Symmetry and duality in Lévy markets

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.

Keywords:

• Lévy processes
• Girsanov theorem
• Smile
• Numeraire
• Duality
• Symmetry

Acknowledgments

We thank comments from two anonymous referees, the conference participants at The 2004 North American Winter Meeting of the Econometric Society and III World Congress of The Bachelier Finance Society, and Álvaro Cartea for helpful discussion. The usual disclaimer applies. J. Fajardo also thanks CNPq and PRONEX E-26/171.193/2003 for financial support.