Semi-analytical solutions for barrier and american options written on a time-dependent ornstein–uhlenbeck process

In this article, we develop semi-analytical solutions for the barrier (perhaps, time-dependent) and American options written on the underlying stock that follows a time-dependent Ornstein–Uhlenbeck process with a lognormal drift. Semi-analytical means that given the time-dependent interest rate, continuous dividend and volatility functions, one need to solve a linear (for the barrier option) or nonlinear (for the American option) Volterra equation of the second kind (or a Fredholm equation of the first kind). After that, the option prices in all cases are presented as one-dimensional integrals of combination of the preceding solutions and Jacobi theta functions. We also demonstrate that computationally our method is more efficient than the backward finite difference method traditionally used for solving these problems, and can be as efficient as the forward finite difference solver while providing better accuracy and stability.