Underlying assets for which options complete the market

This paper shows that investigations on the spanning power of options in spaces of integrable and continuously distributed payoffs can be conducted in the space of Lebesgue integrable claims on [0,1]. It is proved that there are infinite many underlying assets for which options span spaces of integrable claims. It is also shown that options on a single underlyer fail to complete the spaces of continuous contingent claims that are defined over a noncompact state-space.