August 31st, 2016, 9:02 pm
To the extent that you are dealing with an analytical solution (which usually means you are in a Black-Scholes world), your intuition is pretty much right in that there is no pricing problem, per se. But, even there you will have a lurking hedging problem in the relatively unlikely case that you end up in that awkward corner of your state space where your delta goes to infinity (not to mention the higher order Greeks). Once you move out of analytical solutions and into numerical ones, I suspect that our local PDE experts (of which we have several) will tell you that the "local" discontinuity can induce problems of convergence more widely. I am deliberately being imprecise here, since I do not count myself among said experts.