Module 3 - Homework II

Problem 1
Suppose that on March 31, you observe the following data on American options that expire on July 19 of the same year. All questions for this problem pertain to the options whose prices are given below.
          119.5      125     Jul     58   3.25      114     4.50
a) If you owned the call, would you exercise it today?
b) What would your profits be, if you owned a put and exercised it today?
c) What would your profits be, if you bought the put today and immediately exercised it?
d) Could the prices given above prevail in a frictionless and efficient market? Which option's price should adjust and in what direction should it adjust as arbitrageurs move in?


Problem 2
Suppose that on March 31, you observe the following (incomplete) data on prices of European options on the common stock of a non-dividend paying firm Netdrape Inc. The options expire on July 19 of the same year. Assume that you could borrow or lend risklessly at 5% p.a.
          48.5   45   Jul    8   8.50   0    ......
          48.5   50   Jul   46   6.25   0    ......
          48.5   55   Jul   63   ....   1    9.67
a) Complete the table, i.e., provide the missing prices.
b) Suppose that one of your customers wants to buy a July straddle (definition from class notes) on Netdrape at strike 50 from you. What price would you quote your customer and how would you hedge the resulting short straddle position, if strike 50 calls are traded, but not strike 50 puts?
c) Part (c) is optional, but you are strongly encouraged to take a shot!. The answer involves some trial and error experimentation that is best done on a spreadsheet. This is obviously not something I will give in an exam, but attempting it will improve your understanding of the binomial model.
     (c1) Assume that the price of Netdrape Inc. follows a binomial
	  process, and can take one of two values u*S or d*S on July
          19, where u = 1/d and S is the current market price of
	  Netdrape's stock. What value of u would produce a value of
	  $ 6.25 for the strike 50 call option? 
     (c2) Based on your answer to (c1), what is Netdrape's annual
	  return volatility? 
          (Hint: Once you solve for u, you also have the volatility)
Problem 3
Suppose that the shares of Australia Offline Inc. (AOL) have an annual return volatility of 32% and that they trade at a price of $ 120. Suppose that you can borrow or lend risklessly at the rate of 12% p.a. Your job (at Leeson Inc., an investment bank ) is to price European options with 91 days to maturity on AOL stock, using a single step binomial tree of the sort discussed in the class on 4 /10/96. AOL does not pay dividends.
     (a) Price strike 125 call and put options using the binomial tree.

     (b) Describe the composition of the portfolios that replicate the
         options in (a), and verify that these actually do the
         replication accurately.

     (c) Suppose that the maturity of the options in (a) increases to
	 120 days. How would your answers to (a) change and why?
	 Qualitatively discuss, and then recompute your answers to (a)
	 & (b) to verify that your reasoning is correct.

     (d) Suppose that AOL unexpectedly decides to pay a dividend of 
         $ 2.25 per share 45 days from now. What would be the effect
	 on your answers in (a)? Provide a qualitative discussion, no
         numberwork is necessary. 

     (e) Suppose that you are asked to sell a 125 strike straddle with
	 a 91 day maturity. What would be price of the straddle and
	 how would you hedge it? Can you use the binomial model to
	 price the straddle without pricing the individual options
	 that comprise the straddle?