Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints

TL;DR

This paper introduces a unified Gaussian polynomial volatility model for joint SPX-VIX calibration, demonstrating that a conventional one-factor Markovian model can outperform more complex rough and non-rough models in fitting implied volatility surfaces.

Contribution

It develops a generic calibration method using functional quantization and neural networks, and identifies a simple Markovian model that outperforms complex models in market data fitting.

Findings

A Markovian model achieves excellent fit to SPX-VIX implied volatilities.

The proposed calibration method is fast and accurate across models.

The simple Markovian model outperforms rough and non-rough models in all market conditions.

Abstract

We consider the joint SPX-VIX calibration within a general class of Gaussian polynomial volatility models in which the volatility of the SPX is assumed to be a polynomial function of a Gaussian Volterra process defined as a stochastic convolution between a kernel and a Brownian motion. By performing joint calibration to daily SPX-VIX implied volatility surface data between 2012 and 2022, we compare the empirical performance of different kernels and their associated Markovian and non-Markovian models, such as rough and non-rough path-dependent volatility models. In order to ensure an efficient calibration and a fair comparison between the models, we develop a generic unified method in our class of models for fast and accurate pricing of SPX and VIX derivatives based on functional quantization and Neural Networks. For the first time, we identify a \textit{conventional one-factor Markovian continuous stochastic volatility model} that is able to achieve remarkable fits of the implied volatility surfaces of the SPX and VIX together with the term structure of VIX futures. What is even more remarkable is that our conventional one-factor Markovian continuous stochastic volatility model outperforms, in all market conditions, its rough and non-rough path-dependent counterparts with the same number of parameters.