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

Abstract: We develop a non-parametric, semimartingale optimal transport, calibration methodology for local volatility models with stochastic interest rate. The method finds a fully calibrated model which is the closest, in a way that can be defined by a general cost function, to a given reference model. We establish a general duality result which allows to solve the problem by optimising over solutions to a second order fully non-linear Hamilton-Jacobi-Bellman equation. 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. As an example, we apply the method to a sequential calibration problem, where a Vasicek model is already given for the interest rates and we seek to calibrate a stock price's local volatility model with volatility coefficient depending on time, the underlying and the short rate process, and the two processes driven by possibly correlated Brownian motions. The equity model is calibrated to any number of European options prices.