Symmetry and duality in Levy markets

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/future; 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 Chestier (American put call symmetry. preprint. 1996). Some empirical evidence is shown, supporting that. ill general. markets are not symmetric.