Neural Joint S&P 500/VIX Smile Calibration

16 Pages Posted: 27 Dec 2022 Last revised: 30 Jan 2024

See all articles by Julien Guyon

Julien Guyon

Ecole des Ponts ParisTech

Scander Mustapha

Princeton University

Date Written: December 22, 2022


We calibrate neural stochastic differential equations jointly to S&P 500 smiles, VIX futures, and VIX smiles. Drifts and volatilities are modeled as neural networks. Minimizing a suitable loss allows us to fit market data for multiple S&P 500 and VIX maturities. A one-factor Markovian stochastic local volatility model is shown to fit both smiles and VIX futures within bid-ask spreads. The joint calibration actually makes it a pure path-dependent volatility model, confirming the findings in [Guyon, 2022, The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S&P 500 Skew].

Keywords: joint S&P 500/VIX smile calibration, neural stochastic differential equations, path-dependent volatility

JEL Classification: G13

Suggested Citation

Guyon, Julien and Mustapha, Scander, Neural Joint S&P 500/VIX Smile Calibration (December 22, 2022). Available at SSRN: or

Julien Guyon (Contact Author)

Ecole des Ponts ParisTech ( email )


Scander Mustapha

Princeton University ( email )

Joseph Henry House
Princeton, NJ 08542
United States

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