1}/Γ(α)\) and \(α\in(1/2,1)\), has a positive atom at zero at every positive time. Consequently, zero is hit with positive probability before every positive time horizon. This rules out any Feller-type condition making the zero boundary inaccessible in the fractional rough Heston model.