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
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 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 .
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?
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