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.