Options, Futures, and Other Derivatives

John C. Hull

Chapter 26 Interest Rate Derivatives: The Standard Market Models - all with Video Answers

Educators

Chapter Questions

Problem 1

A company caps 3 -month LIBOR at $10 \%$ per annum. The principal amount is 20 million dollar, On a reset date, 3 -month LIBOR is $12 \%$ per annum. What payment would this lead to under the cap? When would the payment be made?

Sam Limsuwannarot

Numerade Educator

07:07

Problem 2

Explain why a swap option can be regarded as a type of bond option.

Shivani Sharma

Numerade Educator

01:35

Problem 3

Use the Black's model to value a l-year European put option on a 10 -year bond. Assume that the current value of the bond is $\$ 125,$ the strike price is $\$ 110,$ the 1 -year interest rate is $10 \%$ per annum, the bond's forward price volatility is $8 \%$ per annum, and the present value of the coupons to be paid during the life of the option is $\$ 10$.

Neel Faucher

Numerade Educator

00:57

Problem 4

Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a 5 -year cap.

Joshua Eastwood

Numerade Educator

03:26

Problem 5

Calculate the price of an option that caps the 3 -month rate, starting in 15 months' time, at $13 \%$ (quoted with quarterly compounding) on a principal amount of 1,000 .The forward interest rate for the period in question is 12 % per annum (quoted with quarterly compounding, the 18 -month risk-free interest rate (continuously compounded) is $11.5 % per annum, and the volatility of the forward rate is 12 % per annum.

James Kiss

Numerade Educator

03:04

Problem 6

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

Mike Stern

Numerade Educator

02:38

Problem 7

Calculate the value of a 4 -year European call option on bond that will mature 5 years from today using Black's model. The 5 -year cash bond price is $\$ 105,$ the cash price of a 4-year bond with the same coupon is 102, the strike price is 100, the 4 -year risk-free interest rate is 10 \%per annum with continuous compounding, and the volatility for the bond price in 4 years is 2 \% per annum.

Sarah Vo

Numerade Educator

03:04

Problem 8

If the yield volatility for a 5 -year put option on a bond maturing in 10 years time is specified 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 $7 \%$

Mike Stern

Numerade Educator

00:49

Problem 9

What other instrument is the same as a 5 -year zero-cost collar where the strike price of the cap equals the strike price of the floor? What does the common strike price equal?

Daniel Cisneros

Numerade Educator

01:31

Problem 10

Derive a put-call parity relationship for European bond options.

Shazia Naz

Numerade Educator

05:57

Problem 11

Derive a put-call parity relationship for European swap options.

Mohamed Mohamed

Numerade Educator

03:13

Problem 12

Explain why there is an arbitrage opportunity if the implied Black (flat) volatility of a cap is different from that of a floor. Do the broker quotes in Table 26.1 present an arbitrage opportunity?

Md.Daniyal Arshad

Numerade Educator

03:10

Problem 13

When a bond's price is lognormal can the bond's yield be negative? Explain your answer.

Kaylee Mcclellan

Numerade Educator

01:39

Problem 14

What is the value of a European swap option that gives the holder the right to enter into a 3 -year annual-pay swap in 4 years where a fixed rate of $5 \%$ is paid and $\mathrm{LIBOR}$ is received? The swap principal is 10 million. Assume that the yield curve is flat at $5 \%$ per annum with annual compounding and the volatility of the swap rate is 20 \% Compare your answer with that given by DerivaGem.

Alejandro Ruiz

Numerade Educator

02:36

Problem 15

Suppose that the yield $R$ on a zero-coupon bond follows the process $$d R=\mu d t+\sigma d z$$ where $\mu$ and $\sigma$ are functions of $R$ and $t,$ and $d z$ is a Wiener process. Use Itô's lemma to show that the volatility of the zero-coupon bond price declines to zero as it approaches maturity.

Andrew Davis

Numerade Educator

01:40

Problem 16

Carry out a manual calculation to verify the option prices in Example 26.2

Manik Pulyani

Numerade Educator

03:26

Problem 17

Suppose that the 1 -year, 2 -year, 3 -year, 4 -year, and 5 -year 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 to determine:
(a) The S-year flat volatility for caps and floors
(b) The floor rate in a zero-cost 5 -year collar when the cap rate is $8 \%$

James Kiss

Numerade Educator

03:38

Problem 18

Show that $V_{1}+f=V_{2},$ where $V_{1}$ is the value of a swap option to pay a fixed rate of $s_{K}$ and receive LIBOR between times $T_{1}$ and $T_{2}, f$ is the value of a forward swap to receive a fixed rate of $s_{K}$ and pay LIBOR between times $T_{1}$ and $T_{2}$, and $V_{2}$ is the value of a swap option to receive a fixed rate of $s_{K}$ between times $T_{1}$ and $T_{2}$. Deduce that $V_{1}=V_{2}$ when $s_{k}$ equals the current forward swap rate.

Rakvi .

Numerade Educator

02:36

Problem 19

Suppose that zero rates are as in Problem 26.17 . Use DerivaGem to determine the value of an option to pay a fixed rate of $6 \%$ and receive LIBOR on a 5 -year swap starting in 1 year. Assume that the principal is $\$ 100$ million, payments are exchanged semiannually, and the swap rate volatility is 21 %.

Kayleah Tsai

Numerade Educator

01:04

Problem 20

Describe how you would (a) calculate cap flat volatilities from cap spot volatilities and
(b) calculate cap spot volatilities from cap flat volatilities.

Ashley High

Numerade Educator

07:43

Problem 21

Consider an 8 -month European put option on a Treasury bond that currently has 14.25 years to maturity. The current cash bond price is $\$ 910,$ the exercise price is $\$ 900,$ and the volatility for the bond price is $10 \%$ per annum. A coupon of $\$ 35$ will be paid by the bond in 3 months. The risk-free interest rate is 8 % for all maturities up to 1 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.

John Lee

Numerade Educator

03:26

Problem 22

Calculate the price of a cap on the 90 -day LIBOR rate in 9 months' time when the principal amount is $\$ 1,000 .$ Use Black's model and the following information:
(a) The quoted 9 -month Eurodollar futures price =92 . (Ignore differences between futures and forward rates.
(b) The interest rate volatility implied by a 9 -month Eurodollar option =15 % per annum.
(c) The current 12 -month interest rate with continuous compounding =7.5 % per annum.
(d) The cap rate =8% per annum. (Assume an actual/360 day count.)

James Kiss

Numerade Educator

04:03

Problem 23

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 5 -year swap starting in 4 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 with that given by DerivaGem.

James Kiss

Numerade Educator

01:35

Problem 24

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

Monica Miller

Numerade Educator

01:33

Problem 25

Use the DerivaGem software to value a European swap option that gives you the right in 2 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 I-year, 2 -year, 5 -year, and 10 -year zero-coupon interest rates (continuously compounded) 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 swap option might be used by a corporation. What bond option is equivalent to the swap option?

Carson Merrill

Numerade Educator