It means the numerical scheme is wrong, at the very least. So, it's not correct.
IMO calling something a 'negative probability' is mathematical heresy
You have to resolve the root cause of the problem.
Negative probabilities in quantum theory have no physical meaning. They are mathematical artifacts resulting from obstinately applying the classical probability formalism to quantum models, where it doesn't apply by construction.
I see similar analogy in this case, where the issue (seems to be) caused by a non-monotonic explicit finite difference scheme?
big dividend in the underlying
Yep. This is called convection dominance and well-known in fluid dynamics. A structural solution is my exponentially fitted finite difference scheme for PDE. I wonder if it can be applied in a lattice context?
See Articles April 22 2013 for this method