  Semi-static trading strategies make frequent appearances in mathematical
finance, where dynamic trading in a liquid asset is combined with static
buy-and-hold positions in options on that asset. We show that the space of
outcomes of such strategies can have very poor closure properties when all
European options for a fixed date $T$ are available for static trading. This
causes problems for optimal investment, and stands in sharp contrast to the
purely dynamic case classically considered in mathematical finance.
