Adjoint expansions in local Lévy models

We propose a novel method for the analytical approximation in local volatility models with Lèvy jumps. In the case of Gaussian jumps, we provide an explicit approximation of the transition density of the underlying process by a heat kernel expansion: the approximation is derived in two ways, using PIDE techniques and working in the Fourier space. Our second and main result is an expansion of the characteristic function for a local volatility model with general Lévy jumps. Combined with standard Fourier methods, such an expansion allows to obtain efficient and accurate pricing formulae. Numerical tests confirm the effectiveness of the method.