loan to fund a project at the following terms. Johnson fears a possible rise in the LIBOR rate by
December and wants to use the December Eurodollar futures contract to hedge this risk. The
contract expires December 20, 1999, has a U.S. $1 million contract size, and a discount yield of 7.3
Johnson will ignore the cash flow implications of marking to market, initial margin requirements,
and any timing mismatch between exchange-traded futures contract cash flows and the interest
payments due in March.
September 20, 1999 December 20, 1999 March 20, 2000
ā¢ Borrow $100 million at ā¢ Pay interest for first ā¢ Pay back principal
September 20 LIBOR1200 three months plus interest
basis points (BPS) ā¢ Roll loan over at
ā¢ September 20 LIBOR 5 7% December 20
Loan Initiated First Loan Payment (9%) Second Payment
and Futures Contract Expires and Principal
a. Formulate Johnsonā™s September 20 floating-to-fixed-rate strategy using the Eurodollar futures
contracts discussed in the preceding text. Show that this strategy would result in a fixed-rate loan,
assuming an increase in the LIBOR rate to 7.8 percent by December 20, which remains at 7.8
percent through March 20. Show all calculations.
Johnson is considering a 12-month loan as an alternative. This approach will result in two additional
uncertain cash flows, as follows.
Loan Initiated First Payment (9%) Second Payment Third Payment Fourth Payment
9/20/99 12/20/99 3/20/00 6/20/00 9/20/00
b. Describe the strip hedge that Johnson could use and explain how it hedges the 12-month loan
(specify number of contracts). No calculations needed.
05-152 Webpages pp2 12/5/05 9:14 AM Page 44
44 Web Problems
1. CFA Examination Level III
You have decided to buy protective put options to protect the U.S. stock holdings of one of Global
Advisers Companyā™s (GAC) portfolios from a potential price decline over the next three months.
You have researched the stock index options available in the United States and have assembled the
Stock Current Strike Trading
Index Index Underlying Value Price Put Volume
Option Value of One Put of Put Premium of Puts
S&P 100 $365.00 $100 times index 365 $10.25 10,000
S&P 500 390.00 $100 times index 390 11.00 4,000
NYSE 215.00 $100 times index 215 6.25 1,000
(For each stock index option, the total cost of one put is the put premium times 100.)
Beta vs. S&P 500 Correlation with Portfolio
Portfolio 1.05 1.00
S&P 100 0.95 0.86
S&P 500 1.00 0.95
NYSE 1.03 0.91
a. Using all relevant data from the preceding tables, calculate for each stock index option both the
number and cost of puts required to protect a $7,761,700 diversified equity portfolio from loss.
Show all calculations.
b. Recommend and justify which stock index option to use to hedge the portfolio, including refer-
ence to two relevant factors other than cost.
You know that it is very unlikely that the current stock index values will be exactly the same as the
put strike prices at the time you make your investment decision.
c. Explain the importance of the relationship between the strike price of the puts and the current
index values as it affects your investment decision.
d. Explain how an option pricing model may help you make an investment decision in this situation.
2. CFA Examination Level II
A stock index is currently trading at 50.00. The annual index standard deviation is 20 percent. Paul
Tripp, CFA, wants to value two-year index options using the binomial model. To correctly value the
options, he needs the following formulas. The annual risk-free interest rate is 6 percent. Assume no
dividends are paid on any of the underlying securities in the index.
FORMULAS FOR OPTION VALUATION
e rDt 2 D
U 5 es"Dt D5 Pu 5
U 5 1.2214 erDt 5 1.06184
U 5 up movement factor
D 5 down movement factor
Pu 5 probability of an upward price movement
05-152 Webpages pp2 12/5/05 9:14 AM Page 45
5.00 Percent 6.00 Percent 7.00 Percent
Period 1 0.95123 0.94176 0.93239
Period 2 0.90484 0.88692 0.86936
Period 3 0.86071 0.83527 0.81058
a. Construct a two-period binomial price tree for the stock index.
b. Calculate the value of a European-style index call option with an exercise price of 60.00.
c. Calculate the value of a European-style index put option with an exercise price of 60.00.
3. Suppose the current contract price of a futures contract on Commodity Z is $46.50 and the expira-
tion date is in exactly six months (i.e., T 5 0.5). The annualized risk-free rate over this period is 5.45
percent and the volatility of futures price movement is 23 percent, which is equal to that of the un-
a. Calculate the values for both a call option and a put option for this futures contract, assuming
both have an exercise price of $46.50 and a six-month expiration date.
b. Suppose the market prices for these contracts agree with the values you computed in Part a. You
decide to buy the call option and sell the put option. What sort of position have you just created?
Under what circumstances (i.e., for what view of subsequent market conditions) would it make
sense for an investor to create such a position?
