Exercise: FDM stability — violating the CFL constraint

The explicit FDM scheme is stable when $\Delta t \le 1/(\sigma^2 M^2)$ (where $S$-grid index is $i = 0, \dots, M$).

Tasks

1. Set up the explicit FDM with $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2, M = 100, S_{\max} = 200$.

2. Run with $\Delta t = 0.001$ — well within the stability bound. Verify the price is reasonable.

3. Run with $\Delta t = 0.005$ — close to the bound $1/(\sigma^2 M^2) = 1/(0.04 \cdot 10^4) = 0.0025$. Does the scheme stay stable?

4. Run with $\Delta t = 0.01$ — twice the bound. Plot $V$ vs $S$ at $t = 0$. Describe what goes wrong.

5. CFL number. Define $\text{CFL} = \sigma^2 M^2 \Delta t$. Tabulate the maximum oscillation amplitude at $t = 0$ as a function of CFL $\in \{0.5, 1.0, 1.5, 2.0\}$.