HullFund8eCh21ProblemSolutions

CHAPTER 21

Interest Rate Options

Practice Questions

Problem 21.8.

A bank uses Black’s model to price European bond options. Suppose that an implied price volatility for a 5-year option on a bond maturing in 10 years is used to price a 9-year option on the bond. Would you expect the resultant price to be too high or too low? Explain your answer.

The forward price of a bond that will have one year remaining at the maturity of the forward contract has a lower volatility than that for a bond that will have five years remaining at the maturity of the forward contract. The volatility used to price a nine-year option on a ten-year bond should therefore be less than that used to price a five-year option on a ten-year bond. Using the volatility backed out from the five-year option to price the nine-year option is therefore likely to produce a price that is too high.

Problem 21.9.

Consider a four-year European call option on a bond that will mature in five years. The five-year bond price is $105, the price of a four-year bond with the same coupon as the five-year bond is $102, the strike price of the option is $100, the four-year risk-free interest rate is 10% per annum (continuously compounded), and the volatility of the forward price of the bond underlying the option is 2% per annum. What is the present value of the principal in the four-year bond? What is the present value of the coupons in the four-year bond? What is the forward price of the bond underlying the option? What is the value of the option?

The present value of the principal in the four year bond is 40110067032e -?.=.. The present value of the coupons is, therefore, 1026703234968-.=.. The coupons on the four-year bond are the income on the five-year bond during the life of the option. This means that the forward price of the bond underlying the option is

014(10534968)104475e .?-.=.

The parameters in Black’s model are therefore 0104475F =., 100K =, 01r =., 4T =, and 002=.σ.

212111144010744

d d d ==.=-.=. Th

e price o

f the European call is

014[104475(11144)100(10744)]319e N N -.?..-.=.

or $3.19.

Problem 21.10.

If the yield volatility for a five-year put option on a bond maturing in 10 years time is

specifie d as 22%, how should the option be valued? Assume that, based on today’s interest rates the modified duration of the bond at the maturity of the option will be 4.2 years and the

forward yield on the bond is 7%.

The relationship between the yield volatility and the price volatility is given by equation (21.6). In this case, the price volatility is

.?.?.=.

00742022647%

This is the volatility substituted into equation (21.2).

Problem 21.11.

A corporation knows that in three months it will have $5 million to invest for 90 days at LIBOR minus 50 basis points and wishes to ensure that the rate obtained will be at least

6.5%. What position in exchange-traded interest-rate options should the corporation take?

The rate received will be less than 6.5% when LIBOR is less than 7%. The corporation requires a three-month call option on a Eurodollar futures option with a strike price of 93. If three-month LIBOR is greater than 7% at the option maturity, the Eurodollar futures quote at option maturity will be less than 93 and there will be no payoff from the option. If the

three-month LIBOR is less than 7%, one Eurodollar futures options provide a payoff of $25 per 0.01%. Each 0.01% of interest costs the corporation $500 (500000000001

=,,?.). A total /=contracts are therefore required.

of 5002520

Problem 21.12.

Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a five-year cap.

When spot volatilities are used to value a cap, a different volatility is used to value each caplet. When flat volatilities are used, the same volatility is used to value each caplet within a given cap. Spot volatilities are a function of the maturity of the caplet. Flat volatilities are a function of the maturity of the cap.

Problem 21.13.

What other instrument is the same as a five-year zero-cost collar in which the strike price of the cap equals the strike price of the floor? What does the common strike price equal?

A 5-year zero-cost collar where the strike price of the cap equals the strike price of the floor is the same as an interest rate swap agreement to receive floating and pay a fixed rate equal to the strike price. The common strike price is the swap rate. Note that the swap is actually a forward swap that excludes the first exchange of payments (see Business Snapshot 21.1).

Problem 21.14.

Suppose that the 1-year, 2-year, 3-year, 4-year and 5-year LIBOR/swap zero rates are 6%,

6.4%, 6.7%, 6.9%, and 7%. The price of a 5-year semiannual cap with a principal of $100 at

a cap rate of 8% is $3. Use DerivaGem(with LIBOR discounting) to determine

a. The 5-year flat volatility for caps and floors

b. The floor rate in a zero-cost 5-year collar when the cap rate is 8%

We choose the Caps and Swap Options worksheet of DerivaGem and choose Cap/Floor as the Underlying Type. We enter the 1-, 2-, 3-, 4-, 5-year zero rates as 6%, 6.4%, 6.7%, 6.9%, and 7.0% in the Term Structure table. We enter Semiannual for the Settlement Frequency, 100 for the Principal, 0 for the Start (Years), 5 for the End (Years), 8% for the Cap/Floor Rate, and $3 for the Price. We select Black-European as the Pricing Model and choose the

Cap button. We check the Imply Volatility box and hit Calculate . The implied volatility is 24.79%. We then uncheck Implied Volatility, select Floor, check Imply Breakeven Rate. The floor rate that is calculated is 6.71%. This is the floor rate for which the floor is worth $3. A collar when the floor rate is 6.71% and the cap rate is 8% has zero cost.

