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

A risk manager simulates the price of a call option using a control variate. The simulated payoff X has mean estimate 8.40 and the control Y has simulated mean 5.10, while the control's known true expected value is 5.50. Var(Y) = 4 and Cov(X,Y) = 3. Using the optimal coefficient b = Cov(X,Y)/Var(Y), what is the control-variate adjusted estimate of E[X]?

The adjusted estimate is 8.70. The optimal coefficient is 0.75, and the control's simulated mean fell 0.40 below its true value, so the estimate of X is corrected upward by 0.75 times 0.40, or 0.30, giving 8.40 plus 0.30.

  1. A8.10
  2. B8.40
  3. C8.70Correct
  4. D9.15

Explanation

b = 3/4 = 0.75. Adjusted estimate = X-bar - b(Y-bar - E[Y]) = 8.40 - 0.75(5.10 - 5.50) = 8.40 + 0.30 = 8.70. Using the opposite sign gives 8.10, a sign error since Y was underestimated and X should be adjusted upward as they are positively correlated.

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