  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.
