Our aim is to establish a precise link between prices of European options in Lévy models and PIDEs. We follow a Fourier transform based approach and outline a structural affinity of PIDE and Fourier methods in this context. Our analysis provides a framework that is extendible to more complex problems, such as to PIDEs for pricing barrier options. Since the payoff functions of a wide range of options such as calls or puts, written as functions on the log-price of the underlying, are not in L2 (R), exponentially weighted Sobolev-Slobodeckii spaces are studied. It turns out that the exponential weight corresponds to a shift of the symbol of the Lévy process in the complex plane. We derive natural assumptions on the symbol and its analytic extension, under which the associated evolution problem has a unique weak solution in an exponentially weighted Sobolev-Slobodeckii space. To provide the characterization of option prices as weak solutions of PIDEs in weighted Sobolev-Slobodeckii spaces, a Feynman-Kac formula for weak solutions of PIDEs is proved. Furthermore, an explicit solution in terms of the Fourier transform is derived.      «

Our aim is to establish a precise link between prices of European options in Lévy models and PIDEs. We follow a Fourier transform based approach and outline a structural affinity of PIDE and Fourier methods in this context. Our analysis provides a framework that is extendible to more complex problems, such as to PIDEs for pricing barrier options. Since the payoff functions of a wide range of options such as calls or puts, written as functions on the log-price of the underlying, are not in L2 (R)...     »