  The goal of this work is to study binary market models with transaction
costs, and to characterize their arbitrage opportunities. It has been already
shown that the absence of arbitrage is related to the existence of
\lambda-consistent price systems (\lambda-CPS), and, for this reason, we aim to
provide conditions under which such systems exist. More precisely, we give a
characterization for the smallest transaction cost \lambda_c (called "critical"
\lambda) starting from which one can construct a \lambda-consistent price
system. We also provide an expression for the set M(\lambda) of all probability
measures inducing \lambda-CPS. We show in particular that in the transition
phase "\lambda=\lambda_c" these sets are empty if and only if the frictionless
market admits arbitrage opportunities. As an application, we obtain an explicit
formula for \lambda_c depending only on the parameters of the model for
homogeneous and also for some semi-homogeneous binary markets.
