Global Multi-Maturity SPX-VIX Calibration Beyond Markovian Stitching

We develop a global martingale optimal transport framework for joint S&P 500 (SPX)-VIX smile calibration across multiple maturities, without the conditional independence that Markovian stitching imposes between months. We prove that exact local and global feasibility are equivalent, yet the stitched laws, which carry earlier months only through the current SPX level, can form a strict subset of the global ones, so adjacent smiles cannot identify the dependence across months. We show that relative entropy against the standard Markov reference selects the stitched law, so this dependence must be supplied deliberately. We supply it through a finite-memory reference, in which a latent autoregressive state drives the VIX levels while every VIX marginal stays exact, and calibrate the resulting law on a junction tree whose size grows linearly with the number of maturities. On market data with up to nine SPX maturities, a law with memory and a law without it fit the same quotes but price VIX-conditioned SPX Asian claims differently: on every calendar, memory makes a claim on the maximum VIX cheaper and a claim on the minimum VIX dearer. Varying the memory strength on one calendar moves these differences steadily from zero, so the vanilla quotes, which fix the law of each date, do not determine prices that depend on the joint law across dates.