We study the Leland model for hedging portfolios in the presence of a constant proportional transaction costs coefficient. The modified Leland's strategy recently defined by the second author, contrarily to the classical one, ensures the asymptotic replication of a large class of payoff. In this setting, we prove a limit theorem for the deviation between the real portfolio and the payoff. As Pergamenshchikov did in the framework of the usual Leland's strategy, we identify the rate of convergence and the associated limit distribution. This rate turns out to be improved using the modified strategy and non periodic revision dates.
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