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

Abstract: We develop and implement a non-parametric method for joint exact calibration of a local volatility model and a correlated stochastic short rate model using semimartingale optimal transport. The method relies on the duality results established in Joseph, Loeper, and Obloj, 2023 and jointly calibrates the whole equity-rate dynamics. It uses an iterative approach which starts with a parametric model and tries to stay close to it, until a perfect calibration is obtained. We demonstrate the performance of our approach on market data using European SPX options and European cap interest rate options. Finally, we compare the joint calibration approach with the sequential calibration, in which the short rate model is calibrated first and frozen.