Consistent price systems and face-lifting pricing under transaction costs

Abstract

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion.

Using the constructed price systems, we show, under very general assumptions, the following “face-lifting” result: the asymptotic superreplication price of a European contingent claim g(S_T) equals ĝ(S₀), where ĝ is the concave envelope of g and S_t is the price of the asset at time t. This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.