Limit Theorem on Option Replication Cost with Transaction Costs

Option replication in a discrete-time framework with transaction costs and its limit is discussed. First the notion of an efficient strategy is introduced, and then it is shown that an efficient strategy is the best strategy if it exists. It is also shown that the limit of the cost for option replication is given by a certain formula when the unit of time $\Delta t$ tends to zero and the transaction costs tends to zero on the order of $\sqrt{\Delta t}$.