Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman-Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.