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Let $Y=h(U)$ have finite variance $v$ and mean $I$. An independent-sample Monte Carlo estimator has variance $v/N$, so its root-mean-square random error is $\sqrt{v/N}$. Three techniques reduce the variance coefficient rather than changing this ordinary $N^{-1/2}$ rate.

For antithetic variates, couple $Y=h(U)$ with $Y^{\prime }=h(1-U)$ and average the pair. With $M$ independent pairs, the variance is $(v+\mathrm{Cov}(Y,Y^{\prime }))/(2M)$, compared with $v/(2M)$ for $2M$ independent function evaluations. If $h$ is monotone in a scalar uniform, the covariance is nonpositive: for independent uniforms $U,V$,Opposite fluctuations therefore cancel. The reduction factor at equal evaluation count is $1+\rho$, where $\rho$ is the paired correlation coefficient; it can approach zero, but positive covariance would make the method worse. This is appropriate for monotone or otherwise negatively coupled payoffs, with cheap complementary paths.

For control variates, use a jointly simulated $C$ with known mean $m_C$ and average $Y-\beta (C-m_C)$. It remains an unbiased estimator for fixed $\beta$, andCompleting the square gives ${\beta }_{\ast }=\mathrm{Cov}(Y,C)/\mathrm{Var}C$ and minimal variance $v(1-{\rho }^2)$. For contingent claim simulation, an analytically priced related financial payoff, such as a simpler option or discounted terminal stock, can be a useful control. It works best when correlation has large absolute value and simulation of the control adds little cost. A coefficient fixed from an independent pilot keeps the elementary unbiasedness argument valid; estimating it from the same sample can introduce finite-sample bias.

For stratified sampling, partition the sampling space into strata $A_h$ of probabilities $p_h$, and independently simulate $N_h$ samples conditional on each stratum. The estimator ${\sum }_h{p}_h{\bar{Y}}_h$ is unbiased and has varianceWith proportional allocation $N_h=N{p}_h$, ignoring integer rounding, this equals ${\sum }_h{p}_h{v}_h/N$. The law of total variance shows the reduction from ordinary sampling is $\mathrm{Var}(\mathbb{E}[Y\mid A_h])/N$: fixed representation removes randomness in the stratum counts. At equal per-draw cost, minimizing the displayed variance gives $N_h\propto p_h\sqrt{v_h}$ and minimum $({\sum }_h{p}_h\sqrt{v_h}{)}^2/N$ for positive within-stratum variances. This follows by Cauchy-Schwarz inequality and is useful when a low-dimensional coordinate strongly predicts the financial payoff and conditional simulation is easy.

There is no universal ranking. At comparable cost the variance factors depend respectively on paired covariance, squared control correlation, and within-stratum variation. A nearly perfect control can outperform a weak antithetic pair; an appropriate stratification can be almost exact for a nearly stratum-constant financial payoff. Extra construction and sampling costs must be included in a fair accuracy comparison, and the methods can be combined.

Three useful methods are antithetic variates, a control variate, and stratified sampling. Their purpose is to reduce sampling variance at comparable computational cost, rather than merely to take more samples.

For antithetic variates, generate independent pairs $(U,1-U)$ and average both payoffs. Since the call integrand is increasing, the two values have nonpositive covariance. Indeed, for an independent copy $U^{\prime }$, twice that covariance is the expectation of $[g(U)-g(U^{\prime })][g(1-U)-g(1-U^{\prime })]$, which is nonpositive pointwise. With $N=2m$ evaluations, the paired average has varianceFor a control variate, use the same simulated discounted stock value $Y=e^{-r\tau }{S}_T$, whose known mean is $S$. Replace the discounted call payoff $H$ by $H-\beta (Y-S)$, preserving its mean. Minimizing its quadratic variance givesA predetermined coefficient or an independent pilot estimate preserves exact unbiasedness; fitting the coefficient from the same observations requires care.

For stratified sampling, divide $[0,1]$ into $q$ equal intervals, sample independently and uniformly within each, and average their $q$ payoffs. If $v_j$ is the conditional variance in interval $j$, this unbiased estimator has variance $q^{-2}{\sum }_j{v}_j$. The law of total variance gives $\mathrm{Var}(H)\ge q^{-1}{\sum }_j{v}_j$, so this is at most the variance $\mathrm{Var}(H)/q$ of $q$ ordinary independent samples. Stratification removes randomness in how many samples fall in each region, which is particularly valuable when conditional payoff means vary strongly.

Label batches before consulting the random-number table, so their labels are independent of the test results. A uniformly random permutation of $1,\dots ,16$, restricted to a smaller label set, induces a uniformly random ordering of that set. This provides a simple rejection-and-restriction scheme:

With independent random digits instead, rejection sampling gives the same uniform choices: for A keep only digits $1,\dots ,6$; for C keep only $1,\dots ,5$. For B, use uniform two-digit numbers, retain labels $01,\dots ,15$, and reject repeats within a three-batch sample. Taking residues modulo six or fifteen from a table whose range is not divisible by that number would give unequal probabilities. These are respectively day-stratified simple random sampling, morning/afternoon stratified sampling, and a one-day design using cluster sampling. The printed word “rest” for A is interpreted as “test”, consistently with the surveillance task. If it instead meant leaving one batch untested, choose that omitted batch uniformly and test the other five; this alternative would test 25 batches per week, with inclusion probability $5/6$, and has a different testing budget.

Probability sampling chooses observational units by a specified random mechanism with known inclusion probabilities. This separates design-based representativeness from the convenience of obtaining particular observations. Simple random sampling, stratified sampling and cluster sampling are different designs with different precision and cost properties.