Variance Reduction

Vanilla Monte Carlo converges at the slow rate $1/\sqrt{N}$. Variance reduction techniques shrink the constant in front by exploiting structure.

Antithetic variates use pairs $(X, -X)$ for symmetric distributions. The pair has lower variance than two independent draws when the integrand is monotonic, halving the standard error in the best case.

Control variates use a related quantity $Y$ whose mean is known: estimate $E[g(X) - c(Y - E[Y])]$ for an optimal coefficient $c$. The closer the correlation between $g(X)$ and $Y$, the more variance you remove.

Importance sampling resamples from a different distribution that puts more mass on the important region of the integrand. It's the technique of choice for tail estimation — VaR, rare-event probabilities, deep out-of-the-money options. The trade-off is that a poorly chosen importance distribution can blow up variance.

Each technique adds bookkeeping but can deliver $10$–$100\times$ speedups when the structure cooperates.