- (35) 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 α =
1
3
of the traders be informed insiders, while the remaining 1 − α =
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) (5) Draw the tree diagram, leaving uncertain the action of informed traders when V = V M .
(b) (5) 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. (3) Calculate the conditional probabilities of a buy order at each value V can take and the unconditional probability of a buy.
ii. (3) 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. (4) 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) (10) Now suppose the informed traders always sell when V = V M .
i. (3) Calculate the conditional probabilities of a sell order at each value V can take and the unconditional probability of a sell.
ii. (3) 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. (4) 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) (5) What happens if V M satisfies neither of the bounds you found above? - (40) Consider a special type of lookback option whose payoff at maturity is given by the difference between
the maximum and minimum stock prices attained over the life of the option.
LT = max
t∈[0,T]
St − min
t∈[0,T]
St
Consider pricing this option using the binomial model. Let the current price of stock in Hindsight Inc. be
S0 = 216 and consider an option maturing in n = T = 3 periods, so that ∆t = 1 per period. Suppose for the
sake of simplicity that the risk free rate is r = 0, and that each period the stock price either doubles u = 2, or
falls by half d =
1
u =
1
2
.
(a) (5) Calculate the risk neutral probability of an uptick p.
(b) (5) An important property of lookback options not shared by vanilla calls and puts is that they are path
dependent. That is, their payoffs depend not only on the final stock price but on how the price moved
over the life of the option.
Show that this option is path dependent, that is, find two price paths that end at the same price.
(c) (15) Draw the stock price tree. Note that there are 2n = 8 possible price paths, in contrast to the n+1 = 4
different final prices.
(d) (15) Using backward induction, calculate the initial price of this option. - (25) Let Rocksteady Industries be an all-equity firm in a Modigliani-Miller world with 500 shares outstanding
and a cost of equity of rE = 0.15. Suppose that it has announced a dividend of 12 per share equal to its
expected annual earnings.
1
(a) (5) Calculate the stock price and total value of the firm.
(b) (5) Suppose that Rocksteady realizes lower than expected of earnings of 3, 000, but does not expect this
to impact its future cash flows. It does not want to reduce its dividend, and decides to issue new shares
to make up the difference. How many new shares will be issued and at what price? What is the payoff to
existing shareholders?
(c) (5) Suppose that Debbie owns 100 shares and disagrees with the firm’s policy of maintaining a dividend
of 12, preferring instead to receive a lower payout than have her shares diluted. What could she do to
achieve her desired outcome?
(d) (5) Now suppose that Rocksteady instead realizes higher than expected earnings of 9, 000. It decides to
keep the dividend at 12 and use the remainder to buy back shares. How many shares will be repurchased
and at what price? What is the payoff to the remaining shareholders?
(e) (5) Once again, Debbie is dissatisfied by the firm’s policy and would have preferred a proportionally
higher dividend to the buyback. How many shares should she sell back to the firm to replicate her desired
outcome?
Sample Solution
