  In this article, a three-time levels compact scheme is proposed to solve the
partial integro-differential equation governing the option prices under
jump-diffusion models. In the proposed compact scheme, the second derivative
approximation of unknowns is approximated by the value of unknowns and their
first derivative approximations which allow us to obtain a tri-diagonal system
of linear equations for the fully discrete problem. Moreover, consistency and
stability of the proposed compact scheme are proved. Due to the low regularity
of typical initial conditions, the smoothing operator is employed to ensure the
fourth-order convergence rate. Numerical illustrations for pricing European
options under Merton and Kou jump-diffusion models are presented to validate
the theoretical results.
