 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/s/s2/b/solution

For the independent standard normals stipulated in this part, simulate

n

values and average the exceedance indicators:

z_{0} = \frac{1}{n} k = 1 \sum n 1_{{X_{k} > c}} .

Each indicator is Bernoulli with probability

z_{c}

, so expectation and independence give

E z_{0} = z_{c}, v_{0} = Var (z_{0}) = \frac{z _{c} ( 1 - z _{c} )}{n} .

Thus the Monte Carlo estimator is unbiased.

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