Pricing Multi-asset Options with an External Barrier

An external barrier of an option contract is a stochastic variable which determines whether the option is knocked in or out when the value of the variable is above or below some predetermined level, but itself is not the price of an asset which underlies the option. In this paper, we present analytic formulation for the valuation of European options on one or multiple assets with single external barrier, where the barrier level can be exponential. As the domain of the problem becomes semi-infinite due to the presence of the external barrier, we employ the method of images to find the Green function of the governing differential equation. An efficient and accurate fractional step finite difference scheme is proposed for the numerical valuation of these barrier options.