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Symmetry and duality in Lévy markets

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Author Info
JosÉ Fajardo
Ernesto Mordecki

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Abstract

The aim of this paper is to introduce the notion of symmetry in a Lévy 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 Lévy 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 Lévy 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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Publisher Info
Article provided by Taylor and Francis Journals in its journal Quantitative Finance.

Volume (Year): 6 (2006)
Issue (Month): 3 (June)
Pages: 219-227
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Handle: RePEc:taf:quantf:v:6:y:2006:i:3:p:219-227

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Related research
Keywords: Lévy processes; Girsanov theorem; Smile; Numeraire; Duality; Symmetry;

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

  1. 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. [Downloadable!] (restricted)
  2. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April. [Downloadable!]
  3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144. [Downloadable!] (restricted)
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  4. 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. [Downloadable!] (restricted)
  5. Fajardo, J. & Farias, A., 2003. "Generalized Hyperbolic Distributions and Brazilian Data," Finance Lab Working Papers flwp_57, Finance Lab, Ibmec São Paulo. [Downloadable!]
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Ilya Molchanov & Michael Schmutz, 2009. "Semi-static hedging under exchangeability type conditions," Quantitative Finance Papers 0901.4914, arXiv.org, revised Mar 2009. [Downloadable!]
  2. José Fajardo & Ernesto Mordecki, 2009. "Skewness Premium with Lévy Processes," CREATES Research Papers 2009-10, School of Economics and Management, University of Aarhus. [Downloadable!]
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