Very Fast Algorithms for Barrier Option Pricing and the Ballot Problem

Combinatorial methods prove extremely useful towards designing blazingly fast yet simple algorithms for pricing European-style barrier options. Closed-form formulae to standard European-style barrier options can then be easily derived. Combinatorial formulae under the trinomial model are also presented. The common practice in the literature compares algorithms based on their respective numbers of time steps towards convergence. We illustrate the pitfalls of this custom by evaluating the performance of our binomial model-based algorithm and the trinomial tree algorithm, whose superiority over the binomial model is widely accepted. Contrary to common beliefs, however, our algorithm emerges as a clear winner. In fact, the performance gap is two orders of magnitude. Also shattered is the myth that the binomial model converges extremely slowly when the current stock price is very close to the barrier.