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Market makingMedium

A contract pays $100$ or $0$ with equal probability. A trader receives a signal that is correct with probability $0.8$ . The market maker quotes a bid of $45$ and an ask of $55$ . The trader buys $1$ unit at the ask when the signal says $100$ and sells $1$ unit at the bid when it says $0$ . What is the market maker's expected loss per trade, as a positive number?

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Hint

Find the trader's expected value of the contract given the signal.

Solution

After a signal of $100$ the contract is worth $80$ in expectation, so buying at $55$ gains $25$ .
After a signal of $0$ the contract is worth $20$ in expectation, so selling at $45$ gains $25$ .
The market maker expects to lose $25$ per trade.

Answer: $25$

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