Where does the price go after two buys in a row?

Question

An asset is worth VH = 110 or VL = 90, equally likely, midpoint \mu = 100, half-gap \delta = 10. A fraction \alpha = 0.4 of traders are informed; the rest are 50/50 noise. You are a competitive maker who re-quotes after every trade. Two buy orders arrive back to back. Trace your fair value after each. Does the second buy move it as much as the first?

Accepted Answer

Each trade is a Bayesian update on P(VH), and your fair value is E[V] = 90 + 20\,P(VH) (since E[V] = P(VH)\cdot 110 + (1-P(VH))\cdot 90). The likelihood ratio for a buy is \frac{P(	ext{buy}\mid VH)}{P(	ext{buy}\mid VL)} = \frac{(1+\alpha)/2}{(1-\alpha)/2} = \frac{1.4}{0.6} = \frac{7}{3}. Start: P(VH) = 0.5, prior odds 1:1, fair value 100. After the first buy: odds become 1 	imes 	frac{7}{3} = 	frac{7}{3}, so P(VH) = 	frac{7}{7+3} = 0.7. Fair value = 90 + 20(0.7) = 104. (This matches the \mu + \alpha\delta = 100 + 4 shortcut, which holds only from the 50/50 start.) After the second buy: …

Where does the price go after two buys in a row?

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