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In this paper we consider a discrete-time finite sample space financial model with penalty costs on short selling positions. We start by presenting a necessary and sufficient condition for the non-existence of arbitrage opportunities. This reduces to the existence of a martingale measure for the case in which the penalties are zero. Next we consider the problem of contingent claim valuation. Our main result states that, under certain conditions, for every contingent there will be a seller price and a buyer price, with a perfect portfolio replication for each of them. Again when the penalty costs on short selling positions are zero, our conditions coincide with the traditional condition for the market to be complete. An explicit and constructive procedure for obtaining hedging strategies, not necessarily in the binomial framework, is presented.

Contingent claim valuation with penalty costs on short selling positions