Semi-closed form prices of barrier options in the Hull-White model

TL;DR

This paper develops semi-closed form pricing formulas for barrier options within the Hull-White model, utilizing integral transform and heat potential methods, which outperform traditional finite difference approaches in efficiency and accuracy.

Contribution

It extends existing methods to infinite domains and introduces a semi-closed form solution that combines numerical Volterra equation solving with integral representation.

Findings

Method is more efficient than finite difference methods.

Provides better accuracy and stability in pricing barrier options.

Extends pricing techniques to infinite domain problems.

Abstract

In this paper we derive semi-closed form prices of barrier (perhaps, time-dependent) options for the Hull-White model, ie., where the underlying follows a time-dependent OU process with a mean-reverting drift. Our approach is similar to that in (Carr and Itkin, 2020) where the method of generalized integral transform is applied to pricing barrier options in the time-dependent OU model, but extends it to an infinite domain (which is an unsolved problem yet). Alternatively, we use the method of heat potentials for solving the same problems. By semi-closed solution we mean that first, we need to solve numerically a linear Volterra equation of the first kind, and then the option price is represented as a one-dimensional integral. Our analysis shows that computationally our method is more efficient than the backward and even forward finite difference methods (if one uses them to solve those problems), while providing better accuracy and stability.