A Collection of Results on a Feynman-Kac Representation of Weak Solutions of PIDEs and on Pricing Barrier and Lookback Options in Lévy Models

Feynman-Kac formulas establish a fundamental link between conditional expectations and deterministic partial integro differential equations (PIDEs). In the context of option pricing in Lévy models, this relation has recently led to the development of various numerical methods to calculate prices via solving PIDEs. We give the precise link between certain conditional expectations and weak solutions of the corresponding PIDEs in Sobolev-Slobodeckii spaces. We apply the main result to price barrier options in (time-inhomogeneous) Lévy models and illustrate this by numerical results using a wavelet-Galerkin method. We look at the characterization of option prices via solutions of PIDEs from two sides. In view of efficient numerical solutions, we concentrate on the formulation as parabolic equations in Sobolev-Slobodeckii spaces. Interpreting these equations as pseudo differential equations provides an appropriate access, when starting from Lévy models. A classification of Lévy processes according to their Fourier transforms is obtained. The article provides a short description of parts of the results obtained in Glau (2010).      «

Feynman-Kac formulas establish a fundamental link between conditional expectations and deterministic partial integro differential equations (PIDEs). In the context of option pricing in Lévy models, this relation has recently led to the development of various numerical methods to calculate prices via solving PIDEs. We give the precise link between certain conditional expectations and weak solutions of the corresponding PIDEs in Sobolev-Slobodeckii spaces. We apply the main result to price barrier...     »