Calibration of local-stochastic volatility models by optimal transport

In this paper, we study a semi-martingale optimal transport problem and its application to the calibration of local-stochastic volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimize our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimization problem, for which we provide a PDE formulation along with its dual counterpart. Then we solve numerically the dual problem, which involves a fully non-linear Hamilton–Jacobi–Bellman equation. The method is tested by calibrating a Heston-like LSV model with simulated data and foreign exchange market data.