Consistent price systems and face-lifting pricing under transaction costs

TL;DR

This paper demonstrates that in markets with transaction costs, continuous price processes with full support admit consistent price systems, leading to a face-lifting result where superreplication prices equal the concave envelope of the claim.

Contribution

It proves the existence of consistent price systems for a broad class of models with transaction costs and extends face-lifting results to processes with conditional full support.

Findings

Consistent price systems exist for continuous processes with full support.

Superreplication prices equal the concave envelope of the payoff.

Results apply to Markovian and non-Markovian models, including fractional Brownian motion.

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 $\hat{g}(S_0)$, where $\hat{g}$ 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.