Homework III
Question 1: Options
Consider the following (incomplete) option price data for U&V Block
Inc., recorded at market close on April 5, 1996. The company pays no
dividends and all options listed below are European. Assume that the
riskless rate is 5% p.a.
54.75 50 Jul 18 ....... 12 4.50
54.75 55 Jul 72 ....... 85 ......
54.75 60 Jul 52 ....... 55 ......
1. Using a two-period binomial model, show that U&V's stock return
volatility is about 61%.
2. Compute all missing option prices in the above table.
3. Describe the composition of the portfolio that replicates the
strike 55 call, and show how it evolves across the binomial
tree. In particular, explain why the hedge ratio increases or
decreases depending on whether the stock went up or down in
the first period.
4. Repeat part 3 for a put at strike 55.
5. Compute the difference between the call and put hedge ratios
at each stage in the tree (i.e., compute delta for C less
delta for P without adjusting signs of the latter). It should
be constant. Explain your answer using put-call parity.
An Optional Question
6. A future is a (European-style) contract that simply pays the
difference between the stock price and an agreed upon strike
price to its owner at expiry. If the difference is positive,
the owner receives it; if it is negative, the owner pays the
difference. Price a strike 55 futures contract using the
binomial tree set up above. Explain the composition of the
portfolio that replicates the future at each stage in the
tree.
(Hint: Write out the cashflows from the future at maturity and
work backwards, replicating the future using a portfolio
of stocks and bonds)
Question 2: The Black-Scholes Model
The following price data are taken from The Wall Street Journal
dated April 17, 1996. Assume that the riskless borrowing/lending rate
is 5% p.a., that all options are European and that none of the
underlying stocks pay dividends.
Bank of New York
47 50 Jul 108 1.375 .... ....
BankAmerica
74.25 70 Jul 50 6.25 .... ....
Netscape
54.5 55 Jul 96 7.125 .... ....
1. Compute the missing prices.
2. Which of the two sectors is more volatile -- high-tech or
banking -- on the basis of the prices reported above?
3. Price July calls and puts at (i) strike 65 for Netscape;
(ii) strike 75 for BankAmerica; (iii) strike 55 for Bank of
New York. In each case, describe the portfolio that
replicates the call or the put. Use the Black-Scholes model
for computing the option prices.
4. Repeat (3), but now using a one period and a
two-period binomial model. How accurate, in your
assessment, are the binomial model prices compared to the
Black-Scholes prices?
You may want to use the option price calculator for solving this
problem.