= Solution
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 $U'$, twice that covariance is the expectation of $[g(U)-g(U')][g(1-U)-g(1-U')]$, which is nonpositive pointwise. With $N=2m$ evaluations, the paired average has <variance>
$$
\frac{\operatorname{Var}(g(U))+\operatorname{Cov}(g(U),g(1-U))}{N}\leq\frac{\operatorname{Var}(g(U))}{N}.
$$
For 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 gives
$$
\beta^*=\frac{\operatorname{Cov}(H,Y)}{\operatorname{Var}(Y)},\qquad
\operatorname{Var}(H-\beta^*(Y-S))=\operatorname{Var}(H)(1-\operatorname{Corr}(H,Y)^2).
$$
A 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> $q^{-2}\sum_jv_j$. The <law of total variance> gives $\operatorname{Var}(H)\geq q^{-1}\sum_jv_j$, so this is at most the <variance> $\operatorname{Var}(H)/q$ 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.
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