570 Part Six Options
bre44380_ch21_547-572.indd 570 10/05/15 12:53 PM
d. Use your answer to part (c) to calculate the option delta (1) today; (2) next period if the
stock price rises; and (3) next period if the stock price falls. Show at each point how you
would replicate a call option with a levered investment in the company’s stock.- Option delta Suppose you construct an option hedge by buying a levered position in delta
shares of stock and selling one call option. As the share price changes, the option delta
changes, and you will need to adjust your hedge. You can minimize the cost of adjustments if
changes in the stock price have only a small effect on the option delta. Construct an example
to show whether the option delta is likely to vary more if you hedge with an in-the-money
option, an at-the-money option, or an out-of-the-money option. - Option risk
a. In Section 21-3 we calculated the risk (beta) of a six-month call option on Google stock
with an exercise price of $530. Now repeat the exercise for a similar option with an exer-
cise price of $450. Does the risk rise or fall as the exercise price is reduced?
b. Now calculate the risk of a one-year call on Google stock with an exercise price of $530.
Does the risk rise or fall as the maturity of the option lengthens? - Option exercise Other things equal, which of these American options are you most likely
to want to exercise early?
a. A put option on a stock with a large dividend or a call on the same stock.
b. A put option on a stock that is selling below exercise price or a call on the same stock.
c. A put option when the interest rate is high or the same put option when the interest rate is low.
Illustrate your answer with examples. - Option exercise Is it better to exercise a call option on the with-dividend date or on the
ex-dividend date? How about a put option? Explain. - Warrants Use the Black–Scholes program from the Beyond the Page feature to value the
Owens Corning warrants described in Section 21-4. The standard deviation of Owens Corn-
ing stock was 41% a year and the interest rate when the warrants were issued was 5%. Owens
Corning did not pay a dividend. Ignore the problem of dilution. - Pension fund insurance Use the Black–Scholes program to estimate how much you should
be prepared to pay to insure the value of your pension fund portfolio for the coming year.
Make reasonable assumptions about the volatility of the market and use current interest rates.
Remember to subtract the present value of likely dividend payments from the current level of
the market index.
CHALLENGE- Option delta Use the put-call parity formula (see Section 20-2) and the one-period bino-
mial model to show that the option delta for a put option is equal to the option delta for a call
option minus 1. - Option delta Show how the option delta changes as the stock price rises relative to the
exercise price. Explain intuitively why this is the case. (What happens to the option delta if
the exercise price of an option is zero? What happens if the exercise price becomes indefi-
nitely large?) - Dividends Your company has just awarded you a generous stock option scheme. You sus-
pect that the board will either decide to increase the dividend or announce a stock repurchase
program. Which do you secretly hope they will decide? Explain. (You may find it helpful to
refer back to Chapter 16.) - Option risk Calculate and compare the risk (betas) of the following investments: (a) a share
of Google stock; (b) a one-year call option on Google; (c) a one-year put option; (d) a portfolio
consisting of a share of Google stock and a one-year put option; (e) a portfolio consisting of a
share of Google stock, a one-year put option, and the sale of a one-year call. In each case assume
that the exercise price of the option is $530, which is also the current price of Google stock.
BEYOND THE PAGEmhhe.com/brealey12eTry it!
The Black-Scholes
model