Here is a relatively simple question about PDE's pricing.
Assume that we are within the BS framework and moreover that interest rate is zero. The price $V(t,S_t)$ of the digital is known to be $\Phi(d2)$.
Now consider the BS PDE and solve it backwards using the explicit method with the straightforward boundary conditions:
$$V(T,S_T) = payoff(S_T)$$ $$V(t,0) =0, \qquad \text{ for } 0\leq t \leq T$$ $$V(t,S_t) = 1, \qquad \text{ for } S_t \text{ large}$$
The numerical solution yields a price that is close enough to BS, but when I plotted the price against the spot at time zero I get this picture, where the PDE solution is the one in black and the red one is a numerical approximation to the SDE:
Some details: (Volatility $=25\%$, Strike $=100$, r $=0$)
Question: How can the staircase-like behaviour of the PDE, instead of a roughly strictly increasing one, be explained mathematically ?
I am assuming it is because the payoff is discontinuous (also pricing other derivatives that were continuous worked just fined), but I would like a semi-rigorous mathematical explanation.
Thanks in advance.