Mean Square Error for the Leland-Lott Hedging Strategy: Convex Pay-Off

Abstract

Leland's approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim VT using the classical Black Scholes formulae with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e. a sequence of models with the transaction costs coefficients kn and n is the number of the portfolio revision dates. The enlarged volatility, in general, depends on n except the case which was investigated in details by Lott to whom belongs the first rigorous result on convergence of the approximating portfolio value to the pay-off. In this paper we consider only the Lott case alpha= 1/2. We prove first, for an arbitrary pay-off VT = G(ST ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n^1/2 in L2 and find the first order term of asymptotics. We are working in the setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t_ni=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. We show that the sequence of approximate error converges in law to a random variable which is the terminal value of a component of two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.

Keywords: Black-Scholes formula, European option, transaction costs, Leland strategy, Lott strategy, approximate hedging, Martingale limit theorem, diffusion approximation

JEL Classification: G11, G13