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. However, for short positions, Boyle and Vorst have to impose three additional conditions. The authors' first aim in this paper is to remove Boyle and Vorst's conditions for the replication of short calls and puts. Boyle and Vorst's algorithm calculates the current holdings in stocks and bonds in terms of those at the following period. This is unlike the case of no transaction costs where the current cost of the option can be calculated directly from the costs at the following period. The authors' second aim is to show that even in the case of transactions costs the cost of replication can be directly calculated also. As a by-product, the authors are able to derive upper bounds for the cost of replication which are valid for long positions and also for short positions when two of Boyle and Vorst's additional conditions hold. The authors' third aim is to show that the time of computation using the backward recursion can be halved. This seems to to be a new observation, even in the case of no transactions costs.