Mean Square Error for the Leland–Lott Hedging Strategy

Abstract

AbstractThe Leland strategy of approximate hedging of the call-option under proportional transaction costs prescribes to use, at equidistant instants of portfolio revisions, the classical Black–Scholes formula but with a suitably enlarged volatility. An appropriate mathematical framework is a scheme of series, i.e. a sequence of models Mn with the transaction costs coefficients kn depending on n, the number of the revision intervals. The enlarged volatility $\widehat{\sigma}_n$, in general, also depends on n. Lott investigated in detail the particular case where the transaction costs coefficients decrease as n-1/2 and where the Leland formula yields $\widehat{\sigma}_n$ not depending on n. He proved that the terminal value of the portfolio converges in probability to the pay-off. In the present note we show that it converges also in L2 and find the first order term of asymptotics for the mean square error. The considered setting covers the case of non-uniform revision intervals. We establish the asymptotic expansion when the revision dates are $t_i^n = g(i/n)$ where the strictly increasing scale function g : [0, 1] → [0, 1] and its inverse f are continuous with their first and second derivatives on the whole interval or g(t) = 1 - (1 - t)β, β ≥ 1.