How does the control variate technique work in Monte Carlo?

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AcadiFi · Accepted Answer

Control variates exploit a correlated auxiliary payoff Y with known analytical value E[Y]. If X is your target estimator (arithmetic Asian), choose Y (geometric Asian) that's highly correlated with X and has analytic price. Your control variate estimator is X_CV = X - beta*(Y - E[Y]) where beta = Cov(X,Y) / Var(Y). The estimator remains unbiased (since E[Y - E[Y]] = 0) and variance becomes Var(X)*(1 - rho^2). For an arithmetic Asian call on Rivermark Energy strike $96, geometric and arithmetic averages are correlated ~0.99, giving variance reduction over 99%. Concretely: simulate N paths, compute arithmetic and geometric averages per path, compute beta from pilot run or during simulation, apply correction. Where raw MC needs 1,000,000 paths for 0.5% std error, control variate needs maybe 5,000. Other common controls: vanilla European closed-form for basket option (correlation ~0.95), bond price for swaption in short-rate simulation, geometric basket for arithmetic basket. The key requirement is known analytical E[Y] and high correlation with target. ```mermaid graph TD A[Simulate
X target] --> C[X - beta*
Y-E Y] B[Simulate
Y control] --> C D[Known
analytic E Y] --> C ```

How does the control variate technique work in Monte Carlo?

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