  Calibration of stochastic local volatility (SLV) models to their underlying
local volatility model is often performed by numerically solving a
two-dimensional non-linear forward Kolmogorov equation. We propose a novel
finite volume (FV) discretization in the numerical solution of general 1D and
2D forward Kolmogorov equations. The FV method does not require a
transformation of the PDE. This constitutes a main advantage in the calibration
of SLV models as the pertinent PDE coefficients are often nonsmooth. Moreover,
the FV discretization has the crucial property that the total numerical mass is
conserved. Applying the FV discretization in the calibration of SLV models
yields a non-linear system of ODEs. Numerical time stepping is performed by the
Hundsdorfer-Verwer ADI scheme to increase the computational efficiency. The
non-linearity in the system of ODEs is handled by introducing an inner
iteration. Ample numerical experiments are presented that illustrate the
effectiveness of the calibration procedure.
