financial economic question

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 , V L = 1 , or some middle
value V M . 44

Let α = 1 of the traders be
informed insiders, while the remaining 1 − α = 2 are uninformed noise traders.
33

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 uncondi- tional

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 .

payoff when V = V M and use this to find the lowest value of V M at which the

(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 uncondi-

tional 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 .

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?