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Hedging with a second contract

Market makingHard

Price changes of contracts B and C each have variance $1$ and covariance $0.5$ . The covariance of the price change of contract A with B is $0.8$ and with C is $0.6$ . A market maker holds $1$ unit of A and sells b units of B and c units of C to minimize the variance of the combined position. What is c? Give an exact fraction.

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Hint

The minimizing amounts solve linear equations built from the covariances.

Solution

The conditions are $b + 0.5 c = 0.8$ and $0.5 b + c = 0.6$ .
The determinant is $1 - 0.25 = 0.75$ , so $b = 0.5 \div 0.75 = \frac{2}{3}$ .
Then $c = 0.6 - 0.5 \times \frac{2}{3} = \frac{4}{15}$ .

Answer: $\frac{4}{15}$

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