  In this paper we develop a semi-closed form solutions for the barrier
(perhaps, time-dependent) and American options written on the underlying stock
which follows a time-dependent OU process with a log-normal drift. This model
is equivalent to the familiar Hull-White model in FI, or a time dependent OU
model in FX. Semi-closed form means that given the time-dependent interest
rate, continuous dividend and volatility functions, one need to solve
numerically a linear (for the barrier option) or nonlinear (for the American
option) Fredholm equation of the first kind. After that the option prices in
all cases are presented as one-dimensional integrals of combination of the
above solutions and Jacobi theta functions. We also demonstrate that
computationally our method is more efficient than the backward finite
difference method used for solving these problems, and can also be as efficient
as the forward finite difference solver while providing better accuracy and
stability.
