1. Consider a variation of the Glosten-Milgrom sequential trade model where the asset's value V can take

three values. Suppose that the true value of stock in Trident Corporation can be, with equal probability, either

V H = 3/4 , V L = 1/4 , or some middle value V ^M .

Let alpha =1/3 of the traders be informed insiders, while the remaining and 1-alpha= 2/3 are uninformed noise traders. Assume as always that informed traders always buy when V = V H and sell when V = V L , while uninformed traders buy or sell with equal probability. The focus of this problem is the traders' behavior when V = V ^M.

(a) Draw the tree diagram, leaving uncertain the action of informed traders when V = V ^M .

(b) Show that there is no value of V M for which informed traders randomize between buying and selling.

(c) (10) Suppose that informed traders always buy when V = V ^M .

i. Calculate the conditional probabilities of a buy order at each value V can take and the unconditional probability of a buy.

ii. Using Bayes' rule, calculate the posterior probabilities of V taking on each value conditional on a buy, and compute the ask price as a function of V M .

iii. Find the informed trader's payoff when V = V M and use this to find the lowest value of V M at

which the trader is willing to buy.

(d) Now suppose the informed traders always sell when V = V M .

i. Calculate the conditional probabilities of a sell order at each value V can take and the unconditional probability of a sell.

ii. Using Bayes' rule, calculate the posterior probabilities of V taking on each value conditional on a sell, and compute the bid price as a function of V M .

iii. Find the informed trader's payoff when V = V M and use this to find the highest value of V M at

which the trader is willing to sell.

(e) What happens if V M satisfies neither of the bounds you found above?