Multi-step reflection principle and barrier options

Abstract

This paper examines a class of barrier options, multi-step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression for option prices of this type under the Black-Scholes model by deriving the multi-step reflection principle, that is, the multi-step boundary-crossing probability of Brownian motion. Multi-step barrier options are not only useful in that they can handle barriers of different levels and time steps but can also approximate options with arbitrary barriers. Moreover, they can be applied to pricing barrier options under jump-diffusion models.