Finite Difference Methods for Barrier Options

Abstract

Abstract This article describes the numerical solution of partial differential equations (PDEs) in the context of computational finance for the pricing of barrier options. Barrier options cease to exist (knockout barrier options) or only come into existence (knock‐in barrier options) when some observable market parameter crosses a designated level (the barrier). They have become ubiquitous in financial contracts for virtually all asset classes. A brief discussion of continuously monitored barrier option pricing using PDEs describes the use of coordinate transformations to obtain smooth, rapid convergence, highly desirable properties that are difficult to obtain using traditional lattice methods. In the majority of barrier option contracts, barrier monitoring frequency is at most daily, for example, at daily market close. For these options, pricing as if continuously monitored is almost always a gross approximation. Gradients of option value with respect to the underlying can become very strong near the barrier(s). The majority of the article is devoted to a more detailed discussion of PDE methods for the pricing of discretely monitored barrier options, including customization of the finite difference grid and time discretizations. Numerical examples illustrate how easy‐to‐implement grid structuring techniques can yield orders of magnitude reduction in finite difference truncation error and smooth, monotonic convergence.