Contract A pays 1 if event E happens, and contract B pays 1 if event F happens. The probabilities are P(E) = 0.5, P(F) = 0.5 and P(E ∩ F) = 0.4. A market maker holds 10 units of A and sells h units of B to minimize the variance of the total payout. What is h?

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Event contract hedge

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Contract A pays $1$ if event E happens, and contract B pays $1$ if event F happens. The probabilities are $P (E) = 0.5$ , $P (F) = 0.5$ and $P (E \cap F) = 0.4$ . A market maker holds $10$ units of A and sells h units of B to minimize the variance of the total payout. What is h?

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Hint

The best hedge is the covariance divided by the variance of the hedging contract.

Solution

The covariance is $0.4 - 0.5 \times 0.5 = 0.15$ and the variance of B is $0.5 \times 0.5 = 0.25$ .
The hedge per unit of A is $0.15 \div 0.25 = 0.6$ .
For $10$ units of A, $h = 10 \times 0.6 = 6$ .

Answer: $6$

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