Working in a binomial framework, Boyle and Vorst (1992) derive self-financing strategies perfectly replicating the final payoffs to long positions in European call and put options, assuming proportional transactions costs on trades in the stocks. The initial cost of such a strategy yields, by an arbitrage argument, an upper bound for the option price. A lower bound for the option price is obtained by replicating a short position. The authors' first aim in this paper is to clarify the conditions under which there is a unique replicating strategy for an arbitrary contingent claim. Following Stettner (1997) and Rutkowski (1998), the authors work in the framework of asymmetric proportional transactions costs, which includes not only the model of Boyle and Vorst but also that of Bensaid, Lesne, Pages and Scheinkman (1992). The authors first clarify the conditions in the case of a one-period model and then extend them to the multi-period case, yielding a result which extends slightly a result of Rutkowski. Even when a contingent claim has a unique replicating portfolio, there may exist super replicating portfolios of lower cost. However, Bensaid, Lesne, Pages and Scheinkman gave super replicating portfolio. These results were generalised by Stettner and Rutkowski to the case of asymmetric transactions costs. Here the authors provide a further slight generalisation and give what seems to be a simpler proof. There are two results here: one gives a class of contingent claims for which the cost of the replicating portfolio does not exceed the cost of any super replicating portfolio and the other gives conditions on the level of transactions costs such that for any contingent claim the cost of the replicating portfolio does not exceed the cost of any super replicating portfolio. Finally the authors conjecture a necessary and sufficient condition in terms of computable "terminal probabilities" for the latter to hold for a given contingent claim when the conditions on the transactions costs do not hold.