An investor with constant absolute risk aversion trades a risky asset with general Itô‐dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading‐order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.