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Smallest order closed sublattices and option spanning

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  • Niushan Gao
  • Denny H. Leung

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 $\overline{Y}^o$ 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.

Suggested Citation

  • Niushan Gao & Denny H. Leung, 2017. "Smallest order closed sublattices and option spanning," Papers 1703.09748, arXiv.org.
  • Handle: RePEc:arx:papers:1703.09748
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    References listed on IDEAS

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    1. Galvani, Valentina & Troitsky, Vladimir G., 2010. "Options and efficiency in spaces of bounded claims," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 616-619, July.
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    5. Galvani, Valentina, 2009. "Option spanning with exogenous information structure," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 73-79, January.
    6. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
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