Exercise: Monte Carlo convergence rate for an ATM call

Black-Scholes parameters: $S_0 = K = 100$, $T = 1$, $r = 0.05$, $\sigma = 0.2$. Closed-form call price: \10.4506$.

Tasks

1. Implement basic Monte Carlo pricing for the European call. For each of $N \in \{10^3, 10^4, 10^5, 10^6\}$, compute $\hat V_0$ and the 95% CI half-width.

2. Plot the half-width against $N$ on a log-log scale. Verify the slope is $-1/2$.

3. Compute the absolute error $|\hat V_0 - 10.4506|$ for each $N$. Does the error decrease at the same rate as the CI half-width?

4. Exit polling intuition. Compare the MC sample-size requirement with: an election poll with 1000 respondents has a margin of error of $\sim 3\%$. How does this compare with the precision MC gives for $N = 1000$?