AS.640.440 Financial Economics
1. (40) Consider modifying the Glosten and Milgrom (1985) model to allow for the possibility that sell orders need not come from traders who actually hold the stock, and thus may represent short selling. Let the asset’s true value be VH = 1 or VH = 0 with equal probability, with a proportion α = 4/1 of traders being informed insiders who receive a perfect signal of the asset’s value and the remaining 1 — α = 4/3 uninformed liquidity traders who receive no such signal.
We will use a variant of the model that accounts for short selling, pioneered by Diamond and Verrecchia (1987), by assuming that of all traders who wish to submit a sell order, only a fraction h = 3/1 of them hold the stock and can proceed without complications. The remaining 1 — h = 3/2 do not own any shares and thus would need to short sell, potentially facing regulatory obstacles preventing them from doing so. No similar restrictions exist for buyers.
(a) (5) Draw the game tree, leaving undetermined for now what potential short sellers would do.
(b) (5) Suppose that short selling is completely unrestricted, so that both informed and uninformed traders will do so. Without doing the full calculation, find the bid-ask spread.
(c) (5) Now suppose that short selling is restricted but not prohibited, and that the cost of doing so is such that only informed traders are willing to, with uninformed traders who would otherwise be short selling simply declining to trade. Diamond and Verrecchia consider this, and not unrestricted short selling, to be the normal state of the market. Find the conditional probabilities of buy B and sell S orders given that the true value is high VH and low VL.
(d) (5) Using Bayes’ rule, calculate the posterior probabilities of high VH and low VL values upon observing buy B and sell S orders.
(e) (5) Find the bid-ask spread when short sales are restricted but not prohibited. How does this compare to the spread when short sales are completely unrestricted?
(f) (5) Now suppose that short selling is completely banned. Find the conditional probabilities of either type of order given each possible stock value.
(g) (5) Calculate the posterior probabilities of each value upon observing each type of order under the ban.
(h) (5) Find the bid-ask spread under a short sale ban. How does this compare to the other cases and what does this say about the effect of such a ban on liquidity?
2. (35) 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.
Consider pricing this option using the binomial model. Let the current price of stock in Hindsight Inc. be S0 = 108 and consider an option maturing in n = T = 3 periods, so that Δt = 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 triples u = 3, or falls to a third d = u/1 = 3/1.
(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 final price but yield different payoff.
(c) (5) Show that there are also two price paths that end at different final prices yet yield the same payoff.
(d) (10) Draw the stock price tree. Note that there are 2n = 8 possible price paths, in contrast to the n + 1 = 4 different final prices, so this should be drawn as a fully branching tree where the number of nodes doubles with each step.
(e) (10) Using backward induction, calculate the initial price of this option.
3. (25) Let Rocksteady Industries be an all-equity firm in a Modigliani-Miller world with 1000 shares outstanding and a cost of equity of rE = 0.06. Suppose that it has announced a dividend of 9 per share equal to its expected annual earnings.
(a) (5) Calculate the stock price and total value of the firm.
(b) (5) Suppose that this year, Rocksteady realizes earnings of only 6,000, which falls short of expectations, but does not expect this to signal lower earnings in future years. 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?
(c) (5) Suppose that Debbie owns 300 shares and disagrees with the firm’s policy of maintaining a dividend of 9, 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 this year, Rocksteady instead realizes higher than expected earnings of 12, 000, but again does not expect this to signal higher earnings in the future. It decides to keep the dividend at 9 and use the remainder to buy back shares. How many shares will be repurchased and at what price?
(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?