Calibration of Local Volatility Models with Stochastic Interest Rates using Optimal Transport

We develop a non-parametric, optimal transport driven, calibration methodology for local volatility models with stochastic interest rate. The method finds a fully calibrated model which is the closest to a given reference model. We establish a general duality result which allows to solve the problem via optimising over solutions to a non-linear HJB equation. We then apply the method to a sequential calibration setup: we assume that an interest rate model is given and is calibrated to the observed term structure in the market. We then seek to calibrate a stock price local volatility model with volatility coefficient depending on time, the underlying and the short rate process, and driven by a Brownian motion which can be correlated with the randomness driving the rates process. The local volatility model is calibrated to a finite number of European options prices via a convex optimisation problem derived from the PDE formulation of semimartingale optimal transport. Our methodology is analogous to Guo, Loeper, and Wang, 2022 and Guo, Loeper, Obloj, et al., 2022a but features a novel element of solving for discounted densities, or sub-probability measures. We present numerical experiments and test the effectiveness of the proposed methodology.