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Options, Futures, and Other Derivatives

John C. Hull

Chapter 29

Interest rate derivatives: The standard market models - all with Video Answers

Educators

Chapter Questions

01:32

Problem 1

A company caps 3-month LIBOR at $2 \%$ per annum. The principal amount is $$\$ 20$$ million. On a reset date, 3-month LIBOR is $4 \%$ per annum. What payment would this lead to under the cap? When would the payment be made?

Sam Limsuwannarot
Sam Limsuwannarot
Numerade Educator

Problem 2

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

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Problem 3

Use the Black's model to value a 1-year European put option on a 10-year bond. Assume that the current cash price of the bond is $$\$ 125$$, the strike price is $$\$ 110$$, the 1-year risk-free 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 $$\$ 10$$.

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Problem 4

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

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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
James Kiss
Numerade Educator

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 your answer.

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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 $$\$ 102$$ and both bonds have a principal of $$\$ 100$$. The strike price of the option is $$\$ 100$$, the 4-year risk-free interest rate is $10 \%$ per annum with continuous compounding, and the forward bond price volatility for the bond underlying the option is $2 \%$ per annum.

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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 $7 \%$.

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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
Daniel Cisneros
Numerade Educator

Problem 10

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

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Problem 11

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

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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.

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03:10

Problem 13

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

Kaylee Mcclellan
Kaylee Mcclellan
Numerade Educator

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 LIBOR is received? The swap principal is $$\$ 10$$ million. Assume that the LIBOR/swap yield curve is used for discounting and is flat at $5 \%$ per annum with annual compounding and that the volatility of the swap rate is $20 \%$. How does the value change if all swap rates are are $5 \%$ and all OIS rates are $4.7 \%$. Compare your answer with that given by DerivaGem.

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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.

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Problem 16

Carry out a manual calculation to verify the option prices in Example 29.2.

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Problem 17

Suppose that all risk-free (OIS) zero rates are $6.5 \%$ (continuously compounded). The price of a 5 -year semiannual cap with a principal of $$\$ 100$$ and a cap rate of $8 \%$ (semiannually compounded) is $$\$ 3$$. Use DerivaGem to determine:
(a) The implied 5-year flat volatility for caps and floors
(b) The floor rate in a zero-cost 5 -year collar when the cap rate is $8 \%$.
Assume that all 6-month LIBOR forward rates are $6.7 \%$ with semiannual compounding.

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Problem 18

Show that $V_1+f=V_2$, where $V_1$ is the value of a swaption 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 swaption 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.

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Problem 19

Suppose that risk-free zero rates and LIBOR forward rates are as in Problem 29.17. 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 $21 \%$.

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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.

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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.

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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 with LIBOR discounting 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 risk-free interest rate with continuous compounding $=7.5 \%$ per annum.
(d) The cap rate $=8 \%$ per annum. (Assume an actual $/ 360$ day count.)

James Kiss
James Kiss
Numerade Educator

Problem 23

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 forward swap rate is $8 \%$ with annual compounding and its volatility is $25 \%$ per annum. The principal is $$\$ 1$$ million and risk-free (OIS) rates for all maturities are $7.8 \%$ (with continuous compounding). Use Black's model to price the swaption. Compare your answer to that given by DerivaGem.

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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 $7 \%$ and $5 \%$, respectively. All 3-month LIBOR forward rates are $6 \%$ per annum (with quarterly compounding). The flat volatility is $20 \%$. Assume that the principal is $$\$ 100$$ and the risk-free (OIS) term structure is flat at $5.8 \%$.

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Problem 25

Use the DerivaGem software to value a European swaption 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 continuously compounded 1-year, 2-year, 5-year, and 10-year risk-free (OIS) zero rates are $5 \%$, $6 \%, 6.5 \%$, and $7 \%$, respectively. Assume a principal of $$\$ 100$$. The forward swap rate is $7 \%$ (compounded semiannually) and its volatility is $15 \%$ per annum. Give an example of how the swaption might be used by a corporation. What bond option is equivalent to the swaption?

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