Smallest order closed sublattices and option spanning

2017/03/28 by Gao, Niushan, Leung, Denny H. · 1 citation
#06F30 #46A40 #54F05 #FOS: Economics and business #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Finance (q-fin.MF)

paper · doi:10.48550/arxiv.1703.09748

Abstract

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 Y^o be the order closure of Y consisting of all order limits of nets of elements from Y. Then 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 Y^oo. Moreover, if X is a σ-order complete Banach lattice, then the condition that 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.

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