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Spanning, Valuation and Options

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Abstract

We model the space of marketed assets as a Riesz space of commodities. In this setting, two alternative characterizations are given of the space of continuous options on a bounded asset, s, with limited liability. The first characterization represents every continuous option on s as the uniform limit of portfolios of calls on s. The second characterization represents an option as a continuous sum (or integral) of Arrow-Debreu securities, with respect to s. The pricing implications of these representations are explored. In particular, the Breeden-Litzenberger pricing formula is shown to be a direct consequence of the integral representation theorem.

Suggested Citation

  • Donald J. Brown & Stephen A. Ross, 1988. "Spanning, Valuation and Options," Cowles Foundation Discussion Papers 873, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:873
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    References listed on IDEAS

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    1. Ross, Stephen A, 1978. "A Simple Approach to the Valuation of Risky Streams," The Journal of Business, University of Chicago Press, vol. 51(3), pages 453-475, July.
    2. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    3. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
    4. Bick, Avi, 1982. "Comments on the valuation of derivative assets," Journal of Financial Economics, Elsevier, vol. 10(3), pages 331-345, November.
    5. Avi Bick., 1982. "Comments on the Valuation of Derivative Assets," Research Program in Finance Working Papers 125, University of California at Berkeley.
    6. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    7. Jarrow, Robert A., 1986. "A characterization theorem for unique risk neutral probability measures," Economics Letters, Elsevier, vol. 22(1), pages 61-65.
    8. N/A, 1970. "Note," Review of Radical Political Economics, Union for Radical Political Economics, vol. 2(4), pages 1-1, October.
    9. Green, Richard C. & Jarrow, Robert A., 1987. "Spanning and completeness in markets with contingent claims," Journal of Economic Theory, Elsevier, vol. 41(1), pages 202-210, February.
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    Cited by:

    1. Aliprantis, Charalambos D. & Harris, David & Tourky, Rabee, 2007. "Riesz estimators," Journal of Econometrics, Elsevier, vol. 136(2), pages 431-456, February.
    2. Bowman, David & Faust, Jon, 1997. "Options, Sunspots, and the Creation of Uncertainty," Journal of Political Economy, University of Chicago Press, vol. 105(5), pages 957-975, October.
    3. Jarno Talponen & Lauri Viitasaari, 2014. "Multidimensional Breeden-Litzenberger representation for state price densities and static hedging," Papers 1401.6383, arXiv.org.
    4. Baptista, Alexandre M., 2003. "Spanning with American options," Journal of Economic Theory, Elsevier, vol. 110(2), pages 264-289, June.
    5. Tian, Weidong, 2014. "Spanning with indexes," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 111-118.
    6. 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.
    7. Jarno Talponen & Minna Turunen, 2017. "Lean derivation of the CRR pricing formula," Papers 1705.00212, arXiv.org, revised May 2017.
    8. Aliprantis, Charalambos D. & Tourky, Rabee, 2002. "Markets that don't replicate any option," Economics Letters, Elsevier, vol. 76(3), pages 443-447, August.
    9. Gianluca Cassese, 2017. "Asset pricing in an imperfect world," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 539-570, October.
    10. Galvani, Valentina, 2009. "Option spanning with exogenous information structure," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 73-79, January.
    11. Charalambos Aliprantis & Donald J. Brown & Werner, J., 1997. "Incomplete Derivative Markets and Portfolio Insurance," Cowles Foundation Discussion Papers 1126R, Cowles Foundation for Research in Economics, Yale University.
    12. Cerreia-Vioglio, S. & Maccheroni, F. & Marinacci, M., 2015. "Put–Call Parity and market frictions," Journal of Economic Theory, Elsevier, vol. 157(C), pages 730-762.
    13. Aliprantis, C. D. & Brown, D. J. & Werner, J., 2000. "Minimum-cost portfolio insurance," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1703-1719, October.
    14. Jarno Talponen & Lauri Viitasaari, 2013. "Note on multidimensional Breeden-Litzenberger representation for state price densities," Papers 1305.5963, arXiv.org, revised Jan 2014.
    15. Aliprantis, Charalambos D. & Polyrakis, Yiannis A. & Tourky, Rabee, 2002. "The cheapest hedge," Journal of Mathematical Economics, Elsevier, vol. 37(4), pages 269-295, July.
    16. Niushan Gao & Denny H. Leung, 2017. "Smallest order closed sublattices and option spanning," Papers 1703.09748, arXiv.org.
    17. Alexandre Baptista, 2007. "On the Non-Existence of Redundant Options," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 31(2), pages 205-212, May.
    18. Protter, Philip, 2001. "A partial introduction to financial asset pricing theory," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 169-203, February.
    19. Jarno Talponen, 2013. "Matching distributions: Asset pricing with density shape correction," Papers 1312.4227, arXiv.org, revised Dec 2016.
    20. Alexandre Baptista, 2000. "Options and Efficiency in Multiperiod Security Markets," Econometric Society World Congress 2000 Contributed Papers 0299, Econometric Society.
    21. Galvani, Valentina, 2007. "A note on spanning with options," Mathematical Social Sciences, Elsevier, vol. 54(1), pages 106-114, July.
    22. Galvani, Valentina, 2007. "Underlying assets for which options complete the market," Finance Research Letters, Elsevier, vol. 4(1), pages 59-66, March.
    23. Christos Kountzakis & Ioannis Polyrakis, 2006. "The completion of security markets," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 29(1), pages 1-21, May.
    24. Ernesto Savaglio & Stefano Vannucci, 2014. "Strategy-proofness and single-peackedness in bounded distributive lattices," Papers 1406.5120, arXiv.org.

    More about this item

    Keywords

    Securities; portfolios; assets; arbitrage; marketed assets;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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