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FRM Exam Part I · Simulation and Bootstrapping

Antithetic and Control Variates: Variance Reduction in Monte Carlo

Updated 11 October 2026 · Fact-checked

Variance reduction techniques make a Monte Carlo estimate more precise without adding draws. Antithetic variates pair each draw with its mirror image so errors offset. Control variates subtract a correlated quantity with a known expected value. Both work by exploiting correlation, which cuts the standard error of the estimate.

Understand Variance Reduction Techniques

A Monte Carlo estimate is an average of simulated outcomes. Its standard error is σ ÷ √N, where σ is the standard deviation of one outcome and N is the number of draws. To halve the error by brute force, you need four times as many draws. That is slow and costly.

Variance reduction takes a different route. Instead of raising N, you lower the variance of what you average. The estimate stays unbiased, but it is tighter around the true value.

Antithetic variates: for every random draw Z, you also use −Z. For a normal draw, both are equally valid. If the payoff moves in a monotonic way with Z, the two payoffs are negatively correlated. Averaging them cancels part of the noise. The variance of the pair average is (σ² ÷ 2) × (1 + ρ), where ρ is the correlation between the two payoffs. The more negative ρ is, the bigger the gain.

Control variates: suppose you want E[X], and you have a second variable Y that is correlated with X and whose true expected value is known exactly. You simulate both. You then adjust X by the error seen in Y: X* = X − b(Y − E[Y]). If Y came out too high in your sample, you push X down by the matching amount. A typical example is pricing an arithmetic Asian option using the geometric Asian option, which has a closed-form price, as the control.

The key idea is the same in both: use correlation. Antithetic variates use negative correlation between paired draws. Control variates use correlation between the target and a known benchmark. Neither needs more draws, though each draw costs a little more work.

Key formulas to remember

Standard error of a Monte Carlo estimate
SE = σ ÷ √N
Quadrupling N halves SE. This is the baseline that variance reduction improves on.
Antithetic pair variance
Var[(X₁ + X₂) ÷ 2] = (σ² ÷ 2) × (1 + ρ)
X₁ and X₂ have the same variance σ². ρ is their correlation. ρ < 0 helps; ρ = 0 gives no gain over two independent draws.
Control variate adjusted estimator
X* = X − b(Y − E[Y])
E[Y] must be known exactly. The adjusted estimator stays unbiased because E[Y − E[Y]] = 0.
Optimal control coefficient
b* = Cov(X, Y) ÷ Var(Y) = ρ × σX ÷ σY
This is the regression slope of X on Y.
Variance after optimal control
Var(X*) = Var(X) × (1 − ρ²)
Gain depends on the size of ρ, not its sign. ρ = 0.9 removes 81% of the variance.

How to solve Variance Reduction Techniques questions

Use this method for any question on variance reduction in Monte Carlo.

  1. 1Identify the technique. Mirrored draws such as Z and −Z mean antithetic. A second variable with a known mean means control variate.
  2. 2Write down what is given: variances, the correlation ρ, and any known E[Y].
  3. 3For antithetic, compute the pair variance (σ² ÷ 2)(1 + ρ). Compare it with the independent-pair variance σ² ÷ 2.
  4. 4For control variate, find b* = Cov(X, Y) ÷ Var(Y) if it is not given.
  5. 5Apply X* = X − b(Y − E[Y]) using the sample value of Y, or Var(X*) = Var(X)(1 − ρ²) if the question asks about variance.
  6. 6Convert to standard error if needed: take the square root and divide by √N.
  7. 7Check sense: variance must fall, and the result must stay unbiased.

Quickest way: Variance ratio shortcut

When to use it: Use when the question asks how much variance falls, or which technique is better, and gives a correlation.

  1. Control variate: remaining variance fraction = 1 − ρ². Standard error fraction = √(1 − ρ²).
  2. Antithetic: remaining fraction versus independent pair = 1 + ρ. Use ρ as the correlation between the two mirrored payoffs.
  3. Compare the numbers, pick the smaller fraction, and match it to the option.
  4. If ρ is positive for antithetic, there is no benefit; the answer is usually that variance rises.

Common mistakes in Variance Reduction Techniques

  • Saying variance reduction works by increasing the number of draws.

    Students link lower error only with larger N.

