Exercise: The Rannacher trick

Crank-Nicolson can produce spurious oscillations near non-smooth payoffs. The Rannacher fix is to perform 2-4 fully implicit steps before switching to CN.

Tasks

1. Pure CN, no Rannacher. Price a digital cash-or-nothing call (payoff $\mathbb{1}_{\{S_T > K\}}$) using CN with $M = 100, N = 100$. Plot $V(S, t = 0)$. What goes wrong near $S = K$?
2. Rannacher-corrected. Apply 4 implicit steps at the start, then CN. Compare $V$ near $K$.
3. Explanation. Why does fully implicit damp oscillations but CN doesn't? Hint: amplification factor of high-frequency Fourier mode for each scheme.
4. Other smooth payoffs. Does the digital problem also occur for a vanilla call? For an Asian call (smooth in average)?