Problem 21.15.

Show that V 1 + f = V 2 where 1V is the value of a swaption to pay a fixed rate of K R and receive LIBOR between times 1T and 2T , f is the value of a forward swap to receive a fixed

rate of K R and pay LIBOR between times 1T and 2T , and 2V is the value of a swaption

to receive a fixed rate of K R between times 1T and 2T . Deduce that 12V V = when K R equals the current forward swap rate.

We prove this result by considering two portfolios. The first consists of the swap option to receive K R ; the second consists of the swap option to pay K R and the forward swap.

Suppose that the actual swap rate at the maturity of the options is greater than K R . The swap

option to pay K R will be exercised and the swap option to receive K R will not be

exercised. Both portfolios are then worth zero since the swap option to pay K R is

neutralized by the forward swap. Suppose next that the actual swap rate at the maturity of the options is less than K R . The swap option to receive K R is exercised and the swap option to

pay K R is not exercised. Both portfolios are then equivalent to a swap where K R is

received and floating is paid. In all states of the world the two portfolios are worth the same at time 1T . They must therefore be worth the same today. This proves the result. When K R equals the current forward swap rate 0f = and 12V V =. A swap option to pay fixed is therefore worth the same as a similar swap option to receive fixed when the fixed rate in the swap option is the forward swap rate.

Problem 21.16.

Explain why there is an arbitrage opportunity if the implied Black (flat) volatility for a cap is different from that for a floor. Do the broker quotes in Table 21.1 present an arbitrage opportunity?

The put –call parity relationship in Business Snapshot 21.1 is

cap swap floor +=

must hold for market prices. It also holds for Black’s model. An argument similar to that in the appendix to Chapter 19 shows that the implied volatility of the cap must equal the implied volatility of the call. If this is not the case there is an arbitrage opportunity. The broker quotes in Table 21.1 do not present an arbitrage opportunity because the cap offer is always higher than the floor bid and the floor offer is always higher than the cap bid.

Problem 21.17.

Suppose that zero rates are as in Problem 21.14. Use DerivaGem (with LIBOR discounting) to determine the value of an option to pay a fixed rate of 6% and receive LIBOR on a

five-year swap starting in one year. Assume that the principal is $100 million, payments are exchanged semiannually, and the swap rate volatility is 21%.

We choose the Caps and Swap Options worksheet of DerivaGem and choose Swap Option as the Underlying Type. We enter 100 as the Principal, 1 as the Start (Years), 6 as the End (Years), 6% as the Swap Rate, and Semiannual as the Settlement Frequency. We choose Black-European as the pricing model, enter 21% as the Volatility and check the Pay Fixed button. We do not check the Imply Breakeven Rate and Imply Volatility boxes. The value of the swap option is 5.63.

Further Questions

Problem 21.18

Consider an eight-month European put option on a Treasury bond that currently has 14.25 years to maturity. The bond principal is $1,000. The current cash bond price is $910, the exercise price is $900, and the volatility of the forward bond price is 10% per annum. A coupon of $35 will be paid by the bond in three months. The risk-free interest rate is 8% for all maturities up to one year. Use Black’s model to determine the price of the option.

Consider both the case where the strike price corresponds to the cash price of the bond and the case where it corresponds to the quoted price.

The present value of the coupon payment is

008025353431e -.?.=. Equation (21.2) can therefore be used with 0088120(9103431)92366F e .?/=-.=., 008r =., 010=.σ and 06667T =.. When the strike price is a cash price, 900K = and

12103587002770

d d d ==.=-.=.

The option price is therefore

00806667900(02770)87569(03587)1834e N N -.?.-.-.-.=.

or $18.34.

When the strike price is a quoted price 5 months of accrued interest must be added to 900 to get the cash strike price. The cash strike price is 900350833392917+?.=.. In this case

12100319001136

d d d ==-.=-.=-.

and the option price is

0080666792917(01136)87569(00319)3122e N N -.?...-..=.

or $31.22.

