A comparison principle between rough and non-rough Heston models—with applications to the volatility surface

We present a number of related comparison results, which allow one to compare moment explosion times, moment generating functions and critical moments between rough and non-rough Heston models of stochastic volatility. All results are based on a comparison principle for certain non-linear Volterra integral equations. Our upper bound for the moment explosion time is different from the bound introduced by Gerhold, Gerstenecker and Pinter [Moment explosions in the rough Heston model. Decisions in Economics and Finance, 2019, 42, 575–608] and tighter for typical parameter values. The results can be directly transferred to a comparison principle for the asymptotic slope of implied variance between rough and non-rough Heston models. This principle shows that the ratio of implied variance slopes in the rough versus non-rough Heston model increases at least with power-law behavior for small maturities.