American contingent claims under small proportional transaction costs

American options are considered in the binary tree model under small proportional transaction costs. Dynamic programming type algorithms, which extend the Snell envelope construction, are developed for computing the ask and bid prices (also known as the upper and lower hedging prices) of such options together with the corresponding optimal hedging strategies for the writer and for the seller of the option. Representations of the ask and bid prices of American options in terms risk-neutral expectations of stopped option payoffs are also established in this setting.