Chapter Twenty Two
ANSWERS TO QUESTIONS
1. The objective is the temporary removal of some or all the market risk associated with
a portfolio. Portfolio protection techniques are generally more economic in terms of
commissions and managerial time than the sale and eventual replacement of portfolio
components.
2. You can “fine-tune” the likely price behavior of a portfolio by altering its market risk
(as measured by the position delta). Writing options will, in addition, alter the income
stream the portfolio produces.
3. Delta indicates the quantity of a particular option necessary to replicate the underlying
asset. There is some quantity of call options, for instance, which (for modest
movements in the underlying asset price) behaves like the underlying asset itself.
4. Buying a protective put to protect the value of a long stock position.
5. The value of the stock index is merely a marker representing the current value of the
underlying basket of stocks. While delivery conceivably could occur with a stock
index, it is not necessary. Because of the guaranty provision of the contract provided
by the Options Clearing Corporation, index option users can settle their positions in
cash.
6. Protective puts provide protection against large price declines, whereas covered calls
provide only limited downside protection. Covered calls bring in the option premium,
while the protective put requires a cash outlay.
7. An S&P500 stock index futures contract has a delta of 1.0 because the S&P is the
market, practically speaking. Short futures contracts add negative deltas to a
portfolio.
8. Yes. There is likely to be substantial unsystematic risk in the five-security portfolio,
however, which the futures contract could not hedge.
9. The higher the portfolio beta the greater the anticipated price volatility of the
portfolio, and so the greater the number of options or futures that are needed to hedge
the portfolio.
10. As time passes, the value of a stock index futures contract will decline even if the
underlying stock index does not change. This is because of the time value of money.
The short hedger benefits from a price decline, so the declining basis works to their
benefit.
Chapter Twenty Two
11. A put has a negative delta; so does a short position in stock. The right quantity of
shares sold short will behave the same as a particular put. A put contract with a delta
of -0.450, for instance, behaves the same as 45 shares sold short.
12. Dynamic hedging is a process in which the position delta of a portfolio is routinely
adjusted to return it to some target level. The passage of time and the changing value
of the underlying asset are two primary reasons why the portfolio needs periodic
revision to stay on track.
13. Insurance connotes a contractual arrangement. In the market crash of 1987 the market
did not behave as arbitrage arguments suggest it was “supposed” to. Insurance
provides protection with little risk of insurance company default. Investment
portfolios based on delta and the behavior of market prices are more risky than a
contractual insurance contract. Some people who thought their portfolios were
insured found that they were not.
14. No; this is commonly done.
15. Hedging involves reducing or eliminating risk. Once the risk is removed, any further
hedging amounts to speculation.
ANSWERS TO PROBLEMS
1. (10000 x 1.0) + (50 x 100 x -0.181) = 9095
2. 10000 + (N x 100 x -.181) = 0
N = 552 contracts
552 contracts x $0.50 x 100 = $27,600
3. 10000 - (100 x .819 x 100) = 1810
4. HR 
Portfolio Value 1 $2,500,000
1
 
x1.12 x
 386.34
Contract Value 
325 x$100
.223
Round to 386 contracts
1. Calculate the new hedge ratio:
$2,450,000
1

 344.61
325x$100  .245
Chapter Twenty Two
 Sell 42 puts
6.
$2,500,000
x1.12  31.32
357.60 x$250
 Sell 31 or 32 contracts
7 - 9. Student response
10. a.
b.
c.
d.
The value declines to converge on the intrinsic value.
The value declines to converge on zero.
The value declines to converge on the intrinsic value.
The value declines to converge on zero.
11. The call delta equals one plus the put delta. Stated another way, the absolute values
of the put and call deltas sum to 1.0.