Consistent pricing of bivariate interest rate exotics via constrained Schrödinger optimal transport

We develop a modeling framework for pricing bivariate interest rate exotic derivatives that maintains consistency across three interconnected markets: CMS spread options and the two underlying CMS option markets that define the spread. Our approach also enables the computation of no-arbitrage bounds for exotic derivatives given observable market prices in the spread option and underlying CMS option markets. The method relies on solving the dual Lagrangian of a constrained version of the Shrödinger optimal transport problem and we demonstrate the practical applicability of our framework through concrete numerical examples that illustrate both the pricing methodology and the computation of no-arbitrage bounds. The approach offers a robust tool for pricing complex interest rate derivatives while ensuring consistency with liquid market instruments.