Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets

Abstract

We conduct an empirical analysis of rough and classical stochastic volatility models to the SPX and VIX options markets. Our analysis focusses primarily on calibration quality and is split in two parts. In part one, we perform a historical calibration to SPX options over the years 2004–2019 of a selection of models that include the one-factor rough Bergomi and rough Heston models. In part two, we consider three calibration dates with low, typical, and high volatility, examine a wide selection of models, and calibrate to both SPX options as well as jointly to SPX and VIX options. The key results are as follows: The rough Bergomi and rough Heston models fail to create a term structure of smile effect that is sufficiently pronounced for SPX options. Moreover, we discover that short-expiry SPX smiles generally are more symmetric than long-expiry smiles, a feature we neither find that these models can reproduce. We propose an alternative volatility model driven by two Ornstein-Uhlenbeck processes that uses a non-standard transformation function. Calibrating it to SPX options, we obtain almost perfect fits, and calibrating it jointly to SPX and VIX options, we obtain very decent fits. This suggests, contrary to what one might be led to believe based on much of the existing literature, that the joint SPX-VIX calibration problem is largely solvable with classical two-factor volatility, all without roughness and jumps.