The space of outcomes of semi-static trading strategies need not be   closed
  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. 
Beatrice Acciaio, Martin Larsson, Walter Schachermayer
q-fin.MF