Working in a binomial framework, Boyle and Vorst derived 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. However, even when a contingent claim has a unique replicating portfolio, there may exist super replicating portfolios of lower cost. Nevertheless, Bensaid, Lesne, Pages and Scheinkman gave conditions under which the cost of the replicating portfolio does not exceed the cost of any super replicating portfolio. These results were generalised by Stettner and Rutkowski to the case of asymmetric transcations costs. Palmer gave a further slight generalisation with what seemed to be a simpler proof. It is known from these results that no super replicating portfolio for long positions in calls and puts can have a lower cost than the replicating portfolio. However, even when a short call or put has a unique replicating portfolio, there may exist super replicating portfolios of lower cost when transactions costs are sufficiently large. Then a lower bound for the call or put price would be the negative of the least possible cost of such a super replicating portfolio. So it is important to be able to calculate this cost. Now the cost of the replicating portfolio can easily be calculated by backward recursion. However, as there are possibly infinitely many super replicating portfolios, it is not immediately obvious how the least possible cost of a super replicating portfolio can be efficiently calculated. The aim of this paper is to show how this cost can be calculated in the one-period case.contemplating priv Ơ