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Options, Futures, and Other Derivatives

John C. Hull

Chapter 29

Interest Rate Derivatives: HJM and LMM - 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
00:18

Problem 2

Prove the relationship between the drift and volatility of the forward rate for the multifactor version of HJM in equation (29.6).

David Collins
David Collins
Numerade Educator
01:42

Problem 3

"When the forward rate volatility $s(t, T)$ in $\mathrm{LMM}$ is constant, the Ho-Lee model results." Verify that this is true by showing that LMM gives a process for bond prices that is consistent with the Ho-Lee model in Chapter 28

Adriano Chikande
Adriano Chikande
Numerade Educator
01:42

Problem 4

"When the forward rate volatility, $s(t, T),$ in $\mathrm{LMM}$ is $\sigma e^{-a[T-t)},$ the Hull-White model results." Verify that this is true by showing that LMM gives a process for bond prices that is consistent with the Hull-White model in Chapter 28

Adriano Chikande
Adriano Chikande
Numerade Educator
00:48

Problem 5

What is the advantage of LMM over BGM?

Mirza Aslam Beig
Mirza Aslam Beig
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
02:33

Problem 7

Show that equation (29.10) reduces to (29.4) as the $\delta_{i}$ tend to zero.

Raushan Kumar
Raushan Kumar
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
02:27

Problem 9

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

Ahmed Kamel
Ahmed Kamel
Numerade Educator
01:49

Problem 10

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

Kaylee Mcclellan
Kaylee Mcclellan
Numerade Educator
05:38

Problem 11

Prove equation (29.15).

Amany Waheeb
Amany Waheeb
Numerade Educator
01:04

Problem 12

Prove the formula for the variance $V(T)$ of the swap rate in equation (29.17).

Dominador Tan
Dominador Tan
Numerade Educator
05:38

Problem 13

Prove equation (29.19)

Amany Waheeb
Amany Waheeb
Numerade Educator
03:26

Problem 14

In an annual-pay cap, the Black volatilities for caplets with maturities $1,2,3,$ and 5 years 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 maturity 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 to estimate flat volatilities for $2-, 3-, 4,5-$ and $6-$ year caps.

James Kiss
James Kiss
Numerade Educator
06:44

Problem 15

In the flexi cap considered in Section 29.2 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?

Kevin Morgan
Kevin Morgan
Numerade Educator