American Contingent Claims With Physical Delivery Under Small Proportional Transaction Costs

American options exercised by physical delivery of a portfolio of cash and underlying stock are considered in the binary tree model under small proportional transaction costs. Dynamic programming type recursive algorithms are developed for computing the ask and bid prices of such options, extending the Snell envelope construction. Representations of the ask and bid prices of American options with physical delivery in terms of maximax and, respectively, maximin martingale expectations of stopped option payoffs are also established in this setting.