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Antithetic variates deliberately couple unbiased estimators of the same expectation with negative covariance. Their average can have smaller variance than an average using independent draws. For example, evaluating a monotone integrand at $U$ and $1-U$, where $U$ has a uniform distribution, gives opposite fluctuations while preserving both marginal distributions.

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.

Antithetic variates preserve each draw's marginal law but couple two estimates negatively. For an average of identically distributed $H,H^{\prime }$,Negative covariance improves on two independent evaluations at the same computational budget.

Here generate independent $U_1,\dots ,U_r\sim U(0,1)$ and pair $x_j=k{U}_j$ with $x_j^{\prime }=k(1-U_j)$. Both are uniform on $[0,k]$. The antithetic Cauchy-tail integration on a finite interval estimator isIt is unbiased. Put $h(u)=f(ku)$. This decreases, whereas $h(1-u)$ increases. For an independent copy $U^{\prime }$ of $U$, the covariance identity givesEach product is nonpositive, and for $k>0$ it is strictly negative off the diagonal. Thereforewhich is the variance using $2r$ independent evaluations in part (v). Thus the comparison accounts for the doubled number of probability density function evaluations in each pair.

Negative covariance can be exploited through antithetic variates, but it must belong to an actual joint sampling scheme. Two separately independent samples do not inherit negative covariance merely because one can imagine a coupling of their marginal distributions. Interpret the data as $h=\min (n,m)$ independent matched pairs $(X_i,Y_i)$ with covariance ${\sigma }_{XY}<0$, together with independent unmatched observations if the sample sizes differ. Other known pairing schemes can be handled by substituting their actual covariance of the two means.

Let $A=\overline{X}$, $B=\overline{Y}$, and putFor any fixed $w$, the combination $wA+(1-w)B$ is an unbiased estimator of the common mean. Its variance is $w^2{V}_X+(1-w{)}^2{V}_Y+2w(1-w)C$. Differentiation, or completing the square, gives the minimum-variance combination of unbiased sample means:With $C<0$, both weights are positive. This is efficiency among fixed linear unbiased combinations, not a universal claim about every possible estimator for arbitrary distributions $F,G$.

For an immediately implementable exactly unbiased estimator without known moments, use fixed $w=n/(n+m)$, the pooled sample mean. Its variance iswhich shows directly the gain from negative covariance. For any fixed weight, estimate $V_X,V_Y$ using $s_X^2/n,s_Y^2/m$, and $C$ using $h{s}_{XY}/(nm)$, with $s_{XY}$ the usual unbiased sample covariance of the matched pairs. Substitute them into the variance quadratic. At least two matched pairs are needed to estimate their covariance nonparametrically.

If the unknown optimal weights are desired while retaining exact unbiasedness, a cross-fitted combination of unbiased sample means gives an algorithm using the supplied data. Split the independent pairs and unmatched observations into two folds, keeping each pair together and retaining observations of both types in each fold. Estimate the moments from fold 2 to choose the weight for the fold-1 sample means, adjusting the variance terms for fold 1's sample sizes; do the converse for fold 2. Clip estimated weights to $(0\le w\le 1)$, to keep them bounded when training moment estimates are unstable. Average the two held-out combinations with deterministic weights, for example one half each for balanced folds.

Conditional on its training fold, each held-out pair of sample means has expectation $\theta$, and its weights sum to one. Each held-out combination, and hence their average, is therefore exactly an unbiased estimator. Under consistent moment estimation with increasing balanced folds, the weights approach $w_{\ast }$ and recover its first-order variance advantage. In contrast, estimating a weight from the same data to which it is applied need not yield an unbiased estimator.

Estimate the variance of this fitted-weight procedure by a bootstrap that resamples matched pairs as blocks and unmatched observations separately, repeats the entire split/weight/combination procedure, and takes the sample variance of the resulting estimates. This preserves the negative covariance and includes variability of the fitted weights. Resampling the $x$ and $y$ values independently would erase the feature responsible for the improvement. The bootstrap estimate has its usual large-sample qualification; the fixed-weight plug-in variance formula above is the simpler alternative when weights are predetermined.