Problem 21.19.

Use the DerivaGem software to value a five-year collar that guarantees that the maximum and minimum interest rates on a LIBOR-based loan (with quarterly resets) are 7% and 5% respectively. The LIBOR zero curve (continuously compounded) is currently flat at 6% and is used for discounting. The flat volatility is 20%. Assume that the principal is $100.

We use the Caps and Swap Options worksheet of DerivaGem. To set the zero curve as flat at 6% with continuous compounding, we need only enter 6% for one maturity. To value the cap we select Cap/Floor as the Underlying Type, enter Quarterly for the Settlement Frequency,

100 for the Principal, 0 for the Start (Years), 5 for the End (Years), 7% for the Cap/Floor Rate, and 20% for the Volatility. We select Black-European as the Pricing Model and choose the Cap button. We do not check the Imply Breakeven Rate and Imply Volatility boxes. The value of the cap is 1.565. To value the floor we change the Cap/Floor Rate to 5% and select the Floor button rather than the Cap button. The value is 1.072. The collar is a long position in the cap and a short position in the floor. The value of the collar is therefore

156510720493.-.=.

Problem 21.20

Suppose that the LIBOR yield curve is flat at 8% with annual compounding. A swaption gives the holder the right to receive 7.6% in a five-year swap starting in four years. Payments are made annually. The volatility of the forward swap rate is 25% per annum and the principal is $1 million. Use Black’s model to price the swaption. Compare your answer to that given by DerivaGem.

The payoff from the swaption is a series of five cash flows equal to max[00760]R .-, in millions of dollars where R is the five-year swap rate in four years. The value of an annuity that provides $1 per year at the end of years 5, 6, 7, 8, and 9 is 95129348108

i i ==..∑ The value of the swaption in millions of dollars is therefore

2129348[0076()008()]N d N d ..--.-

where

2103526d ==. and

2201474d ==-. The value of the swaption is

29348[0076(01474)008(03526)]0039554N N ...-.-.=.

or $39,554. This is the same answer as that given by DerivaGem. Note that for the purposes of using DerivaGem the zero rate is 7.696% continuously compounded for all maturities.

Problem 21.21.

Calculate the price of a cap on the three-month LIBOR rate in nine months’ tim e for a principal amount of $1,000. Use Black’s model and the following information:

Quoted nine-month Eurodollar futures price = 92

Interest-rate volatility implied by a nine-month Eurodollar option = 15% per annum 12-month risk-free interest rate with continuous compounding = 7.5% per annum Cap rate = 8% per annum

The quoted futures price corresponds to a forward rate is 8% per annum with quarterly

compounding and an actual/360 day count. (We do not worry about the convexity adjustment discussed in Section 6.4 because the futures contract has a relatively short life.) This is 8365360811%?/=. with an actual/actual day count. The parameters for Black’s model are therefore: 00811k F =., 008K =., 10075k r +=., 015k =.σ, 075k t =., and 1100k t +=..

21220170100402d d ==.==. and the call price, c []007510251000008(01701)008(00402)110c e N N -.?=.?,?..-..=.

Problem 21.22.

Use the DerivaGem software to value a European swaption that gives you the right in two years to enter into a 5-year swap in which you pay a fixed rate of 6% and receive floating. Cash flows are exchanged semiannually on the swap. The continuously compounded 1-year, 2-year, 5-year, and 10-year LIBOR/swap rates (which are also used for discounting) are 5%, 6%, 6.5%, and 7%, respectively. Assume a principal of $100 and a volatility of 15% per annum. Give an example of how the swaption might be used by a corporation. What bond option is equivalent to the swaption?

We choose the third worksheet of DerivaGem and choose Swap Option as the Underlying Type. We enter 100 as the Principal, 2 as the Start (Years), 7 as the End (Years), 6% as the Swap Rate, and Semiannual as the Settlement Frequency. We also enter the zero curve information. We choose Black-European as the pricing model, enter 15% as the Volatility and check the Pay Fixed button. We do not check the Imply Breakeven Rate and Imply

Volatility boxes. The value of the swaption is 4.606. For a company that expects to borrow at LIBOR plus 50 basis points in two years and then enter into a swap to convert to five-year fixed-rate borrowings, the swap guarantees that its effective fixed rate will not be more than

6.5%. The swaption is the same as an option to sell a five-year 6% coupon bond for par in two years.