Approximate Hedging of Contingent Claims Under Transaction Costs

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black–Scholes formula with a suitably enlarged volatility for a periodically revised portfolio whose terminal value approximates the pay-off of the call option. In subsequent studies, Lott, Kabanov and Safarian, and Gamys and Kabanov provided a rigorous mathematical analysis and established that the hedging portfolio approximates this pay-off in the case where the transaction costs decrease to zero as the number of revisions tends to infinity. The arguments used heavily the explicit expressions given by the Black–Scholes formula leaving open the problem whether the Leland approach holds for more general options and other types of price processes. In this paper we show that for a large class of the pay-off functions Leland's method can be successfully applied. On the other hand, if the pay-off function h(x) is not convex, then this method does not work