The Joint S&P 500/VIX Smile Calibration Puzzle Solved

Abstract

Since VIX options started trading in 2006, many researchers have tried to build a model that jointly and exactly calibrates to the prices of S&P 500 (SPX) options, VIX futures and VIX options. So far the best attempts, which used parametric continuous-time jump-diffusion models on the SPX, only produced an approximate fit. In this article we solve this longstanding puzzle using a completely different approach: a nonparametric discrete-time model. Given a VIX future maturity T1, we build a joint probability measure on the SPX at T1, the VIX at T1, and the SPX at T2 = T1 + 30 days which is perfectly calibrated to the SPX smiles at T1 and T2, and the VIX future and VIX smile at T1. Our model satisfies the martingality constraint on the SPX as well as the requirement that the VIX at T1 is the implied volatility of the 30-day log-contract on the SPX. We prove by duality that the existence of such a model means that the SPX and VIX markets are jointly arbitrage-free. The joint calibration puzzle is cast as a dispersion-constrained martingale transport problem which is solved using (an extension of) the Sinkhorn algorithm, in the spirit of De March and Henry-Labordere (2019). The algorithm identifies joint SPX/VIX arbitrages should they arise. Our numerical experiments show that the algorithm performs very well in both low and high volatility regimes. Finally we explain how to handle the fact that the VIX future and SPX option monthly maturities do not perfectly coincide, and how to extend the two-maturity model to include all available monthly maturities.