Exercise: Crank-Nicolson convergence rate

For Crank-Nicolson, total error is $O(\Delta t^2 + \Delta S^2)$. To achieve the optimal balance, take $\Delta t \propto \Delta S$.

Tasks

1. With $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2, S_{\max} = 400$, run CN for $M \in \{50, 100, 200, 400\}$ with $N = M$. Tabulate error vs Black-Scholes.

2. Verify $O(M^{-2})$ convergence: error should drop by 4x for each doubling of $M$.

3. Compare with $N = M^2$ (over-resolved in time): does the error change much? What does this tell you about which dimension dominates?

4. Compare with $N = M/4$ (under-resolved in time): when does the time error start to dominate?