= Solution
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 $Y'=h(1-U)$ and average the pair. With $M$ independent pairs, the <variance> is $(v+\operatorname{Cov}(Y,Y'))/(2M)$, compared with $v/(2M)$ for $2M$ independent function evaluations. If $h$ is monotone in a scalar uniform, the <covariance> is nonpositive: for independent uniforms $U,V$,
$$
2\operatorname{Cov}(h(U),h(1-U))
=\mathbb E[(h(U)-h(V))(h(1-U)-h(1-V))]\le0.
$$
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 $\beta$, and
$$
\operatorname{Var}(Y-\beta C)=v-2\beta\operatorname{Cov}(Y,C)+\beta^2\operatorname{Var}C.
$$
Completing the square gives $\beta_*=\operatorname{Cov}(Y,C)/\operatorname{Var}C$ 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 $\sum_hp_h\bar Y_h$ is unbiased and has <variance>
$$
\sum_h\frac{p_h^2v_h}{N_h},\qquad v_h=\operatorname{Var}(Y\mid A_h).
$$
With proportional allocation $N_h=Np_h$, ignoring integer rounding, this equals $\sum_hp_hv_h/N$. The <law of total variance> shows the reduction from ordinary sampling is $\operatorname{Var}(\mathbb E[Y\mid A_h])/N$: 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 $(\sum_hp_h\sqrt{v_h})^2/N$ 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.
\b[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.
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