Very fast algorithms for implied barriers and moving-barrier options pricing

Two closely related O(nlogn)-time tree algorithms under the Black–Scholes model are presented, where n denotes the tree’s number of time steps. The first finds the implied step barrier that matches the barrier-hitting probabilities exactly. In the constant-barrier case, the implied barrier is surprisingly accurate even for small ns; indeed, n=1 gives good results in typical situations. The second prices options with a time-dependent barrier (i.e., moving-barrier options). In practice, both algorithms are one to three orders faster than the standard algorithms even when n is moderate. As a consequence, large portfolios or datasets can finally be studied in a timely manner. Both algorithms can be easily tailored to handle barriers that are continuously monitored, discretely monitored, a mixture of both, or even when the model parameters are all time varying.