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FRM Part I · FRM Exam Part I · Simulation and Bootstrapping

A risk manager uses antithetic variates when simulating a stock price with standard normal draws. For each draw Z that generates a payoff X(Z), the method also uses -Z to generate X(-Z) and averages the pair. Under which condition does this technique reduce the variance of the estimator most effectively?

Antithetic variates work best when the payoffs from Z and -Z are negatively correlated. The variance of the pair average is proportional to one plus the correlation, so more negative correlation gives larger variance reduction, as with monotonic payoffs.

  1. AX(Z) and X(-Z) are negatively correlatedCorrect
  2. BX(Z) and X(-Z) are positively correlated
  3. CX(Z) is independent of the random seed
  4. DThe number of paths is below 100

Explanation

The variance of the pair average is (Var(X)/2)(1 + rho), where rho is the correlation between X(Z) and X(-Z). The more negative rho is, the larger the reduction. Positive correlation increases variance relative to the independent case.

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