Joint calibration to SPX and VIX options with signature-based models

Paper: arXiv 2301.13235

Authors: Christa Cuchiero, Guido Gazzani, Janka Möller, Sara Svaluto-Ferro

Abstract

We consider a stochastic volatility model where the dynamics of the volatility are described by a linear function of the (time extended) signature of a primary process which is supposed to be a polynomial diffusion. We obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial diffusion is again a polynomial diffusion. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. We also propose a Fourier pricing approach for both VIX and SPX options exploiting that the signature of the augmented primary process is an infinite dimensional affine process. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly sophisticated mathematical framework utilizing path signatures, polynomial diffusions, and infinite-dimensional affine processes to tackle the joint SPX-VIX calibration problem without relying on rough paths or jumps. While the theoretical derivation and joint calibration results are strong and novel, the empirical section focuses mainly on calibration quality rather than extensive out-of-sample backtesting or providing open-source code.

Research Flowchart

  flowchart TD
    A["Research Goal: Jointly calibrate SPX & VIX options without jumps or rough volatility"] --> B["Inputs: SPX & VIX market options data"]
    B --> C["Model Formulation: Signature of Polynomial Diffusion for volatility & augmented log-price"]
    C --> D["Fourier Pricing Framework: Exploit infinite-dimensional affine property of signatures"]
    D --> E["Efficient Calibration: Offline signature sampling + Online optimization"]
    E --> F["Key Outcome: Highly accurate joint calibration to SPX & VIX options"]