The additive method for upper bounds for Bermudan options is rephrased in terms of buyer's and seller's prices. It is shown how to deduce Jamshidian's upper bound result in a simple fashion from the additive method, including the case of possibly zero final pay-off. Both methods are improved by ruling out exercise at sub-optimal points. It is also shown that it is possible to use sub-Monte Carlo simulations to estimate the value of the hedging portfolio at intermediate points in the Jamshidian method without jeopardizing its status as upper bound.
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