Deck 28: The Heath-Jarrow-Morton HJM and Libor Market Model LMM

Question 1

In the LMM, which of the following are martingales?

A) Libor rates.
B) Libor returns.
C) Zero-coupon rates.
D) Yields.

Answer 1

Libor rates.

Question 2

Consider a two-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process with two shock terms: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.01 \pm 0.01$ , where the forward rate movements are equiprobable. What this means is that the forward rate may move up by either 0.02 with probability 1/4, or move down by 0.02 with probability 1/4, or remain the same with probability 1/2. What is the price of a put option on a $100 notional, 6.5% coupon bond, with a strike price of $100?

A) 0.25
B) 0.54
C) 0.77
D) 0.96

Answer 2

0.96

Question 3

In the Libor Market Model (LMM) which of the following is not a beneficial feature of the model?

A) The LMM only requires one factor for each rate.
B) The model is easily calibrated to the Black model caplet/floorlet prices.
C) The evolution of Libor rates is modeled directly.
D) It is easily implemented on a recombining tree.

Answer 3

It is easily implemented on a recombining tree.

Question 4

The Libor Market Model is most often implemented by means of

A) A binomial tree.
B) A trinomial tree.
C) Monte Carlo simulation.
D) A closed-form approximation equation for derivative prices.

Question 5

Consider a two-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process with two shock terms: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.01 \pm 0.01$ , where the forward rate movements are equiprobable. Compare this to a one-factor HJM model where $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ . Which of the following statements is most valid?

A) ATMF bond option prices will be higher in the one-factor model.
B) ATMF bond option prices will be lower in the one-factor model.
C) ATMF bond option prices will be equal in the one- and two-factor models.
D) There is insufficient information to determine the answer.

Question 6

Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. What is the price of a $100 notional one-year floor on the one-year forward rate at a strike rate of 7%.

A) 0.85
B) 0.89
C) 0.95
D) 1.00

Question 7

Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. What is the price of a one-year call option on a two-year 6.5% coupon bond, with a strike price of $100 ex-coupon?

A) 0.50
B) 0.55
C) 0.60
D) 0.65

Question 8

Consider a one-factor HJM model on a binomial model with time steps of one year and a probability of an up shift of $q$ . Each $j$ -th forward rate has constant volatility $\sigma ( j )$ . For each forward period of one year what is the risk-neutral drift term in the $n$ -th period equal to?

A) $\alpha ( n ) = \ln \left[ q \exp \left( \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) + ( 1 - q ) \exp \left( - \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) \right] - \sum _ { j = 1 } ^ { n - 1 } \alpha ( j )$
B) $\alpha ( n ) = \ln [ q \exp ( \sigma ( n ) ) + ( 1 - q ) \exp ( - \sigma ( n ) ) ]$
C) $\alpha ( n ) = \exp \left[ q \ln \left( \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) + ( 1 - q ) \ln \left( - \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) \right] - \sum _ { j = 1 } ^ { n - 1 } \alpha ( j )$
D) $\alpha ( n ) = \exp [ q \ln ( \sigma ( n ) ) + ( 1 - q ) \ln ( - \sigma ( n ) ) ]$

Question 9

In the HJM model, one of the striking features is that the risk-neutral drifts in the model are functions of only the

A) The forward rates.
B) The yields.
C) The volatility of forward rates.
D) The risk premia for interest-rate risk.

Question 10

Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. What is the $\alpha$ value for $f ( 1,1 )$ ?

A) $- 0.0001$
B) $- 0.0002$
C) $+ 0.0001$
D) $+ 0.0002$

Question 11

Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. What is the price of a one-year put option on a two-year 6.5% coupon bond, with a strike price of $100 ex-coupon?

A) 1.00
B) 1.26
C) 1.54
D) 1.67

Question 12

The HJM model is implemented by depicting the evolution of which rates in particular?

A) Yields.
B) Zero-coupon rates.
C) Forward rates.
D) Discount functions.

Question 13

Consider a two-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process with two shock terms: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.01 \pm 0.01$ , where the forward rate movements are equiprobable. What this means is that the forward rate may move up by either 0.02 with probability 1/4, or move down by 0.02 with probability 1/4, or remain the same with probability 1/2. What is the price of a call option on a $100 notional, 6.5% coupon bond, with a strike price of $100?

A) 0.25
B) 0.31
C) 0.43
D) 0.55

Question 14

Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. Consider the price of one-year call and put options on a two-year 6.5% coupon bond, with a strike price of $100 ex-coupon. The difference between the call and put prices will be

A) $6.5 e ^ { - 0.06 } + 106.5 e ^ { - 0.06 - 0.07 } - 100 e ^ { - 0.06 }$
B) $6.5 e ^ { - 0.06 } + 106.5 e ^ { - 0.06 - 0.07 } - 100 e ^ { - 0.06 } + 6.5 e ^ { - 0.06 }$
C) $106.5 e ^ { - 0.06 - 0.07 } - 100 e ^ { - 0.06 } - 6.5 e ^ { - 0.06 }$
D) $106.5 e ^ { - 0.06 - 0.07 } - 100 e ^ { - 0.06 }$

Question 15

Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. What is the price of a $100 notional one-year cap on the one-year forward rate at a strike rate of 7%.

A) 0.87
B) 0.95
C) 1.05
D) 1.10

Question 16

Swap rates in the SMM are, under the risk-neutral forward measure

A) Normal.
B) Lognormal.
C) Exponential.
D) None of the above.

Question 17

The numeraire in the Swap Market Model (SMM) is

A) The price of the longest maturity bond (i.e., the numeraire under forward measure).
B) The total of discount functions to the longest swap maturity.
C) The money market account.
D) The value of the fixed side of the swap.

Question 18

Which of the following is not a valid property of the Heath-Jarrow-Morton (HJM) interest-rate framework?
(a) The model may be calibrated to be consistent with any initial yield curve.
(b) The tree version of the model has rates of all remaining maturities at each node of the tree.
(c) The model fits volatilities of rates of all maturities.
(d) The model is a one-factor model.

Question 19

Consider a two-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process with two shock terms: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.01 \pm 0.01$ , where the forward rate movements are equiprobable. What this means is that the forward rate may move up by either 0.02 with probability 1/4, or move down by 0.02 with probability 1/4, or remain the same with probability 1/2. What is the risk-neutral drift ( $\alpha$ ) for $f ( 1,1 )$ ?

A) $- 0.0002$
B) $- 0.0001$
C) $+ 0.0001$
D) $+ 0.0002$

Question 20

Which of the following is not necessarily a beneficial feature of the HJM binomial tree class of models?

A) The drift term is obtained in analytical form as a function of the volatilities.
B) The binomial tree carries the entire term structure of forward rates at each node.
C) The tree is recombining because the drift terms are available in analytical form.
D) The approach requires only drawing the tree out to the maturity of the option, and not to the maturity of bonds underlying a bond option.