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 optimise our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimisation problem, for which we provide a dual formulation. We then solve numerically the dual problem, which involves a fully nonlinear Hamilton--Jacobi--Bellman equation. The method is tested by calibrating a Heston-like LSV model with simulated data.
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