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Symmetry and duality in Levy markets

  • JosE Fajardo
  • Ernesto Mordecki

The aim of this paper is to introduce the notion of symmetry in a Levy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put-call duality. Symmetric Levy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices. Put-call duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov's theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov's theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Levy process, answering a question raised by Carr and Chesney (American put call symmetry, preprint, 1996). Some empirical evidence is shown, supporting that, in general, markets are not symmetric.

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Article provided by Taylor & Francis Journals in its journal Quantitative Finance.

Volume (Year): 6 (2006)
Issue (Month): 3 ()
Pages: 219-227

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Handle: RePEc:taf:quantf:v:6:y:2006:i:3:p:219-227
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  1. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
  2. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  3. Fajardo, J. & Farias, A., 2003. "Generalized Hyperbolic Distributions and Brazilian Data," Finance Lab Working Papers flwp_57, Finance Lab, Insper Instituto de Ensino e Pesquisa.
  4. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
  5. Orlin Grabbe, J., 1983. "The pricing of call and put options on foreign exchange," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 239-253, December.
  6. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
  7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  8. Schroder, Mark, 1999. "Changes of Numeraire for Pricing Futures, Forwards, and Options," Review of Financial Studies, Society for Financial Studies, vol. 12(5), pages 1143-63.
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