How does a “delayed” barrier feature affect the price of a convertible bond?

This paper presents a Partial Differential Equation (PDE)–based framework for pricing convertible bonds with a soft call feature. Unlike “one-touch” callable convertible bonds, soft-call provisions are defined through an averaging mechanism over a moving window. To model this averaging feature rigorously, two auxiliary clock variables are introduced to track the effective “real time” in the soft-call contract, which leads to two four-dimensional PDE systems. These high-dimensional PDE systems are first reduced to three dimensions and then further simplified to two dimensions by exploiting the structure of the admissible domain and applying a coordinate rotation. After this dimensional reduction, an integral equation (IE) approach is employed to compute the price of convertible bonds with a soft call. The resulting framework allows us to examine the impact of the delayed barrier effect induced by the soft-call feature.