Actuarial Outpost SOA sample question 75
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 Investment / Financial Markets Old Exam MFE Forum

#1
06-25-2017, 10:19 PM
 mistersunnyd Member SOA Join Date: Aug 2016 Studying for how to find a job Posts: 146
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(X-bar)(1-rho^2), but why do they not use the formula Var(X*) = Var(X-bar) + beta^2(Var(Y-bar)) - 2betaCov(X-bar,Y-bar)? Is it because Var(Y-bar) 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?
#2
06-25-2017, 11:19 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,256

If you solve for the beta that minimizes the variance and plug in you should get the given formula after simplification.

 Tags monte carlo