Exercise: Explicit FDM vs Black-Scholes

For $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2$ (BS price 10.4506):

Tasks

1. Compute the FDM price for $M \in \{50, 100, 200, 400, 800\}$ with $S_{\max} = 4 S_0 = 400$ and $\Delta t$ at $90\%$ of the CFL bound.

2. Tabulate error vs $M$. Verify the convergence is $O(\Delta S^2) = O(1/M^2)$.

3. Effect of $S_{\max}$. Vary $S_{\max} \in \{2 S_0, 4 S_0, 8 S_0\}$ at fixed $M = 200$. How does the error change?
4. At-the-money vs deep ITM. Compute the FDM error for $K = 100$ vs $K = 50$ at $S_0 = 100$. Where is the FDM more accurate?