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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.

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File URL: http://cowles.econ.yale.edu/P/cd/d08b/d0873.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 873.

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Length: 20 pages
Date of creation: Jun 1988
Date of revision:
Publication status: Published in Economic Theory (1991), 1(1): 3-12
Handle: RePEc:cwl:cwldpp:873

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Securities; portfolios; assets; arbitrage; marketed assets;

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References

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  1. 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.
  2. Bick, Avi, 1982. "Comments on the valuation of derivative assets," Journal of Financial Economics, Elsevier, vol. 10(3), pages 331-345, November.
  3. Avi Bick., 1982. "Comments on the Valuation of Derivative Assets," Research Program in Finance Working Papers 125, University of California at Berkeley.
  4. 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 621-51, October.
  5. 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 453-75, July.
  6. 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.
  7. 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.
  8. Jarrow, Robert A., 1986. "A characterization theorem for unique risk neutral probability measures," Economics Letters, Elsevier, vol. 22(1), pages 61-65.
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Citations

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Cited by:
  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.
  2. Galvani, Valentina, 2009. "Option spanning with exogenous information structure," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 73-79, January.
  3. C. D. Aliprantis & D. Brown & J. Werner, 1999. "Minimum-Cost Portfolio Insurance," Discussion Paper Serie A 599, University of Bonn, Germany.
  4. Baptista, Alexandre M., 2003. "Spanning with American options," Journal of Economic Theory, Elsevier, vol. 110(2), pages 264-289, June.
  5. David Bowman & Jon Faust, 1995. "Options, sunspots, and the creation of uncertainty," International Finance Discussion Papers 510, Board of Governors of the Federal Reserve System (U.S.).
  6. Aliprantis, C. D. & Harris, David & Tourky, Rabee, 2004. "Riesz Estimators," Purdue University Economics Working Papers 1170, Purdue University, Department of Economics.
  7. Aliprantis, Charalambos D. & Tourky, Rabee, 2002. "Markets that don't replicate any option," Economics Letters, Elsevier, vol. 76(3), pages 443-447, August.
  8. Jarno Talponen & Lauri Viitasaari, 2014. "Multidimensional Breeden-Litzenberger representation for state price densities and static hedging," Papers 1401.6383, arXiv.org.
  9. Galvani, Valentina, 2007. "Underlying assets for which options complete the market," Finance Research Letters, Elsevier, vol. 4(1), pages 59-66, March.
  10. Galvani, Valentina, 2007. "A note on spanning with options," Mathematical Social Sciences, Elsevier, vol. 54(1), pages 106-114, July.
  11. Protter, Philip, 2001. "A partial introduction to financial asset pricing theory," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 169-203, February.
  12. Ernesto Savaglio & Stefano Vannucci, 2014. "Strategy-proofness and single-peackedness in bounded distributive lattices," Papers 1406.5120, arXiv.org.
  13. Christos Kountzakis & Ioannis Polyrakis, 2006. "The completion of security markets," Decisions in Economics and Finance, Springer, vol. 29(1), pages 1-21, 05.
  14. Aliprantis, Charalambos D. & Polyrakis, Yiannis A. & Tourky, Rabee, 2002. "The cheapest hedge," Journal of Mathematical Economics, Elsevier, vol. 37(4), pages 269-295, July.
  15. 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.
  16. Ioannis Polyrakis & Foivos Xanthos, 2011. "Maximal submarkets that replicate any option," Annals of Finance, Springer, vol. 7(3), pages 407-423, August.
  17. Jarno Talponen, 2013. "On static hedging, real options and valuation of cash flows with skewed distributions," Papers 1312.4227, arXiv.org.
  18. Alexandre Baptista, 2000. "Options and Efficiency in Multiperiod Security Markets," Econometric Society World Congress 2000 Contributed Papers 0299, Econometric Society.
  19. Alexandre Baptista, 2007. "On the Non-Existence of Redundant Options," Economic Theory, Springer, vol. 31(2), pages 205-212, May.

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