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?