    Fix: Remember the point: same N, lower variance. The techniques change what is averaged, not how many draws are used.

  • Using a control variate whose expected value is unknown or estimated.

    Students see correlation and forget the second requirement.

    Fix: The control needs a known E[Y], often from a closed-form price. Otherwise the adjustment adds bias or noise.

  • Thinking the sign of ρ matters for control variates.

    Students copy the antithetic rule that negative correlation helps.

    Fix: For control variates, variance falls by the factor (1 − ρ²), so ρ = −0.8 helps as much as ρ = +0.8. Only b* changes sign.

  • Applying antithetic variates to a non-monotonic payoff and expecting gains.

    Students assume mirroring always creates negative correlation.

    Fix: Gains need the payoff to move monotonically in Z. A payoff symmetric in Z, such as Z², gives ρ = 1 between the pair and no gain.

  • Treating the pair variance as σ² ÷ 2 regardless of ρ.

    Students assume independence between the two draws.

    Fix: Include the factor (1 + ρ). Antithetic pairs are deliberately dependent.

Worked examples

Example 1

A Monte Carlo estimate uses antithetic pairs. Each payoff has variance 16. The correlation between the payoff at Z and at −Z is −0.60. What is the variance of the average of one antithetic pair, and by what percentage is it lower than that of an average of two independent draws?

Show the solution
  1. Pair variance = (σ² ÷ 2)(1 + ρ) = (16 ÷ 2) × (1 − 0.60).
  2. = 8 × 0.40 = 3.2.
  3. Two independent draws give 16 ÷ 2 = 8.
  4. Reduction = (8 − 3.2) ÷ 8 = 4.8 ÷ 8 = 0.60, or 60%.

Answer: The pair variance is 3.2, which is 60% lower than the 8 from two independent draws.

Example 2

You simulate the payoff X of an exotic option. Var(X) = 25. A control variate Y has a known mean and Var(Y) = 9, and Cov(X, Y) = 13.5. What is the optimal coefficient b*, and what is the variance of the adjusted estimator?

Show the solution
  1. b* = Cov(X, Y) ÷ Var(Y) = 13.5 ÷ 9 = 1.5.
  2. σX = 5, σY = 3, so ρ = 13.5 ÷ (5 × 3) = 0.90.
  3. Var(X*) = Var(X)(1 − ρ²) = 25 × (1 − 0.81).
  4. = 25 × 0.19 = 4.75.
  5. Check using the formula Var(X) − Cov² ÷ Var(Y) = 25 − 182.25 ÷ 9 = 25 − 20.25 = 4.75.

Answer: b* = 1.5 and the adjusted variance is 4.75, down from 25.

Exam tips

  • Questions are often conceptual. Know the one-line difference: antithetic uses mirrored draws; control uses a correlated variable with a known mean.
  • Expect a correlation given in the question. Plug it into (1 + ρ) for antithetic or (1 − ρ²) for control.
  • Watch for traps: a positive ρ in an antithetic setup means no benefit, and an unknown control mean means the technique cannot be used.
  • Remember that both techniques keep the estimator unbiased. An option claiming bias is almost always wrong.
  • Link to the baseline: SE = σ ÷ √N. Variance reduction lowers σ instead of raising N.

Practice questions from Simulation and Bootstrapping

Variance Reduction Techniques: frequently asked questions

What is the difference between antithetic and control variates?

Antithetic variates pair each random draw with its mirror image to create negatively correlated payoffs that offset each other. Control variates use a second, correlated variable with a known expected value to correct the estimate. Antithetic needs no extra model; control needs a good benchmark.

Do variance reduction techniques add bias?

No. Both leave the estimator unbiased when applied correctly. Antithetic draws are each valid draws, and the control adjustment has an expected value of zero when E[Y] is known exactly.

Why does the control variate gain depend on ρ²?

The optimal adjustment removes the part of X explained by Y, like a regression. The unexplained share of variance is 1 − ρ². So a correlation of 0.9 or −0.9 removes 81% of the variance.

How can I reduce variance in Monte Carlo without more simulations?

Use antithetic variates, control variates, or related methods such as stratified sampling. FRM Part I focuses on antithetic and control variates. All aim to lower the variance of each averaged outcome so the standard error falls at the same N.