Martingale Optimal Transport and Martingale Schrödinger Bridges for Calibration of Stochastic Volatility Models
Antonios Zitridis
TL;DR
This paper develops a rigorous, theory-driven framework for calibrating stochastic volatility models using continuous-time Martingale Optimal Transport (MOT) and Martingale Schrödinger Bridges. It derives duality formulas for MOT and for entropy-minimizing Schrödinger bridges under martingale constraints, leveraging Von Neumann minimax and viscosity solutions of Hamilton–Jacobi equations, with extensions to VIX-inspired dispersion constraints. The results connect calibration problems to explicit characterizations of optimal measures via Girsanov changes of measure and value-function PDEs, and establish existence results for minimizers under structured coefficient assumptions. The work advances the theoretical foundations for model calibration in financial mathematics and links mean-field game techniques to localization of martingale transport problems, with potential implications for joint SPX and VIX calibration without relying on numerical approximations.
Abstract
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.