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Options, Futures, and Other Derivatives

John C. Hull

Chapter 33

HJM, LMM, and multiple zero curves - all with Video Answers

Educators

Chapter Questions

01:17

Problem 1

Explain the difference between a Markov and a non-Markov model of the short rate.

Lucas Finney
Lucas Finney
Numerade Educator
05:24

Problem 2

Prove the relationship between the drift and volatility of the forward rate for the multifactor version of HJM in equation (33.6).

Rowan Ahmed
Rowan Ahmed
Numerade Educator
01:42

Problem 3

"When the forward rate volatility $s(t, T)$ in HJM is constant, the Ho-Lee model results." Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the Ho-Lee model in Chapter 32.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 4

"When the forward rate volatility, $s(t, T)$, in HJM is $\sigma e^{-a(T-t)}$, the Hull-White model results." Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the Hull-White model in Chapter 32 .

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07:23

Problem 5

What is the advantage of LMM over HJM?

Md.Daniyal Arshad
Md.Daniyal Arshad
Numerade Educator
00:52

Problem 6

Provide an intuitive explanation of why a ratchet cap increases in value as the number of factors increase.

Joanna Quigley
Joanna Quigley
Numerade Educator
06:12

Problem 7

Show that equation (33.10) reduces to (33.4) as the $\delta$ tend to zero.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:40

Problem 8

Explain why a sticky cap is more expensive than a similar ratchet cap.

Ronald Prasad
Ronald Prasad
Numerade Educator

Problem 9

Explain why IOs and POs have opposite sensitivities to the rate of prepayments.

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Problem 10

"An option adjusted spread is analogous to the yield on a bond." Explain this statement.

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05:38

Problem 11

Prove equation (33.15).

Amany Waheeb
Amany Waheeb
Numerade Educator
01:04

Problem 12

Prove the formula for the variance $V(T)$ of the swap rate in equation (33.17).

Dominador Tan
Dominador Tan
Numerade Educator

Problem 13

Show that the swap volatility expression (33.19) in Section 33.2 is correct.

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03:26

Problem 14

In an annual-pay cap, the Black volatilities for at-the-money caplets which start in $1,2,3$, and 5 years and end 1 year later are $18 \%, 20 \%, 22 \%$, and $20 \%$, respectively. Estimate the volatility of a 1-year forward rate in the LIBOR Market Model when the time to the start of the period covered by the forward rate is (a) 0 to 1 year, (b) 1 to 2 years, (c) 2 to 3 years, and (d) 3 to 5 years. Assume that the zero curve is flat at $5 \%$ per annum (annually compounded). Use DerivaGem with LIBOR discounting to estimate flat volatilities for $2-, 3-, 4-5-$, and 6-year at-the-money caps.

James Kiss
James Kiss
Numerade Educator
03:25

Problem 15

In the flexi cap considered in Section 33.2 the holder is obligated to exercise the first $N$ in-the-money caplets. After that no further caplets can be exercised. (In the example, $N=5$.) Two other ways that flexi caps are sometimes defined are:
(a) The holder can choose whether any caplet is exercised, but there is a limit of $N$ on the total number of caplets that can be exercised.
(b) Once the holder chooses to exercise a caplet all subsequent in-the-money caplets must be exercised up to a maximum of $N$.
Discuss the problems in valuing these types of flexi caps. Of the three types of flexi caps, which would you expect to be most expensive? Which would you expect to be least expensive?

Hunza Gilgit
Hunza Gilgit
Numerade Educator