Exercise: Compute budget — explicit vs CN

Suppose you have a fixed compute budget of $B$ floating-point operations and want to price a single European call as accurately as possible.

Tasks

1. Explicit cost. Each time step costs $\sim M$ flops; CFL forces $N \ge T \sigma^2 M^2$. Total cost: $C_{\text{exp}} = T\sigma^2 M^3$.
2. CN cost. Each step is a tridiagonal solve, $\sim 5M$ flops (constant factor). $N \sim M$ for matched accuracy. Total: $C_{\text{CN}} = 5 M^2$.
3. Achievable accuracy. With both schemes producing $O(M^{-2})$ spatial error (so error $\sim 1/M^2$):
   • For explicit: $M_{\text{exp}} = (B/(T\sigma^2))^{1/3}$, error $\sim (T\sigma^2/B)^{2/3}$.
   • For CN: $M_{\text{CN}} = (B/5)^{1/2}$, error $\sim 5/B$.
4. Numerical comparison. With $B = 10^9$, $T = 1$, $\sigma = 0.2$:
   • $M_{\text{exp}} = ?$, error $\sim ?$
   • $M_{\text{CN}} = ?$, error $\sim ?$
5. CN advantage scales with $B$. Show the CN/explicit error ratio scales as $B^{-1/3}$ — CN's advantage grows with budget.