Smallest order closed sublattices and option spanning
HTML articles powered by AMS MathViewer
- by Niushan Gao and Denny H. Leung PDF
- Proc. Amer. Math. Soc. 146 (2018), 705-716 Request permission
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.References
- Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782, DOI 10.1090/gsm/050
- C.D. Aliprantis and O. Burkinshaw, Positive operators, Springer, the Netherlands, 2006.
- Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Pure and Applied Mathematics, Vol. 76, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493242
- G. Bakshi and D. Madan, Spanning and derivative-security valuation, Journal of Financial Economics 55 (2000) 205–238.
- Sara Biagini and Marco Frittelli, On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures, Optimality and risk—modern trends in mathematical finance, Springer, Berlin, 2009, pp. 1–28. MR 2648595, DOI 10.1007/978-3-642-02608-9_{1}
- Donald J. Brown and Stephen A. Ross, Spanning, valuation and options, Econom. Theory 1 (1991), no. 1, 3–12. MR 1095150, DOI 10.1007/BF01210570
- Patrick Cheridito and Tianhui Li, Risk measures on Orlicz hearts, Math. Finance 19 (2009), no. 2, 189–214. MR 2509268, DOI 10.1111/j.1467-9965.2009.00364.x
- G. A. Edgar and Louis Sucheston, Stopping times and directed processes, Encyclopedia of Mathematics and its Applications, vol. 47, Cambridge University Press, Cambridge, 1992. MR 1191395, DOI 10.1017/CBO9780511574740
- Valentina Galvani, Option spanning with exogenous information structure, J. Math. Econom. 45 (2009), no. 1-2, 73–79. MR 2477693, DOI 10.1016/j.jmateco.2008.06.004
- Valentina Galvani and Vladimir G. Troitsky, Options and efficiency in spaces of bounded claims, J. Math. Econom. 46 (2010), no. 4, 616–619. MR 2674145, DOI 10.1016/j.jmateco.2010.05.004
- N. Gao, D. Leung, C. Munari, and F. Xanthos, Fatou property, representations, and extensions of risk measures on general Orlicz spaces, (2017), preprint, arXiv:1701.05967
- N. Gao, D. Leung, and F. Xanthos, Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures, (2016), preprint, arXiv:1610.08806
- N. Gao, V. Troitsky, and F. Xanthos, Uo-convergence and its applications to Cesáro means in Banach lattices, Israel J. Math, 220 (2017), no 2, 649-689.
- Niushan Gao and Foivos Xanthos, Unbounded order convergence and application to martingales without probability, J. Math. Anal. Appl. 415 (2014), no. 2, 931–947. MR 3178299, DOI 10.1016/j.jmaa.2014.01.078
- N. Gao and F. Xanthos, On the C-property and $w^*$-representations of risk measures, Mathematical Finance, to appear, arXiv:1511.03159
- N. Gao and F. Xanthos, Option spanning beyond $L_p$-models, Math. Financ. Econ. 11 (2017), no. 3, 383–391. MR 3636612, DOI 10.1007/s11579-017-0185-0
- Petr Hájek, Vicente Montesinos Santalucía, Jon Vanderwerff, and Václav Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26, Springer, New York, 2008. MR 2359536
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- W. A. J. Luxemburg and B. de Pagter, Representations of positive projections. I, Positivity 9 (2005), no. 3, 293–325. MR 2188521, DOI 10.1007/s11117-004-2773-5
- W. A. J. Luxemburg and B. de Pagter, Representations of positive projections. II, Positivity 9 (2005), no. 4, 569–605. MR 2193179, DOI 10.1007/s11117-004-2774-4
- Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
- D. C. Nachman, Spanning and completeness with options, Review of Financial Studies 1(3) (1998), 311–328.
- S. A. Ross, Options and efficiency, Quarterly Journal of Economics 90(1) (1976), 75–89.
Additional Information
- Niushan Gao
- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Canada T1K 3M4
- MR Author ID: 866193
- Email: gao.niushan@uleth.ca
- Denny H. Leung
- Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543
- MR Author ID: 113100
- Email: matlhh@nus.edu.sg
- Received by editor(s): March 28, 2017
- Published electronically: August 30, 2017
- Additional Notes: The first author is a PIMS Postdoctoral Fellow. He also acknowledges support from the National Natural Science Foundation of China (No. 11601443).
The second author was partially supported by AcRF grant R-146-000-242-114. - Communicated by: Thomas Schlumprecht
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 705-716
- MSC (2010): Primary 46A40, 06F30, 54F05
- DOI: https://doi.org/10.1090/proc/13820
- MathSciNet review: 3731703