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

## TL;DR

This paper develops a comparison principle for rough and non-rough Heston models, enabling analysis of their volatility surfaces and providing tighter bounds on moment explosion times, with implications for implied volatility slopes.

## Contribution

It introduces a novel comparison principle based on non-linear Volterra integral equations, offering improved bounds and insights into the behavior of rough versus non-rough Heston models.

## Key findings

- Tighter upper bound for moment explosion time than previous work
- Comparison principle for implied volatility slopes between models
- Power-law increase of implied volatility slope ratio at small maturities

## Abstract

We present a number of related comparison results, which allow 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 (2018) and tighter for typical parameter values. The results can be directly transferred to a comparison principle for the asymptotic slope of implied volatility between rough and non-rough Heston models. This principle shows that the ratio of implied volatility slopes in the rough vs. the non-rough Heston model increases at least with power-law behavior for small maturities.