Exercise: Antithetic variance ratio for ATM call

For the ATM call $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2$:

Tasks

1. With $N = 10{,}000$ pairs, compute the discounted payoffs at $Z$ and $-Z$ for each pair. Estimate $\rho = \text{Corr}(f(Z), f(-Z))$ from the sample.

2. Predict the variance reduction factor as $(1 + \rho)/2$ relative to plain MC at the same total compute (same $N$ payoff evaluations).

3. Verify the prediction by computing the variance of the antithetic estimator and the variance of plain MC at $N = 20{,}000$ samples (same compute as $N = 10{,}000$ pairs).

4. Repeat for varying moneyness. $K \in \{50, 75, 100, 125, 150\}$. Plot $\rho$ as a function of $K/S_0$. Where is antithetic most effective? Where does it stop helping?