The additive method for upper bounds for Bermudanoptions is rephrased in terms of buyer’s and seller’s prices. It isshown how to deduce Jamshidian’s upper bound result in a simplefashion from the additive method, including the case of possiblyzero final pay-off. Both methods are improved by ruling out exerciseat sub-optimal points. It is also shown that it is possible to usesub- Monte Carlo simulations to estimate the value of the hedgingportfolio at intermediate points in the Jamshidian method withoutjeopardizing its status as upper bound.