Leland's Approximations for Concave Pay-Off Functions

In 1985, Leland suggested an approach to pricing 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 which terminal value approximates the pay-off h(ST). In subsequent studies, Lott (for α = 1/2), Kabanov and Safarian proved that for the call-option, i.e. for h(x) = (x-K)+, Leland's portfolios, indeed, approximate the pay-off if the transaction costs coefficients decreases as n-α for α ∈ ]0, 1/2] where n is the number of revisions. These results can be extended to the case of more general pay-off functions and non-uniform revision intervals [1]. Unfortunately, the terminal values of portfolios do not converge to the pay-off if h is not a convex function. In this paper, we show that we can slightly modify the Leland strategy such that the convergence holds for a large class of concave pay-off functions if α = 1/2.