  In this paper a novel contour integral method is proposed for linear
convection-diffusion equations. The method is based on the inversion of the
Laplace transform and makes use of a contour given by an elliptic arc joined
symmetrically to two half-lines. The trapezoidal rule is the chosen integration
method for the numerical inversion of the Laplace transform, due to its
well-known fast convergence properties when applied to analytic functions.
Error estimates are provided as well as careful indications about the choice of
several involved parameters. The method selects the elliptic arc in the
integration contour by an algorithmic strategy based on the computation of
pseudospectral level sets of the discretized differential operator. In this
sense the method is general and can be applied to any linear
convection-diffusion equation without knowing any a priori information about
its pseudospectral geometry. Numerical experiments performed on the
Black-Scholes ($1D$) and Heston ($2D$) equations show that the method is
competitive with other contour integral methods available in the literature.
