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Calibration of stochastic local volatility (SLV) models to their underlyinglocal volatility model is often performed by numerically solving atwo-dimensional non-linear forward Kolmogorov equation. We propose a novelfinite volume (FV) discretization in the numerical solution of general 1D and2D forward Kolmogorov equations. The FV method does not require atransformation of the PDE. This constitutes a main advantage in the calibrationof SLV models as the pertinent PDE coefficients are often nonsmooth. Moreover,the FV discretization has the crucial property that the total numerical mass isconserved. Applying the FV discretization in the calibration of SLV modelsyields a non-linear system of ODEs. Numerical time stepping is performed by theHundsdorfer-Verwer ADI scheme to increase the computational efficiency. Thenon-linearity in the system of ODEs is handled by introducing an inneriteration. Ample numerical experiments are presented that illustrate theeffectiveness of the calibration procedure.

A Finite Volume - Alternating Direction Implicit Approach for the Calibration of Stochastic Local Volatility Models