Mean square error for the Leland–Lott hedging strategy: convex pay-offs

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 formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients kn=k0n−α, where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility \(\widehat{\sigma}_{n}\) in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value \(V^{n}_{T}\) to the pay-off VT. In this paper, we consider only the Lott case α=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 the asymptotics. We are working in a setting with non-uniform revision intervals and 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. We show that the sequence \(n^{1/2}(V_{T}^{n}-V_{T})\) converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.