The aim of this paper is to study the problem of optimality of replicating strategies associated with pricing of American contingent claims in the Cox-Ross-Rubinstein model with proportional transaction costs. We show that a replication of the option is always possible. We give sufficient conditions for the existence of a replicating strategy which is optimal, and also show an example of an optimal replicating strategy that is not optimal in the global sense.

Hedging of the European option in a discrete time financial market with proportional transaction costs is considered. It is shown that for a certain class of options the set of portfolios which allow the seller to pay the claim of the buyer in quite a general discrete time market model is the same as the set of such portfolios under the assumption that the stock price movement is given by a suitable CRR model.

The shortfall risk minimization problem for the investor who hedges a contingent claim is studied. It is shown that in case the nonnegativity of the final wealth is not imposed, the optimal strategy in a finite market model is obtained by super-hedging a contingent claim connected with a martingale measure which is a solution of an auxiliary maximization problem.

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