Illustrate how antithetic variates and control variates are used to cut Monte Carlo sampling error, and explain why they work.

FRM® Part I · Quantitative Analysis · Simulation and Bootstrapping · FRM1.QA.13.c

The idea

Antithetic variates pair each random draw e with its mirror image −e and average the two results. For a monotonic payoff the two values are negatively correlated. Variance = σ² × (1 + ρ) ÷ 2. Here the variance is that of the pair average, σ² is the variance of one value, and ρ is the correlation between the two. It is below σ² ÷ 2, the variance for independent draws, when ρ is negative. An even payoff such as e² has ρ of +1 and gives no gain. A control variate is a related quantity with a known expected value, such as an analytic price of a similar option. Adjusted = Y − b × (C − E[C]). Here Y is the estimate, C is the control, and E[C] is its known mean. The best slope is b = Cov(Y, C) ÷ Var(C). The variance falls by the factor (1 − ρ²), where ρ is the correlation of Y with C. Both work by using correlation so that noise cancels. Exam trap: subtract the control's error, and do not confuse the variance reduction with ρ itself.

Formulas

Variance of an antithetic pair average

Var(pair average) = σ² × (1 + ρ) ÷ 2

σ: the standard deviation of a single payoff f(e)
ρ: the correlation between f(e) and f(−e)

A gain over σ² ÷ 2 only when ρ is negative, as for monotonic payoffs. For an even payoff such as e², ρ is +1 and nothing is gained.

Control variate estimate

Ŷ = Y − b × (C − E[C])

Y: the simulated estimate of the target
b: the coefficient Cov(Y, C) ÷ Var(C)
C: the simulated value of the control
E[C]: the control's known expected value

Subtract the control's error, do not add it. With the optimal b the variance falls by the factor (1 − ρ²), not by ρ.

Optimal control variate coefficient

b = Cov(Y, C) ÷ Var(C)

Cov(Y, C): the covariance between the target and control payoffs on the same draws
Var(C): the variance of the control payoff

Estimated from the same simulated paths. It is the slope of a regression of the target on the control.

A practice question

From the question bank, with the reason each wrong option is wrong.

In Monte Carlo simulation, how are antithetic variates used?

Each random draw is repeated twice so that the sample size doubles at no cost.Why not: Duplicating draws adds no information; the antithetic partner is the mirror -e, not a copy of e.
Each draw e is paired with -e and the two results are averaged. - correct
Draws are generated from two different distributions and the larger result is kept.Why not: Keeping the larger result biases the estimate upward and has nothing to do with antithetic pairing.
Draws are ranked and only those from the tails are used to estimate the quantity.Why not: Tail selection is not antithetic sampling; it would bias an expectation computed over the whole distribution.

Antithetic variates use e and -e together. When the payoff is monotonic in e, f(e) and f(-e) are negatively correlated, so their average has a lower variance than an average of two independent draws.

A harder one

A Monte Carlo analyst uses antithetic variates. The payoff f(e) has variance 16, and the correlation between f(e) and f(-e) is -0.6. What is the variance of the pair average [f(e) + f(-e)] / 2?

3.2 - correct
8.0Why not: Ignores the correlation and treats the two payoffs as independent, which is the variance without antithetic pairing.
12.8Why not: Reverses the sign on the correlation term, making the negative correlation increase the variance.
6.4Why not: Forgets to divide by 2 for the average, so it gives the variance of the sum scaled by 1/2 rather than 1/4.

Var of the average = (sigma² + sigma² + 2 rho sigma²) / 4 = sigma² (1 + rho) / 2 = 16 x 0.4 / 2 = 3.2. Independent draws would give 8.0; the negative correlation cuts it further.

The hardest tier

An analyst prices an arithmetic Asian option by Monte Carlo and gets a mean of 6.30. On the same paths, the geometric Asian option, whose exact value is 5.60, has a simulated mean of 6.10. The covariance between the two simulated payoffs is 4.8 and the variance of the geometric payoff is 8.0. What is the control variate estimate of the arithmetic Asian value, using the optimal coefficient?

5.80Why not: Sets the coefficient to 1 instead of Cov / Var = 0.6.
6.60Why not: Adds the correction instead of subtracting it; the simulated control overshot its true value, so the estimate must be moved down.
5.47Why not: Inverts the coefficient as Var / Cov = 1.67 instead of Cov / Var = 0.6.
6.00 - correct

The optimal coefficient is b = Cov / Var = 4.8 / 8.0 = 0.6. The control overshot its known value by 6.10 - 5.60 = 0.50. Adjusted estimate = 6.30 - 0.6 x 0.50 = 6.00.

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