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Variance left after hedging

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A market maker holds $10$ units of contract A. Price changes of A have a standard deviation of $5$ per unit, and a hedging contract B has a correlation of $0.8$ with A. The market maker hedges with the variance-minimizing amount of B. What variance remains in the combined position?

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Hint

Per unit of A, the remaining variance is the share not explained by B.

Solution

Per unit, the variance of A is $5^{2} = 25$ , and the share unexplained by B is $1 - 0. 8^{2} = 0.36$ .
The remaining variance per unit is $25 \times 0.36 = 9$ .
For $10$ units it is $1 0^{2} \times 9 = 900$ .

Answer: $900$

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