Summary: The paper addresses the problem of pricing American Contingent Claims (ACCs) under constraints on portfolio choice and a higher interest rate for borrowing than for lending. In this paper, the formulae of the upper hedging price \(h_{\text{up}}(K)\) and the lower hedging price \(h_{\text{low}}(K)\) of an ACC are derived by introducing a family of auxiliary frictionless financial markets. Furthermore, the arbitrage-free interval [\(h_{\text{low}}(K),h_{\text{up}}(K)]\) is identified, based on the principle of absence of arbitrage. In the end, for several concrete constraints on the portfolio, explicit computations or estimations of the upper hedging price and the lower hedging price are carried out in the case of American call-option.