Break-even spread on a two-point value

Question

An asset is worth either VH = 120 or VL = 80, each with probability 	frac12. A fraction \alpha = 0.25 of arriving traders are informed (they know V and always trade the profitable side); the other 75\% are noise traders who buy or sell 50/50. You post a bid and an ask before seeing who arrives, and competition drives your expected profit on each side to zero. What bid and ask do you post, and what is the resulting spread?

Accepted Answer

This is Glosten-Milgrom adverse selection. Zero expected profit on each side means your quote equals the value conditional on the trade direction: set the ask = E[V \mid 	ext{buy}] and the bid = E[V \mid 	ext{sell}]. First, the chance of a buy under each true value: - If V = VH: all informed buy, half the noise buys, so P(	ext{buy}\mid VH) = \alpha + (1-\alpha)	frac12 = 	frac{1+\alpha}{2}. - If V = VL: no informed buys, half the noise buys, so P(	ext{buy}\mid VL) = (1-\alpha)	frac12 = 	frac{1-\alpha}{2}. Write \mu = 100 (the midpoint) and \delta = 20 (the half-gap), so VH = \mu + \del…

Break-even spread on a two-point value

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