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Outline the ways in which Monte Carlo sampling error can be reduced.

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

The idea

Monte Carlo sampling error is the standard error of the estimate. Standard error = s ÷ √N. Here s is the standard deviation of the simulated quantity and N is the number of replications. The most direct fix is more replications, but accuracy is costly: halving the error needs four times as many draws. The alternative is to cut the variance of each draw. The main techniques are antithetic variates, control variates, stratified sampling, and importance sampling. Common random numbers help when comparing two similar valuations, since shared draws cancel noise in the difference. Low-discrepancy (quasi-random) sequences cover the sample space more evenly but are not independent, so the usual error formula does not apply. In an exam, check the cost. A technique that halves variance but doubles the run time per draw gains less than it appears.

A practice question

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

In a Monte Carlo simulation with independent draws, the standard error of the estimate is proportional to which of the following?

  1. 1 divided by the number of replications.Why not: Treats the error as shrinking linearly with N; it shrinks only with the square root of N.
  2. The square root of the number of replications.Why not: Gets the direction wrong: more replications reduce the standard error, they do not increase it.
  3. 1 divided by the square of the number of replications.Why not: Assumes error falls with N²; the standard error falls only at the rate of 1/sqrt(N).
  4. The reciprocal of the square root of N. - correct

The standard error of a sample mean of N independent draws is sigma / sqrt(N). Accuracy improves only slowly with more replications, which is why variance reduction techniques matter.

A harder one

A simulation with 2,000 replications has a standard error of 0.60. The desk wants the standard error cut to 0.20, using the same method. How many replications in total are needed?

  1. 6,000Why not: Scales N linearly with the required error reduction; the required factor is the square of the ratio of standard errors.
  2. 3,464Why not: Takes the square root of the ratio of standard errors instead of squaring it.
  3. 18,000 - correct
  4. 16,000Why not: Gives the additional replications to be run, not the total number of replications required.

The standard error scales with 1/sqrt(N). Cutting it by a factor of 3 needs 3² = 9 times as many replications: 2,000 x 9 = 18,000.

The hardest tier

A technique cuts the variance of each replication by 60% but makes each replication twice as expensive to compute. With a fixed computing budget that allows 10,000 ordinary replications, by how much does the standard error of the estimate change compared with ordinary Monte Carlo, closest to?

  1. It falls by 10.6%. - correct
  2. It falls by 60.0%.Why not: Equates the 60% variance reduction with a 60% standard error reduction; the standard error falls with the square root of the variance.
  3. It falls by 36.8%.Why not: Applies the variance reduction but ignores that the doubled cost halves the number of replications affordable.
  4. It falls by 20.0%.Why not: Compares variances (0.4 x 2 = 0.8) instead of standard errors, overstating the improvement.

Only 5,000 replications fit the budget. Variance of the estimator = 0.4 sigma² / 5,000 versus sigma² / 10,000, a ratio of 0.8. The standard error ratio is sqrt(0.8) = 0.894, a fall of 10.6%.

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