Pricing Complex Barrier Options under General Diffusion Processes

Abstract

Many derivations have payoffs that depend on whether a prespecified price “barrier” is hit prior to option maturity. Standard “in” and “out” options are obvious examples, but more complex contingencies, like a debt coverage ratio covenant in a bond indenture, may also be modeled as barrier problems. Such options present serious challenges to numerical valuation techniques, because accuracy depends heavily on where the barrier falls relative to the nodes in a discrete numerical approximation lattice. For good performance, the barrier should coincide with a set of nodes. Standard models typically perform well only for a single barrier of a particular shape, such as an “outstrike” at a constant price. Non-constant volatility for the underlying asset is an insuperable problem, leading to very slow convergence in most models. Here, Tian shows how to restructure a trinomial lattice to allow time-varying volatility, while maintaining the ability to match both an upper and a lower barrier of general form. The result is an accurate numerical approximation that converges very rapidly.