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Fair price of a run of buys

Market makingMedium

The share of buyers in the order flow has a $Beta (7, 3)$ posterior, and each order is a buy with that chance, independently given the share. A contract pays $100$ if the next $2$ orders are both buys. What is its fair value? Give a fraction.

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Hint

The first buy raises your belief about the second. Do not square the mean.

Solution

The first order is a buy with chance $\frac{7}{10}$ .
After that the posterior is $Beta (8, 3)$ , so the second is a buy with chance $\frac{8}{11}$ .
The fair value is $100 \times \frac{7}{10} \times \frac{8}{11} = \frac{560}{11}$ .

Answer: $\frac{560}{11}$

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