  The important application of semi-static hedging in financial markets
naturally leads to the notion of quasi self-dual processes. The focus of our
study is to give new characterizations of quasi self-duality for exponential
L\'evy processes such that the resulting market does not admit arbitrage
opportunities. We derive a set of equivalent conditions for the stochastic
logarithm of quasi self-dual martingale models and derive a further
characterization of these models not depending on the L\'evy-Khintchine
parametrization. Since for non-vanishing order parameter two martingale
properties have to be satisfied simultaneously, there is a non-trivial relation
between the order and shift parameter representing carrying costs in financial
applications. This leads to an equation containing an integral term which has
to be inverted in applications. We first discuss several important properties
of this equation and, for some well-known models, we derive a family of
closed-form inversion formulae leading to parameterizations of sets of possible
combinations in the corresponding parameter spaces of well-known L\'evy driven
models.
