Analysis of valuation formulae and applications to option pricing in Levy models

We discuss the valuation problem for a broad spectrum of derivatives, especially in Levy driven models. The key idea in this approach is to separate from the computational point of view the role of the two ingredients which are the payoff function and the driving process for the underlying quantity. Conditions under which valuation formulae based on Fourier and Laplace transforms hold in a general framework are analyzed. An interesting interplay between the properties of the payoff function and the driving process arises. We also derive the analytically extended characteristic function of the supremum and the infimum processes derived from a Levy process. Putting the different pieces together, we can price lookback and one-touch options in Levy driven models, as well as options on the minimum and maximum of several assets.