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Quote update after a buy

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An asset is worth $110$ or $90$ with equal probability. A trader arrives: with probability $0.25$ the trader is informed and buys if the value is $110$ and sells if it is $90$ ; otherwise the trader is uninformed and buys or sells with probability $\frac{1}{2}$ each. You, a market maker, observe a buy. What is the expected value of the asset given this buy?

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Hint

Find the probability of a buy in each value state, update the probability of the high value, then take the expectation.

Solution

If the value is $110$ , a buy occurs with probability $0.25 + 0.75 \times 0.5 = 0.625$ .
If the value is $90$ , a buy occurs with probability $0.75 \times 0.5 = 0.375$ .
With equal priors, the posterior probability of the high value is $0.625 \div (0.625 + 0.375) = 0.625$ .
The expected value is $90 + 20 \times 0.625 = 102.5$ .
The answer is $102.5$ .

Answer: $102.5$

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