In this paper we review several relationships between prices of put and call options, of both the European and the American type, obtained mainly through Girsanov Theorem, when the asset price is driven by a time-changed Lévy process. This relation is called put-call duality, and includes the relation known as put-call symmetry as a particular case. Necessary and sufficient conditions for put-call symmetry to hold are shown in terms of the triplet of local characteristic of the time-changed

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... process. This way we extend the results obtained by Fajardo and Mordecki (2006b).