Financial Risk Manager Part 1
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A financial analyst is using Monte Carlo simulation to price a complex derivative. They are concerned about the computational cost associated with achieving a high level of accuracy, as a large number of simulations are required to reduce sampling error. The analyst is considering using variance reduction techniques to improve the efficiency of the Monte Carlo simulation. Which of the following statements best describes how antithetic and control variates reduce Monte Carlo sampling error?
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A
Antithetic variates reduce sampling error by increasing the number of independent random samples used in the simulation, while control variates reduce error by replacing complex calculations with simpler approximations.
B
Antithetic variates reduce sampling error by creating pairs of negatively correlated random variables, while control variates reduce error by exploiting the known relationship between the derivative price and a related, analytically tractable variable.
C
Antithetic variates reduce sampling error by smoothing out the probability distribution of the underlying asset, while control variates reduce error by introducing additional randomness into the simulation to explore a wider range of possible outcomes.
D
Antithetic variates reduce sampling error by generating random numbers that are closer to the mean of the distribution, while control variates reduce error by using stratified sampling techniques to ensure better coverage of the sample space.
Explanation:
Explanation
Antithetic variates work by generating pairs of negatively correlated random variables. For each random path generated, a complementary path is created with the opposite sign (or negative correlation). This reduces variance because when one path gives a high value, the complementary path tends to give a low value, and vice versa, leading to cancellation of errors.
Control variates reduce sampling error by exploiting the known relationship between the derivative being priced and a related variable whose value can be computed analytically. The idea is to use the difference between the Monte Carlo estimate and the known analytical value of the control variable to adjust the estimate of the target derivative price.
Why the other options are incorrect:
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Option A: Incorrect - Antithetic variates do not increase the number of independent samples; they create correlated pairs. Control variates do not replace calculations with approximations.
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Option C: Incorrect - Antithetic variates do not smooth distributions, and control variates do not introduce additional randomness.
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Option D: Incorrect - Antithetic variates do not generate numbers closer to the mean, and control variates are not related to stratified sampling.
This variance reduction technique is particularly valuable in quantitative finance where computational efficiency is crucial for complex derivative pricing.
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