1. Explain why Mattel’s managers were able to slowly change decision making over time and what kinds of cognitive errors contributed 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) (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. 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.
2. (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 St − min St t∈[0,T ] t∈[0,T ]
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 fallsbyhalfd=1 =1.
(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.