Pricing barrier options in discrete-time using lattice techniques is not a straightforward exercise. While the use of larger numbers of time steps may produce more accurate values for standard options, the value of a barrier option is extremely sensitive to the number of time steps used. Merely using more time steps will often produce erroneous option values while simultaneously increasing computation time. There is a consequential necessity for closed-form solutions for this class of derivative. This paper outlines how a reformulation of a simple random walk and an application of the reflection principle may be used to find a binomial coefficient that counts the number of ways a sample path will cross a constant barrier. In this way a general discrete closed-form solution, analogous to Cox, Ross and Rubinstein’s result, may be found for some barrier options. The case of a down-and-in call option is examined in detail. In addition, we prove the convergence of this discrete solution to its continuous-time counterpart.
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