Smallest order closed sublattices and option spanning

Abstract

Let Y Y be a sublattice of a vector lattice X X . We consider the problem of identifying the smallest order closed sublattice of X X containing Y Y . It is known that the analogy with topological closure fails. Let Y ¯ o \overline {Y}^o be the order closure of Y Y consisting of all order limits of nets of elements from Y Y . Then Y ¯ o \overline {Y}^o need not be order closed. We show that in many cases the smallest order closed sublattice containing Y Y is in fact the second order closure Y ¯ o ¯ o \overline {\overline {Y}^o}^o . Moreover, if X X is a σ \sigma -order complete Banach lattice, then the condition that Y ¯ o \overline {Y}^o is order closed for every sublattice Y Y characterizes order continuity of the norm of X 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.