Smallest order closed sublattices and option spanning

Let $Y$ be a sublattice of a vector lattice $X$. We consider the problem of identifying the smallest order closed sublattice of $X$ containing $Y$. It is known that the analogy with topological closure fails. Let $\overline{Y}^o$ be the order closure of $Y$ consisting of all order limits of nets of elements from $Y$. Then $\overline{Y}^o$ need not be order closed. We show that in many cases the smallest order closed sublattice containing $Y$ is in fact the second order closure $\overline{\overline{Y}^o}^o$. Moreover, if $X$ is a $\sigma$-order complete Banach lattice, then the condition that $\overline{Y}^o$ is order closed for every sublattice $Y$ characterizes order continuity of the norm of $X$. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.