- (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

Consequently, jus ad bellum comprises several conditions but most importantly: just cause and proportionality. This gives people a guide whether it’s lawful to enter a war or not. However, this is only one part of the theory of the just war. Nevertheless, it can be seen above that jus ad bellum can be debated throughout, showing that there is no definitive theory of a just war, as it is normatively theorised. The second section begins deciphering jus in bello or what actions can we classify as permissible in just wars (Begby et al (2006b), Page 323). First, it is never just to intentionally kill innocent people in wars, supported by Vittola’s first proposition. This is widely accepted as ‘all people have a right not to be killed’ and if a soldier does, they have violated that right and lost their right. This is further supported by “non-combatant immunity” (Frowe (2011), Page 151), which leads to the question of combatant qualification mentioned later in the essay. This is corroborated by the bombing of Nagasaki and Hiroshima, ending the Second World War, where millions were intently killed, just to secure the aim of war. However, sometimes civilians are accidentally killed through wars to achieve their goal of peace and security. This is supported by Vittola, who implies proportionality again to justify action: ‘care must be taken where evil doesn’t outweigh the possible benefits (Begby et al (2006b), Page 325).’ This is further supported by Frowe who explains it is lawful to unintentionally kill, whenever the combatant has full knowledge of his actions and seeks to complete his aim, but it would come at a cost. However, this does not hide the fact the unintended still killed innocent people, sho>

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