Solutions FINAL EXAM 2002 SPRING Sridhar Seshadri B

Transcription

1 Solutions FINAL EXAM 22 SPRING Sridhar Seshadri B Answer all questions. Answer questions and 2 before attempting question 3. The exam is closed book and closed notes (except for 4 pages of notes). Please write clearly, be brief, and show all steps for calculations. If I have asked for an explation no credit will be given for that part unless the explation is correct. Question carries 65 points, question 2 has 35 points. Question 3 has bonus points.. Consider a binomial stock price model with parameters u =.3, d =.77, S =, and T = 3. (The figures for u and d are rounded; assume still that u = /d ). The interest rate is r =.2. a. Compute the price process of an American put option with strike price e = 9. What is the optimal exercising time? (2 pts.) Most students got this correct. If t= price incorrect took off 2 points. Stock Price Process t= t= t=2 t= American Put Price Process r.2 q u.3 u t= t= t=2 t= Early exercise optimal at time =, price = 7.7

2 b. Compute the price process of an up and out call option on the same stock as above with barrier k = and e = 3. This option is similar to a usual call option except that if the historical maximum price of the stock is larger than then the up and out call pays nothing at expiry. Give your result in a tree form. Verify that the time zero price can be written as the expected discounted value of a contingent claim at time 3 (in other words verify that the pricing formula, V = E Q [X/B 3 ] for some appropriate contingent claim X. (5 pts.) t= t= t=2 t= If did not get the out part correct took off 5 points. X = 4.7 w.p. 2 x q (-q)^2 and w.p. (-q)^3 Thus time zero price = (4.7 x 2 x.83 x x.7 3 )/(.2) 3 Many students lost one point because they used 3 q (-q)^2 but the option pays only on two paths, and is out on one of the three paths (at t=). b.is the American put option in part (a) attaible? If so compute the replicating trading strategy. Carefully show the value of the portfolio corresponding to this TS, the gain, the discounted value of the portfolio and the discounted gain all on a tree. What do you conclude from the tree of discounted gains? (2 pts.) Each TS was worth 5 points (there is one TS at t=, one at t=, and two at t=2). H Time = H Cost H H Cost.2553 Time= Stock Price = 3 H Time=2 H Stock Price = Cost

3 The gain calculations were worth 5 points. Gain Disctd Gain Based on above I believe that a risk averse person should sell the option at time and liquidate the portfolio too. 2. In this model we have a single risky security with prices (not discounted) shown below. The interest rate is assumed to be % for the first period, but % if S =8.8 and 9% if S =3.2. Consider the chooser option with t =, T = 2, and e =, i.e., at time the buyer can decide whether the option will be a call or a put option with expiry 2 and strike price

4 rnpm rnpm t= t= put q.5 q2.5.3 t= t=2 t= t= t= 2.74 chooser call a. What decision should the buyer make at time if S = 8.8? What should be the decision if S = 3.2? (5 pts.) Each rnpm carried 3 points. There are 3 rnpm calculations. Keep put if price falls else keep call. b. Compute the price process for the above chooser option (time, and 2 prices). Give your results in a tree form. Is this price process a Q-martingale, explain? (5 pts.) It (the discounted price) is a Q-MTG as the price process is that of a t=3 CC. c. Calculate and show the time zero, time one and time two future and forward price of the stock for delivery at time 2. Is any of the prices a Q-martingale (explain or show briefly why or why not)? (5 pts.) future forward Future price process is a Q-MTG. (one point) Forward price process is not a Q-MTG. (one point) 4

5 3. Short questions ( points) a. Use the tree for question 2. A firm is thinking about issuing a convertible bond. The bond will yield an interest rate equal to that period s interest rate. The interest is paid each period. The face value of the bond at time zero equals $. The holder of this bond has the option to either exchange it for one unit of the stock at time or not to exchange it (thus keep the bond until time period 2). What should be the fair price of the convertible bond at time zero? We keep bond if price = 8.8 else convert. Payoffs are: $ if stock price = 8.8 and $4.2 if stock price = $3.2. Discounted value at t= is: (.5* +.5*4.2)/. = $.45 This is just like a chooser pricing method. Keep the more valuable object! b. Use the tree for question 2. What is the fair time-zero price of the contingent claim that pays $ at time 2? What is the relationship between this price and the forward price for time-2 delivery of the stock at time zero? Explain. The t= price = $.5 + $.5 = Now $/ = $.998 which is the forward price. The reason this is so either one can buy the stock at t= or arrange for getting the forward price with probability one at t=3 (for which one pays the forward price multiplied by.83345) and exchange it for the stock at t=2. Both the strategies are equivalent and should have the same price. The explation was worth 2 points. Happy Summer!! 5