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

Abstract

We consider a stochastic volatility model where the dynamics of the volatility are described by linear functions of the (time extended) signature of a primary underlying process, which is supposed to be some multidimensional continuous semimartingale. Under the additional assumption that this primary process is of polynomial type, we obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial process is again a polynomial process. 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. 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.

From Constant to Rough: A Survey of Continuous Volatility Modeling

“…We conclude our presentation by saying that updating, refining, and improving models still seems an endless race. Right now, new techniques and modeling paradigms are being developed-in particular, we mention [201], which finds strong arguments for fully path-dependent volatility models as well as machine learning and signature methods such as in [202][203][204][205] (the model in the latter actually includes the Gaussian polynomial models, Equation (47), as a special case). Perhaps these novel tools will become the new classics and open new frontiers in understanding the financial market.…”

Signature-based validation of real-world economic scenarios

“…Cuchiero et al (2023b) extend the work of Perez Arribas and develop a new class of asset price models based on the signature of semimartingales allowing to approximate arbitrarily well classical models such as the SABR and the Heston models. Using this new modelling framework, Cuchiero et al (2023a) propose a method to solve the joint S&P 500/VIX calibration problem without adding jumps or rough volatility. Akyildirim et al (2022) introduce a signature-based machine learning algorithm to detect rare or unexpected items in a given data set of time series type.…”

The Quintic Ornstein-Uhlenbeck Volatility Model that Jointly Calibrates SPX & VIX Smiles

“…More recently, Fourier techniques have also found applications in Signature volatility models in Abi Jaber and Gérard [3] and Cuchiero, Gazzani, Möller, and Svaluto-Ferro [12], where the volatility process is modeled as a linear functional of the path-signature of semi-martingales (e.g. a Brownian motion).…”