Pricing multi-step double barrier options by the efficient non-crossing probability

Abstract

This paper considers pricing multi-step double barrier options. The non-crossing probability for a multi-step double boundary is vital in valuing the options. We extend an explicit formula for the non-crossing probability using the solutions to the relevant Fokker–Planck equations. The derived formula not only provides a new look at the non-crossing probability but also outperforms the existing probability formula relying on multivariate normal distribution functions in terms of efficiency by avoiding multi-dimensional numerical integrals. We demonstrate the merits of the new expression of the non-crossing probability via numerical experiments and apply it to valuing multi-step double barrier options.