Option pricing under some Lévy-like stochastic processes

A generalization of the Lèvy model for financial options is considered which employs pseudodifferential operators with symbols depending on the state variables throughout a small parameter ε. Adapting the classical method of the construction of a parametrix by means of the pseudodifferential calculus an approximate solution to the pricing problem is derived and its implication in terms of the volatility smile, even in very stylized models, is obtained.