Exercise: Greeks from the FDM grid

Once the FDM grid $\{V(S_i, t)\}$ is computed, all Greeks come from grid differences.

Tasks

1. Delta: $\Delta(S_0) = (V_{i+1} - V_{i-1})/(2\Delta S)$ where $S_i \approx S_0$. Compute for the ATM call ($S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2, M = 200$).
2. Gamma: $\Gamma(S_0) = (V_{i+1} - 2V_i + V_{i-1})/\Delta S^2$. Compute and compare to the closed form $\phi(d_1)/(S_0\sigma\sqrt{T})$.
3. Theta: Use the value at the next time step on the grid: $\Theta = -(V^{n+1} - V^n)/\Delta t$ at $S_0$.
4. Vega and rho require re-running the FDM with bumped $\sigma$ or $r$ — show how to compute Vega via $(V(\sigma + h) - V(\sigma - h))/(2h)$.
5. Why FDM Greeks are noisier than the price. Argue heuristically: the price has $O(\Delta S^2)$ error, but the gradient has $O(\Delta S)$ error in general. Quantify by computing Delta as a function of $M$ and comparing the relative error.