4. CFA Examination Level III
Ken Webster manages a $100 million equity portfolio benchmarked to the S&P 500 index. Over the
past two years, the S&P 500 index has appreciated 60 percent. Webster believes the market is overval-
ued when measured by several traditional fundamental/economic indicators. He is concerned about
maintaining the excellent gains the portfolio has experienced in the past two years but recognizes that
the S&P index could still move above its current 668 level. Webster is considering the following option
ā¢ Protection for the portfolio can be attained by purchasing an S&P 500 index put with a strike
price of 665 (just out of the money).
ā¢ The put can be financed by selling two 675 calls (farther out of the money, for every put purchased).
ā¢ Because the combined delta of the two calls is less than 1 (that is, 2 3 0.36 5 0.72), the options
will not lose more than the underlying portfolio advances.
The information in the following table describes the two options used to create the collar.
OPTIONS TO CREATE THE COLLAR
Characteristics 675 Call 665 Put
Option price $4.30 $8.05
Option implied volatility 11.00% 14.00%
Optionā™s delta 0.36 0.44
Contracts needed for collar 602 301
ā¢ Ignore transaction costs
ā¢ S&P 500 historical 30-day volatility 5 12.00%
ā¢ Time to option expiration 5 30 days
05-152 Webpages pp2 12/5/05 9:14 AM Page 46
46 Web Problems
a. Describe the potential returns of the combined portfolio (the underlying portfolio plus the option
collar) if after 30 days the S&P 500 index has (1) risen approximately 5 percent to 701.00, (2)
remained at 668 (no change), and (3) declined by approximately 5 percent to 635.
b. Discuss the effect on the hedge ratio (delta) of each option as the S&P 500 approaches the level
for each of the potential outcomes listed in Part a.
c. Evaluate the pricing of each of the following in relation to the volatility data provided: (1) the
put, (2) the call, and (3) the collar.
d. Explain the term wasting asset in the context of the suggested collar strategy and discuss its
effect on Websterā™s management of the portfolio.
CFA 5. CFA Examination Level II (2003)
Michael Weber, CFA, is analyzing several aspects of option valuation, including the determinants of
the value of an option, the characteristics of various models used to value options, and the potential
for divergence of calculated option values from observed market prices.
A. State, and justify with one reason for each case, the expected effect on the value of a call option
on common stock if each of the following changes occurs:
i. The volatility of the underlying stock price decreases
ii. The time to expiration of the option increases
Using the Black-Scholes option-pricing model, Weber calculates the price of a three-month call op-
tion and notices the optionā™s calculated value is different from its market price. A colleague verifies
that Weberā™s methodology and results are correct.
B. With respect to Weberā™s use of the Black-Scholes option-pricing model, and given that his
methodology and results are correct.
i. Discuss one reason why the calculated value of an out-of-the-money European option may
differ from that same optionā™s market price.
ii. Discuss one reason why the calculated value of an American option may differ from that
same optionā™s market price.
1. If the fixed rate on a five-year, plain vanilla swap is currently 8 percent, what would happen if
you (a) bought a five-year cap agreement with an exercise rate of 7 percent and (b) sold a five-year,
7 percent floor agreement? Use the concept of cap-floor-swap parity to describe the kind of position
you have created and discuss whether or not a front-end cash payment would be necessary and
whether you or your counterparty would receive it.
2. Corporation XYZ seeks USD 100 million, five-year fixed-rate funding. The firm is confident that it
can issue a 61ā„2 percent coupon bond (semiannual payments) at par value. Since the five-year, on-the-
run U.S. Treasury issue yields 6.00 percent, this funding could be attained at 50 basis points over
Treasuriesā”a reasonable spread given XYZā™s strong credit rating. The corporate treasurer would
like to explore the possibility of issuing a floating-rate note (FRN), possibly a structured note, and
use the interest rate swap market to create synthetic fixed-rate funding.
As the bank relationship officer working with the treasurer, you check with your Capital Markets
Group to determine that in the current market the following FRNs could be launched for XYZ at par
Type Reset Formula LIBOR = 5.75%: Initial Coupon
LIBOR 1 0.10%
āStraight floaterā 5.85%
12.75% 2 LIBOR
āBull floaterā 7.00
(2 3 LIBOR) 2 6.40%
āBear floaterā 5.10
05-152 Webpages pp2 12/5/05 9:14 AM Page 47
The FRNs are each based on six-month USD LIBOR and make semiannual coupon payments in
arrears. Assume that the flotation costs for the fixed-rate bond and various FRNs are the same (and
can therefore be ignored in the comparison). Also, assume that all rates are quoted on a semiannual
bond basis so that no day-count conversions need be made.
You also check with the Derivatives Group to ascertain the rates at which XYZ could execute
five-year, semiannual settlement plain vanilla swaps:
Bankā™s Bid Bankā™s Offer
TSY + 34 BP TSY + 38 BP
where TSY = 6.00%
a. What specific swap transactions are needed to transform each of the FRNs into a synthetic fixed
b. What synthetic fixed rate can be attained by each of the structures? For which possible levels of
LIBOR is a fixed-rate attained?
c. What other derivative instrument(s) would be needed to obtain a truly fixed rate?