  We generalize ideas in the recent literature and develop new ones in order to
propose a general class of contour integral methods for linear
convection-diffusion PDEs and in particular for those arising in finance. These
methods aim to provide a numerical approximation of the solution by computing
its inverse Laplace transform. The choice of the integration contour is
determined by the computation of a few pseudo-spectral level sets of the
leading operator of the equation. Parabolic and hyperbolic profiles proposed in
the literature are investigated and compared to the elliptic contour originally
proposed by Guglielmi, L\'opez-Fern\'andez and Nino. In summary, the article
  (i) provides a comparison among three different integration profiles;
  (ii) proposes a new fast pseudospectral roaming method;
  (iii) optimizes the selection of time windows on which one may arbitrarily
approximate the solution by no extra computational cost with respect to the
case of a fixed time instant;
  (iv) focuses extensively on computational aspects and it is the reference of
the MATLAB code https://github.com/MattiaManucci/Contour_Integral_Methods.git,
where all algorithms described here are implemented.
