On Pricing Barrier Options

B oyle and Lau (hereafter BL) [1994] have illustrated how a naive application of the binomial option pricing algorithm can lead to significantly biased estimates in the prices of a variety of barrier, capped, and vulnerable options, even when the number of time steps is large. The source of the problem arises from the location of the barrier with respect to adjacent layers of nodes in the lattice. BL show that if the layers of the lattice are set up so that the barrier falls between layers of the lattice, the errors may be quite significant. To avoid these errors, they constrain the time partition so that the resulting lattice has layers that are as close as possible to the barrier. While this procedure reduces the size of errors, refining the partition size may not necessarily produce more precise results. Moreover, the BL procedure may be difficult to implement if the barriers are time-varying or if there are multiple barriers. This article provides a simple and highly efficient algorithm that can be used to price and hedge options that have single barriers that are either at constant levels or time-varying as well as contracts that are subject to multiple barriers. First, a lattice is constructed to pass through the barrier points exactly. Second, the stock price partition and the