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Options, Futures, and Other Derivatives

John C. Hull

Chapter 32

No-arbitrage models of the short rate - all with Video Answers

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Chapter Questions

01:50

Problem 1

What is the difference between an equilibrium model and a no-arbitrage model?

John Nicolle
John Nicolle
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Problem 2

Can the approach described in Section 32.2 for decomposing an option on a couponbearing bond into a portfolio of options on zero-coupon bonds be used in conjunction with a two-factor model? Explain your answer.

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Problem 3

Suppose that $a=0.1, b=0.08$, and $\sigma=0.015$ in Vasicek's model, with the initial value of the short rate being $5 \%$. Calculate the price of a 1 -year European call option on a zerocoupon bond with a principal of $$\$ 100$$ that matures in 3 years when the strike price is $$\$ 87$$.

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Problem 4

Repeat Problem 32.3 valuing a European put option with a strike of $$\$ 87$$. What is the put-call parity relationship between the prices of European call and put options? Show that the put and call option prices satisfy put-call parity in this case.

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Problem 5

Suppose that $a=0.05, b=0.08$, and $\sigma=0.015$ in Vasicek's model with the initial shortterm interest rate being $6 \%$. Calculate the price of a 2.1-year European call option on a bond that will mature in 3 years. Suppose that the bond pays a coupon of $5 \%$ semiannually. The principal of the bond is 100 and the strike price of the option is 99 . The strike price is the cash price (not the quoted price) that will be paid for the bond.

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Problem 6

Use the answer to Problem 32.5 and put-call parity arguments to calculate the price of a put option that has the same terms as the call option in Problem 32.5.

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Problem 7

In the Hull-White model, $a=0.08$ and $\sigma=0.01$. Calculate the price of a 1-year European call option on a zero-coupon bond that will mature in 5 years when the term structure is flat at $10 \%$, the principal of the bond is $$\$ 100$$, and the strike price is $$\$ 68$$.

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Problem 8

Suppose that $a=0.05$ and $\sigma=0.015$ in the Hull-White model with the initial term structure being flat at $6 \%$ with semiannual compounding. Calculate the price of a 2.1-year European call option on a bond that will mature in 3 years. Suppose that the bond pays a coupon of $5 \%$ per annum semiannually. The principal of the bond is 100 and the strike price of the option is 99 . The strike price is the cash price (not the quoted price) that will be paid for the bond.

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03:32

Problem 9

Suppose $a=0.05, \sigma=0.015$, and the term structure is flat at $10 \%$. Construct a trinomial tree for the Hull-White model where there are two time steps, each 1 year in length.

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Natalie Daly
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Problem 10

Calculate the price of a 2-year zero-coupon bond from the tree in Figure 32.4.

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Problem 11

Calculate the price of a 2-year zero-coupon bond from the tree in Figure 32.7 and verify that it agrees with the initial term structure.

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Problem 12

Calculate the price of an 18-month zero-coupon bond from the tree in Figure 32.8 and verify that it agrees with the initial term structure.

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00:41

Problem 13

What does the calibration of a one-factor term structure model involve?

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Vishal Parmar
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Problem 14

Use the DerivaGem software to value $1 \times 4,2 \times 3,3 \times 2$, and $4 \times 1$ European swap options to receive fixed and pay floating. Assume that the 1-, 2-, 3-, 4-, and 5-year interest rates are $6 \%, 5.5 \%, 6 \%, 6.5 \%$, and $7 \%$, respectively. The payment frequency on the swap is semiannual and the fixed rate is $6 \%$ per annum with semiannual compounding. Use the Hull-White model with $a=3 \%$ and $\sigma=1 \%$. Calculate the volatility implied by Black's model for each option.

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Problem 15

Prove equations (32.15), (32.16), and (32.17).

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Problem 16

Construct a trinomial tree for the Ho-Lee model where $\sigma=0.02$. Suppose that the initial zero-coupon interest rate for a maturities of $0.5,1.0$, and 1.5 years are $7.5 \%, 8 \%$, and $8.5 \%$. Use two time steps, each 6 months long. Calculate the value of a zero-coupon bond with a face value of $$\$ 100$$ and a remaining life of 6 months at the ends of the final nodes of the tree. Use the tree to value a 1-year European put option with a strike price of 95 on the bond. Compare the price given by your tree with the analytic price given by DerivaGem.

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Problem 17

A trader wishes to compute the price of a l-year American call option on a 5-year bond with a face value of 100 . The bond pays a coupon of $6 \%$ semiannually and the (quoted) strike price of the option is $$\$ 100$$. The continuously compounded zero rates for maturities of 6 months, 1 year, 2 years, 3 years, 4 years, and 5 years are $4.5 \%, 5 \%$, $5.5 \%, 5.8 \%, 6.1 \%$, and $6.3 \%$. The best-fit reversion rate for either the normal or the lognormal model has been estimated as $5 \%$.
A 1-year European call option with a (quoted) strike price of 100 on the bond is actively traded. Its market price is $$\$ 0.50$$. The trader decides to use this option for calibration. Use the DerivaGem software with 10 time steps to answer the following questions:
(a) Assuming a normal model, imply the $\sigma$ parameter from the price of the European option.
(b) Use the $\sigma$ parameter to calculate the price of the option when it is American.
(c) Repeat (a) and (b) for the lognormal model. Show that the model used does not significantly affect the price obtained providing it is calibrated to the known European price.
(d) Display the tree for the normal model and calculate the probability of a negative interest rate occurring.
(e) Display the tree for the lognormal model and verify that the option price is correctly calculated at the node where, with the notation of Section $32.4, i=9$ and $j=-1$.

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Problem 18

Use the DerivaGem software to value $1 \times 4,2 \times 3,3 \times 2$, and $4 \times 1$ European swap options to receive floating and pay fixed. Assume that the 1-, 2-, 3-, 4-, and 5-year interest rates are $3 \%, 3.5 \%, 3.8 \%, 4.0 \%$, and $4.1 \%$, respectively. The payment frequency on the swap is semiannual and the fixed rate is $4 \%$ per annum with semiannual compounding. Use the lognormal model with $a=5 \%, \sigma=15 \%$, and 50 time steps. Calculate the volatility implied by Black's model for each option.

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Problem 19

Verify that the DerivaGem software gives Figure 32.9 for the example considered. Use the software to calculate the price of the American bond option for the lognormal and normal models when the strike price is 95,100 , and 105 . In the case of the normal model, assume that $a=5 \%$ and $\sigma=1 \%$. Discuss the results in the context of the heavy-tails arguments of Chapter 20.

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Problem 20

Modify Sample Application $\mathrm{G}$ in the DerivaGem Application Builder software to test the convergence of the price of the trinomial tree when it is used to price a 2-year call option on a 5 -year bond with a face value of 100 . Suppose that the strike price (quoted) is 100 , the coupon rate is $7 \%$ with coupons being paid twice a year. Assume that the zero curve is as in Table 32.2. Compare results for the following cases:
(a) Option is European; normal model with $\sigma=0.01$ and $a=0.05$
(b) Option is European; lognormal model with $\sigma=0.15$ and $a=0.05$
(c) Option is American; normal model with $\sigma=0.01$ and $a=0.05$
(d) Option is American; lognormal model with $\sigma=0.15$ and $a=0.05$.

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