Control variates
Quantitative Finance glossary · Numerical Methods & Optimization
Definition
Variance-reduction technique using a correlated random variable Y with known mean mathbbE[Y]: estimator hatθ_CV = barX - β,(barY - mathbbE[Y]), optimal β^* = Cov(X,Y)/Var(Y), yielding variance Var(hatθ_CV) = (1 - ρ_XY^2),Var(barX). Pricing an arithmetic Asian by Monte Carlo using the geometric Asian (closed form under GBM) as a control is the canonical example — variance can drop 100×.
Control variates in plain English
This is another trick to make Monte Carlo simulation less noisy, and the idea is beautifully simple: cheat by using a similar problem you already know the exact answer to. Suppose you're estimating something hard (X) but there's a closely related quantity (Y) whose true average you know by formula. In each simulation, if your estimate of Y came out too high, that same run probably pushed X too high as well, so you subtract off that known error to correct X. It's like weighing yourself on a scale you know reads 2 pounds heavy — you just subtract the 2 pounds. The more tightly the two quantities move together, the more noise cancels; in a classic option-pricing example this can cut the required work by a factor of 100.
Practice Control variates
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