An asset is worth 0 or 100 with equal probability. Each order comes from an informed trader with probability 0.4, who buys when the value is 100 and sells when it is 0, and otherwise from a noise trader who buys or sells with probability 1/2 each. What ask should a competitive risk neutral maker post?

An asset is worth 0 or 100 with equal probability. Each order comes from an informed trader with probability 0.4, who buys when the value is 100 and sells when it is 0, and otherwise from a noise trader who buys or sells with probability 1/2 each. What ask should a competitive risk neutral maker post?

Approach: A competitive quote breaks even on the trade it gets, so set the ask equal to the expected value conditional on a buy arriving, and update with Bayes rule.

70. The ask must equal E[V | buy]. A buy arrives with probability 0.5 * (0.4 + 0.6 * 0.5) = 0.35 jointly with V = 100, and with probability 0.5 * 0.6 * 0.5 = 0.15 jointly with V = 0, so total 0.5. Bayes rule gives P(V = 100 | buy) = 0.35/0.5 = 0.7 and the ask is 70. By symmetry the bid is 30, so the spread is 40, which is pure adverse selection cost and equals the prior range 100 times the informed share 0.4. The zero profit condition holds trade by trade, with the loss to informed traders offset by the profit from noise traders. Pushing the informed share to 1 widens the quote to 0 bid at 100 and the market shuts.

Follow-up: After one buy at 70, what are the maker's next bid and ask if the informed share stays at 0.4?

Key concepts: adverse selection, bayes rule, informed trading, zero profit condition.