Efficient and high accuracy pricing of barrier options under the CEV diffusion

Abstract

Binomial and trinomial lattices are popular techniques for pricing financial options. These methods work well for European and American options, but for barrier options, the need to place a tree node very close to a barrier brings difficulties in their implementation and a large number of time steps are usually required when the barrier is close to the current asset price. A finite difference implementation is simpler and we propose a fourth-order numerical scheme for continuously and discretely monitored barriers. We demonstrate the superior performance of our technique over existing procedures for the Black–Scholes model and we then price barriers under constant elasticity of variance (CEV) diffusion. Continuously monitored barriers under CEV admit an analytical solution but evaluation via this formula is not straightforward. Furthermore, discretely monitored barriers have to be priced numerically. Our main contribution is therefore a highly accurate and efficient numerical scheme for barrier options under CEV and we provide several numerical examples to illustrate the merit of the new technique.