Martingale Optimal Transport and Martingale Schrödinger Bridges for Calibration of Stochastic Volatility Models

Motivated by recent developments in the calibration of stochastic volatility models (SVMs for short), we study continuous-time formulations of martingale optimal transport and martingale Schrödinger bridge problems. We establish duality formulas and also provide alternative proofs, via different techniques, of duality results previously established in the mathematical finance literature. Applications include calibration of SVMs to SPX options, as well as joint calibration to both SPX and VIX options.