On the duality principle in option pricing: semimartingale setting

The purpose of this paper is to describe the appropriate mathematical framework for the study of the duality principle in option pricing. We consider models where prices evolve as general exponential semimartingales and provide a complete characterization of the dual process under the dual measure. Particular cases of these models are the ones driven by Brownian motions and by Lévy processes, which have been considered in several papers. Generally speaking, the duality principle states that the calculation of the price of a call option for a model with price process S=e ^H (with respect to the measure P) is equivalent to the calculation of the price of a put option for a suitable dual model S′=e ^H′ (with respect to the dual measure P′). More sophisticated duality results are derived for a broad spectrum of exotic options. The second named author acknowledges the financial support from the Deutsche Forschungsgemeinschaft (DFG, Eb 66/9-2). This research was carried out while the third named author was supported by the Alexander von Humboldt foundation.