Paper: arXiv 2212.08297

Authors: Eduardo Abi Jaber, Camille Illand, Shaun, Li

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 8.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a strong mathematical framework using Gaussian Volterra processes and functional quantization alongside deep neural networks to address the challenging joint SPX-VIX calibration problem. Its empirical evaluation spans a robust 10-year dataset (2012-2022) and yields a surprising, novel insight: a conventional one-factor Markovian model can outperform rough volatility models with equal parameterization.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br/>Joint SPX-VIX calibration & model comparison<br/><i>(Rough vs. Non-Rough vs. Markovian)</i>"] --> B["<b>Inputs & Data</b><br/>Daily SPX-VIX Implied Volatility Surfaces &<br/>VIX Futures Term Structure (2012–2022)"]
    
    B --> C["<b>Model Formulation</b><br/>Gaussian Polynomial Volatility Models<br/><i>(Volterra process with different kernels)</i>"]
    
    C --> D["<b>Pricing & Calibration Engine</b><br/>Functional Quantization + Deep Neural Networks<br/><i>(Fast, accurate joint pricing framework)</i>"]
    
    D --> E["<b>Empirical Evaluation</b><br/>Out-of-sample fit comparison across<br/>market regimes (2012–2022)"]
    
    E --> F["<b>Key Finding</b><br/>1-Factor Markovian Model outperforms<br/>rough & path-dependent models in all conditions"]