Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 4 / d

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.