Monte Carlo variance reduction: antithetic and control variates
Plain MC error scales as $1/\sqrt{N}$. Two tricks — antithetic pairs (negatively correlated samples) and control variates (a related closed-form anchor) — can drop the variance by an order of magnitude at the same compute cost.
Standard Monte Carlo for a barrier option needs 100,000 paths to get a stable price — variance reduction can hit the same tolerance in 10,000. That is not a coincidence; it is the point. Antithetic variates pair each simulated path with its mirror image so that overestimates on one cancel underestimates on the other. Control variates anchor the estimator to a known closed-form price and remove the shared simulation noise. Importance sampling tilts the distribution so more paths land where the payoff actually matters. Stratified sampling forces paths to cover the probability space evenly rather than clustering by chance. Either of the first two tricks routinely cuts variance by $5{-}20\times$; combined they reach $50{-}100\times$. The math behind all four is covariance arithmetic — no new theory, just cleverer bookkeeping of the randomness you already have.
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Worked example
Compute $E[U^2]$ for $U \sim \mathrm{Uniform}(0, 1)$ by Monte Carlo, with two iid samples each (so the same compute budget). (a) Compute the variance of the plain MC estimator $\hat\mu_{\text{plain}} = (U_1^2 + U_2^2)/2$. (b) Compute the variance of the antithetic estimator $\hat\mu_{\text{anti}} = (U^2 + (1-U)^2)/2$. (c) Compute the ratio (variance-reduction factor). The closed-form answer is $E[U^2] = 1/3$.
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Reflection
Antithetic works when the integrand is monotone in the random input. Control variates work when you have a related variable with a closed form. Why do these tricks tend to *combine* multiplicatively in practice (rBergomi pricing typically uses both), and what kind of integrand defeats both?