  In an incomplete model, where under an appropriate num\'eraire, the stock
price process is driven by a sigma-bounded semimartingale, we investigate the
behavior of the expected utility maximization problem under small perturbations
of the num\'eraire. We establish a quadratic approximation of the value
function and a first-order expansion of the terminal wealth. Relying on a
description of the base return process in terms of its semimartingale
characteristics, we also construct wealth processes and nearly optimal
strategies that allow for matching the primal value function up to the second
order. We also link perturbations of the num\'eraire to distortions of the
finite-variation part and martingale part of the stock price return and
characterize the asymptotic expansions in terms of the risk-tolerance wealth
process.
