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Let $Y$ be a sublattice of a vector lattice $X$. We consider the problem ofidentifying the smallest order closed sublattice of $X$ containing $Y$. It isknown that the analogy with topological closure fails. Let $\overline{Y}^o$ bethe order closure of $Y$ consisting of all order limits of nets of elementsfrom $Y$. Then $\overline{Y}^o$ need not be order closed. We show that in manycases the smallest order closed sublattice containing $Y$ is in fact the secondorder 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 ordercontinuity of the norm of $X$. The present paper provides a general approach toa fundamental result in financial economics concerning the spanning power ofoptions written on a financial asset.

Smallest order closed sublattices and option spanning