Duality and Derivative Pricing with Lévy Processes

Authors: José Fajardo and Ernesto Mordecki

Abstract: The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to the pricing of a one Lévy driven stock in an auxiliary market, baptized as ``dual market''. In this way, we extend the results obtained by Gerber and Shiu (1996) for two dimensional Brownian motion. Also we examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov's Theorem, is called put - call duality. It includes as a particular case, the relation known as put - call symmetry. Necessary and sufficient conditions for put - call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Lévy process.

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