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PIDE and Fourier methods for pricing European options in Lévy models (Kathrin Glau)

Abstract

We concentrate on the relation between time-inhomogeneous Lévy processes and evolution problems that are associated with prices of options such as calls, puts and barrier options. A major concern is to shed light on the structural affinity between the PIDE and the Fourier transform based approach for European options. We characterize  Lévy processes according to the solution spaces of associated parabolic equations. It turns out that for a wide class of processes these spaces are weighted Sobolev-Slobodeckii spaces with different indices. To classify the processes according to these spaces, we define the related Sobolev index of the process. Since it is the most convenient to work with the Fourier transform of Lévy processes,  the classification is done according to the symbol i.e. the characteristic function of the process. In contrast to the criteria provided in the literature, our criteria based on the Sobolev index does not require differentiability conditions of the symbol or smoothness of the Lévy kernel, but purely translates the ellipticity condition on the infinitesimal generator to the symbol.
We derive the Sobolev index for several classes of Lévy processes and compare it to the Blumenthal-Getoor index, which reveals a relation between the Sobolev index and path properties of the process. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the unboundedness of the paths of the process.
The talk is based on joint work with Ernst Eberlein.