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

Abstract

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