Control variates

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The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.[1] [2][3]

Underlying principle[edit]

Let the unknown parameter of interest be , and assume we have a statistic such that the expected value of m is μ: , i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic such that is a known value. Then

is also an unbiased estimator for for any choice of the coefficient . The variance of the resulting estimator is

By differentiating the above expression with respect to , it can be shown that choosing the optimal coefficient

minimizes the variance of , and that with this choice,


is the correlation coefficient of and . The greater the value of , the greater the variance reduction achieved.

In the case that , , and/or are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.

When the expectation of the control variable, , is not known analytically, it is still possible to increase the precision in estimating (for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating is significantly cheaper than computing ; 2) the magnitude of the correlation coefficient is close to unity. [3]


We would like to estimate

using Monte Carlo integration. This integral is the expected value of , where

and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as . Then the estimate is given by

Now we introduce as a control variate with a known expected value and combine the two into a new estimate

Using realizations and an estimated optimal coefficient we obtain the following results

Estimate Variance
Classical estimate 0.69475 0.01947
Control variates 0.69295 0.00060

The variance was significantly reduced after using the control variates technique. (The exact result is .)

See also[edit]


  1. ^ Lemieux, C. (2017). "Control Variates". Wiley StatsRef: Statistics Reference Online: 1--8. doi:10.1002/9781118445112.stat07947.
  2. ^ Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. ISBN 0-387-00451-3 (p. 185)
  3. ^ a b Botev, Z.; Ridder, A. (2017). "Variance Reduction". Wiley StatsRef: Statistics Reference Online: 1--6. doi:10.1002/9781118445112.stat07975.