Pricing European Barrier Options with Partial Differential Equations

Barrier options were first priced by Merton in 1973 using partial differential equation. In this work, we present a closed form formula for pricing European barrier option with a moving barrier that increases with time to expiration. We adopted a three-step approach which include; justifying that barrier options satisfy the Black-Scholes partial differential equation under certain conditions, partial differential equation transformation, and solution using Fourier Transform and method of images. We concluded that all barrier options satisfy the Black-Scholes partial differential equation under different domains, expiry conditions, and boundary conditions. And also that closed form solution for several versions of barrier option exists within the Black-Scholes framework and can be found using this approach.