Dispersion-constrained martingale Schrödinger problems and the exact joint S&P 500/VIX smile calibration puzzle

We solve for the first time a longstanding puzzle of quantitative finance that has often been described as the holy grail of volatility modelling: build a model that jointly and exactly calibrates to the prices of S&P 500 (SPX) options, VIX futures and VIX options. We use a nonparametric discrete-time approach: given a VIX future maturity \(T_{1}\), we consider the set \({\mathcal {P}}\) of all probability measures on the SPX at \(T_{1}\), the VIX at \(T_{1}\) and the SPX at \(T_{2} = T_{1} + 30\) days which are perfectly calibrated to the full SPX smiles at \(T_{1}\) and \(T_{2}\) and the full VIX smile at \(T_{1}\), and which also satisfy the martingality constraint on the SPX as well as the requirement that the VIX is the implied volatility of the 30-day log-contract on the SPX.

By casting the superreplication problem as a dispersion-constrained martingale optimal transport problem, we first establish a strong duality theorem and prove that the absence of joint SPX/VIX arbitrage is equivalent to \({\mathcal {P}}\neq \emptyset \). Should they arise, joint arbitrages are identified using classical linear programming. In their absence, we then provide a solution to the joint calibration puzzle by solving a dispersion-constrained martingale Schrödinger problem: we choose a reference measure and build the unique jointly calibrating model that minimises the relative entropy. We establish several duality results. The minimum-entropy jointly calibrating model is explicit in terms of the dual Schrödinger portfolio, i.e., the maximiser of the dual problem, should the latter exist, and is numerically computed using an extension of the Sinkhorn algorithm. Numerical experiments show that the algorithm performs very well in both low and high volatility regimes.