SOA sample question 75
You are using Monte Carlo simulation to estimate the price of an option X, for
which there is no pricing formula. To reduce the variance of the estimate, you use the control variate method with another option Y, which has a pricing formula.
You are given:
(i) The naive Monte Carlo estimate of the price of X has standard deviation 5.
(ii) The same Monte Carlo trials are used to estimate the price of Y.
(iii) The correlation coefficient between the estimated price of X and that of Y
is 0.8.
Calculate the minimum variance of the estimated price of X, with Y being the
control variate.
In the solution, they just use the formula Var(X*) = Var(Xbar)(1rho^2), but why do they not use the formula Var(X*) = Var(Xbar) + beta^2(Var(Ybar))  2betaCov(Xbar,Ybar)? Is it because Var(Ybar) is not given which makes calculating covariance impossible? Then again, the question didn't say that beta is set to minimize Var(x*), or does that not matter?
