Fast Exact Joint S&P 500/VIX Smile Calibration in Discrete and Continuous Time

19 Pages Posted: 30 Dec 2022 Last revised: 6 Feb 2024

See all articles by Julien Guyon

Julien Guyon

Ecole Nationale des Ponts et Chaussées (ENPC)

Florian Bourgey

Bloomberg LP

Date Written: February 2, 2024


We introduce a novel discrete-time-continuous-time exact calibration method: we first build an S&P 500/VIX jointly calibrated discrete-time model that is later extended to continuous time by martingale interpolation. The benefit is that both steps can be made much faster than the known methods that directly calibrate a continuous-time model. We propose Newton-Sinkhorn and implied Newton algorithms that are much faster than the Sinkhorn algorithm that (Guyon, Risk, April 2020) used to build the first arbitrage-free model exactly consistent with S&P 500 and VIX market data. Using a (purely forward) Markov functional model, we then quickly build an arbitrage-free continuous-time extension of this discrete- time model. Additionally, new model-free bounds on S&P 500 options emphasize the value of the VIX smile information. Extensive numerical tests are conducted.

Keywords: joint S&P 500/VIX smile calibration, Newton-Sinkhorn, implied Newton, martingale interpolation, martingale optimal transport

JEL Classification: G13

Suggested Citation

Guyon, Julien and Bourgey, Florian, Fast Exact Joint S&P 500/VIX Smile Calibration in Discrete and Continuous Time (February 2, 2024). Available at SSRN: or

Julien Guyon (Contact Author)

Ecole Nationale des Ponts et Chaussées (ENPC) ( email )

28, rue des Saints-Peres
75343 Paris Cedex 07

Florian Bourgey

Bloomberg LP

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