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Answer: The application of control variates involves employing a variable with a mean of zero and a strong correlation with the simulated values.
## Explanation **D is correct.** This is the definition of control variates. Control variates involve using a variable that has a known expected value (typically zero) and is strongly correlated with the variable being simulated, which helps reduce variance in Monte Carlo simulations. **A is incorrect.** Antithetic variables must be negatively correlated, not positively correlated, with the simulation variables to reduce Monte Carlo sampling variability. Positive correlation would actually increase variance. **B is incorrect.** Both techniques reduce Monte Carlo sampling error, not bootstrapping error. Bootstrapping is a different statistical technique used for estimating sampling distributions. **C is incorrect.** Control variates are particularly useful when no closed-form solution exists. Monte Carlo simulation is most valuable for complex problems without analytical solutions, and control variates help improve the efficiency of such simulations.
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Which of the following statements is correct regarding the methods typically used to reduce sampling error?
A
Antithetic variables introduce a set of random variables that are positively correlated with the simulation variables to reduce the number of replications.
B
Control variates and antithetic variables both reduce bootstrapping sampling variability for a given number of replications.
C
The use of control variates is limited to simulations in which there is a closed-form solution with which to compare the simulated outcome.
D
The application of control variates involves employing a variable with a mean of zero and a strong correlation with the simulated values.
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