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Mixed flow profit

Market makingEasy

A contract has a fair value of $100$ for uninformed traders. A market maker sells at an ask of $102$ . With probability $0.25$ the buyer is informed and the unit is really worth $106$ . Otherwise the unit is worth $100$ . What is the expected profit per unit sold?

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Hint

Weight each type of buyer by its probability.

Solution

Against an informed buyer the profit is $102 - 106 = - 4$ .
Against an uninformed buyer the profit is $102 - 100 = 2$ .
The expected profit is $0.25 \times (- 4) + 0.75 \times 2 = 0.5$ .

Answer: $0.5$

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