  Long maturity options or a wide class of hybrid products are evaluated using
a local volatility type modelling for the asset price S(t) with a stochastic
interest rate r(t). The calibration of the local volatility function is usually
time-consuming because of the multi-dimensional nature of the problem. In this
paper, we develop a calibration technique based on a partial differential
equation (PDE) approach which allows an efficient implementation. The essential
idea is based on solving the derived forward equation satisfied by P(t; S;
r)Z(t; S; r), where P(t; S; r) represents the risk neutral probability density
of (S(t); r(t)) and Z(t; S; r) the projection of the stochastic discounting
factor in the state variables (S(t); r(t)). The solution provides effective and
sufficient information for the calibration and pricing. The PDE solver is
constructed by using ADI (Alternative Direction Implicit) method based on an
extension of the Peaceman-Rachford scheme. Furthermore, an efficient algorithm
to compute all the corrective terms in the local volatility function due to the
stochastic interest rates is proposed by using the PDE solutions and grid
points. Different numerical experiments are examined and compared to
demonstrate the results of our theoretical analysis.
