Chapter Twenty Four
ANSWERS TO QUESTIONS
1. Writing index calls does not influence the value of the stock portfolio itself in any
way. However, writing calls can result in a financial liability if they close in-the-
money. Calls become in-the-money when the stock market rises, so presumably the
stock portfolio rose as well if the index calls became valuable. It is likely that the
“loss” on the calls is largely offset by a gain on the stock, so in practice the price
appreciation of the total portfolio is limited.
2. The lower the striking price the higher the premium income, but the greater the
likelihood of exercise. Logically, high striking prices yield little income, but are
seldom exercised. High striking prices also have low deltas, which means they
provide only a modest adjustment to position delta. In-the-money options, which
have high deltas, are powerful tools in altering the portfolio risk.
3. You could also write index or equity puts.
4. Individual equity options are useful in hedging company-specific risk. They are also
useful for generating income in small portfolios that are not sufficiently valuable to
provide the necessary collateral for option writing.
5. There is disagreement over which is riskier, writing individual calls or index calls. A
good argument can be made that writing individual equity calls is riskier, as a single
security sometimes has a spectacular one-day gain, rising by perhaps 50% because of
takeover news. The stock market has never experienced a gain of this size. Option
writers are hurt by rapidly rising prices, and because equity options can rise faster
than index options, a good argument can be made that the margin requirements should
be stiffer on individual options. (If the equity option is covered, of course, the risk of
a rapid price rise is eliminated.)
6. First, figure out the market value and beta of the portfolio to be hedged. Then select a
striking price for the put depending on the level of protection desired. Next, calculate
the delta of the desired option. (This may require calculating implied volatility first.)
Finally, calculate the hedge ratio as shown on page 547.
7. At-the-money futures puts and calls sell for the same price.
8. Writing calls is a bearish strategy when done in isolation. From the writer’s
perspective, gains accrue when the value of the underlying asset declines. Combining
this strategy with a long position in the underlying asset reduces the market exposure
in the underlying asset, because the short option will offset some gains that might be
experienced without the options.
Chapter Twenty Four
9. Option prices are not a linear function of the underlying stock price. The 0.5%
decline results in the option falling in value by $0.93, while a 0.5% rise results in an
option price rise of only $0.50. This difference is primarily because the option was
not initially at-the-money.
ANSWERS TO PROBLEMS
1. Student response.
2. The income shortfall in the example beginning on page 536 is $16,624. The AUG
315 calls have a premium of $1.75. The number of contracts is therefore
$1.75 x 100
3. As shown in the example, the market value of the stock portfolio is equal to about 36
OEX contracts. If fewer than 36 contracts were written, the maximum value of the
stock portfolio would be theoretically unlimited. Writing 36 or more contracts puts a
limit on the portfolio price appreciation, with the limit occurring at the option striking
The index can rise from 298.96 to 315.00 (a rise of 5.36%) before the options
become in-the-money. Therefore, the maximum value of the equity portfolio is
($996,975 x 1.0536) + $3,025 = $1,053,438
This assumes the option premium income was spent and that the stock portfolio
tracks the OEX index perfectly.
4. a. 10% rise
stock: $996,975 x 0.10 x 1.08 = 107,673 gain
calls: 100 x 56 x ($3.40 - 20.68) = 96,768 loss
net $10,905 gain
$996,975 + 10,905 + 3,025 = $1,010,905
b. 10% decline
stock: $996,975 x -0.10 x 1.08 = 107,673 loss
calls: 100 x 56 x ($3.40 - 0) = 19,040 gain
Chapter Twenty Four
net $88,633 loss
$996,975 - 88,633 + 3,025 = $911,367
5 - 7. Student response
8. These delta values assume the futures contract sells for 100.
NOV DEC MAR NOV DEC MAR
96 .941 .865 .754 -.055 -.127 -.226
98 .784 .711 .635 -.212 -.281 -.345
100 .503 .503 .501 -.493 -.489 -.479
102 .224 .298 .369 -.772 -.694 -.611
9. At-the-money futures puts and calls should sell for the same price. Therefore, the put
should also sell for $2.
10. With a striking price of 99, an underlying asset price of 100, a riskfree interest rate of
5%, a call premium of $2, and one-twelfth of a year until expiration, use the futures
put-call parity relationship:
P = C - e-Rt(F-K)
= $2 - e-(.05)(.0833)($100 - 99)
= $2.00 - $1.00 = $1.00
Portfolio size 0.035 x $223 million
11. HR = x beta = x 1.0 = 24.02
Futures size 1300 x 250
Assume a long position in 24 SPX futures contracts to remove the cash drag.
12. CFA Guideline Answer (reprinted with permission from the CFA Study Guide,
Association for Investment Management and Research, Charlottesville, VA. All
A. The problem here is to sell equities and reinvest the proceeds with the skilled
fixed-income manager, without changing the split between the existing
allocations to the two asset classes. The solution is to turn to derivative financial
instruments as the means to the end: selling enough of the existing fixed-income
exposure and bringing in enough of the equity exposure to get the desired mix
Chapter Twenty Four
The following are distinct derivatives strategies that the board could use to
increase the Fund’s allocation to the fixed-income manager without changing the
present fixed-income index/equity proportions.
Strategy 1. Use futures. One strategy would be to sell futures on a fixed-income
index and buy futures on an equity index.
Selling the futures eliminates the fixed-income index return and risk, while
keeping the skilled fixed-income manager’s extra return. By being long the equity
index, the portfolio obtains the index return and risk, keeping its exposure to the
Strategy 2. Use swaps. A second strategy would be to use over-the-counter
BI would swap a fixed-income index return for an equity index return in a notional
amount large enough to keep the skilled manager’s extra return while eliminating
the fixed-income market return and replacing it with the equity market return.
Strategy 3. Use option combinations. A third strategy would use put and call
options to create futures-like securities.
Buying put options and selling call options on a fixed-income index, while selling
put options and buying call options on a stock index, would achieve the same
result as the appropriate futures position.
B. The following are advantages and disadvantages of each strategy identified and
explained in Part A:
Strategy 1. Use futures
1. Futures contracts are liquid instruments.
2. Transaction costs are low.
3. Credit risk is negligible because the securities are marked to market daily.
1. If the holding period is long, rollover (transaction) costs are incurred.
2. Standard contract forms are limited, so contracts may not exist on the index or
Chapter Twenty Four
Strategy 2. Use swaps
1. Swaps can be tailored to fit the desired investment horizon, eliminating (or
reducing) rollover costs.
2. Swaps can be contracted for a specific index (like the performance
benchmark) even if there is no futures contract on it.
3. The desired adjustment goal can be accomplished through a single transaction.
1. A counterparty credit risk is created that can be much larger than with other
types of instruments.
2. Swap agreements are illiquid instruments, and disposals can be both difficult
3. Transaction costs are large because of typical “tailoring” of a given swap.
Strategy 3. Use option combinations
1. Transaction costs are low.
2. Credit risks are small.
1. Rollover(s) may be necessary.
2. The “right” option may not be available when needed or at all.
3. Holders may exercise the put option and end the hedge.
A generic disadvantage of any strategy is that returns are automatically eroded by
the costs of establishing and maintaining the strategies, of meeting margin
requirements, if any, and of unwinding them, if necessary.