  This paper deals with the exact calibration of semidiscretized stochastic
local volatility (SLV) models to their underlying semidiscretized local
volatility (LV) models. Under an SLV model, it is common to approximate the
fair value of European-style options by semidiscretizing the backward
Kolmogorov equation using finite differences. In the present paper we introduce
an adjoint semidiscretization of the corresponding forward Kolmogorov equation.
This adjoint semidiscretization is used to obtain an expression for the
leverage function in the pertinent SLV model such that the approximated fair
values defined by the LV and SLV models are identical for non-path-dependent
European-style options. In order to employ this expression, a large non-linear
system of ODEs needs to be solved. The actual numerical calibration is
performed by combining ADI time stepping with an inner iteration to handle the
non-linearity. Ample numerical experiments are presented that illustrate the
effectiveness of the calibration procedure.
