Semi-Analytical Pricing of Barrier Options in the Time-Dependent Heston Model

Abstract

This article develops the generalized integral transform (GIT) method for pricing barrier options in the time-dependent Heston model (also with a time-dependent barrier), whereby the option price is represented in a semi-analytical form as a two-dimensional (2D) integral. This integral depends on the as yet unknown function Φ(t, v), which is the gradient of the solution at the moving boundary S = L(t), and solves a linear mixed Volterra–Fredholm equation of the second kind, also derived in this article. Thus, the authors generalize the one-dimensional (1D) GIT method developed in Itkin, Lipton, and Muravey (2021) and the corresponding articles to the 2D case. In other words, we show that the GIT method can be extended to stochastic volatility models (two drivers with inhomogeneous correlation). As such, this 2D approach naturally inherits all advantages of the corresponding 1D methods—in particular, their speed and accuracy. This result is new and has various applications not only in finance, but also in physics. Numerical examples illustrate the high speed and accuracy of the method compared with the finite-difference approach.