Ways to cut Monte Carlo variance

Question

Which techniques can you use to make a Monte Carlo estimator more efficient, i.e. reduce its variance for a given number of simulations?

Short answer

Antithetic variates, control variates, importance sampling, stratified sampling, moment matching and low-discrepancy (quasi-random) sequences. Antithetic pairs $Z$ with $-Z$ ; control variates subtract a correlated quantity with known mean (e.g. a geometric Asian for an arithmetic one); importance sampling shifts draws toward the region that matters (deep OTM); stratification and Sobol sequences spread draws more evenly than pure random sampling.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 7.3 Numerical Methods, 'Monte Carlo simulation C', p.186; question and answer rewritten by SelfMadeQuant.com.

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