  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.
