Deck 27: Factor Models of the Term Structure

Question 1

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity cap on the one-year interest rate at a strike rate of 8% and a notional of $100?

A) 1.000
B) 1.025
C) 1.050
D) 1.075

Answer 1

1.025

Question 2

A one-factor bond pricing model implies that interest-rates of all maturities are driven by a single source of stochastic randomness. For example the system of interest rates may be described by the following equation: $d r ( T ) = \alpha ( r ( T ) , T ) d t + \sigma ( r ( T ) , T ) d W , \quad \forall T$ where $T$ denotes the maturity of different rates. A single-factor model implies that

A) All rates either move up together or all move down together.
B) The yield curve experience parallel shifts.
C) Instantaneous changes in rates of all maturities are perfectly positively or negatively correlated with each other.
D) Twists in shape of the yield curve are not possible.

Answer 2

Instantaneous changes in rates of all maturities are perfectly positively or negatively correlated with each other.

Question 3

In the Black-Derman-Toy (BDT) model, short rates have

A) Constant volatility for all maturities.
B) Volatility that changes by maturity of the short rate.
C) Volatility that varies by maturity and level of the short rate, i.e., state-dependent volatility.
D) Stochastic volatility.

Answer 3

Volatility that changes by maturity of the short rate.

Question 4

Based on your answers to the previous two questions and a comparison of the prices of the cap and floor, what can you say about the forward rate between one and two years?
(a) The forward rate is roughly equal to the strike rate of the cap and floor. (b) The forward rate is such that the mark-to-market value of the FRA at the strike rate of the cap and floor will be zero.
(c) Both (a) and (b).
(d) There is not enough information to say anything about the forward rate.

Question 5

In the Cox-Ingersoll-Ross or CIR model, interest rates are specified by the following stochastic process: $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ One attractive feature of this process relative to the Vasicek interest rate process $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ is that

A) Interest rates are always non-negative in CIR while they may be negative in the Vasicek model.
B) There are parameter restrictions which guarantee non-negative stochastic interest rates in the CIR model, but there are no such restrictions possible in the Vasicek model.
C) It has extra parameters, so can fit observed yield curves better.
D) It allows for imperfect instantaneous correlation between rates of different maturities, whereas in the Vasicek model, they are perfectly correlated.

Question 6

In the Cox-Ingersoll-Ross (CIR 1985) model, you are given that $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ where $x = 0.5$ , $\theta = 0.06$ , $\sigma = 0.10$ , and the current short rate of interest is $r _ { 0 } = 0.08$ . What is the expected short rate of interest one year hence?

A) 6.6%
B) 7.2%
C) 7.6%
D) 8.2%

Question 7

In the Ho & Lee (1986) model, assume that the initial curve of zero-coupon discount bond prices for one and two years is $0.9434$ and $0.8734$ , respectively. Assume that the probability of an upshift in discount functions is equal to that of a downshift. If the parameter $\delta = 0.95$ , then the price of a one-year zero-coupon bond in the up node after one year will be

A) 0.9282
B) 0.9496
C) 0.9563
D) 0.9678

Question 8

The Ho & Lee (1986) model directly models the following on a binomial tree:

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

Question 9

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity floor on the one-year interest rate at a strike rate of 8% and a notional of $100?

A) 1.000
B) 1.026
C) 1.052
D) 1.078

Question 10

In the Ho & Lee (1986) model, assume that the initial curve of zero-coupon rates for one and two years is 6% and 7%, respectively. Assume that the probability of an upshift in discount functions is equal to that of a downshift. If the parameter $\delta = 0.95$ , then the price of a one-year maturity call option on a two-year $100 face value zero-coupon bond in the up node after one year at a strike of $92 will be

A) 1.10
B) 1.20
C) 1.30
D) 1.40

Question 11

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity call option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

A) 0.80
B) 0.90
C) 1.00
D) 1.10

Question 12

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity put option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

A) 1.00
B) 1.08
C) 1.16
D) 1.24

Question 13

In the Cox-Ingersoll-Ross (1985) model, interest rates are specified by the following stochastic process: $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma _ { } \sqrt { r _ { t } } d W _ { t }$ Implementation of the model to match observed nominal rate processes generally requires of the parameters that

A) $k > 0$
B) $\theta > 0$
C) Both (a) and (b).
D) Neither (a) nor (b).

Question 14

Vasicek (1977) posits a general mean-reverting form for the short-rate: $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ He then derives, in the absence of arbitrage, a restriction on the market price of risk $\lambda$ of any bond, where $( \mu - r ) / \eta = \lambda$ of any bond, with $\mu$ being the instantaneous return on the bond, and $\eta$ being the bond's instantaneous volatility. The derived restriction is that

A) $\lambda$ is a constant.
B) $\lambda$ may be a function of time $t$ , but not of any other time- $t$ information or of the maturity $T$ of the bond.
C) $\lambda$ may be a function of the time- $t$ short rate $r _ { t }$ , but not of current time $t$ or of the bond maturity $T$ .
D) $\lambda$ may be a function of time $t$ and the time- $t$ short rate $r _ { t }$ , but not of the bond maturity $T$ .

Question 15

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . At what strike price will one-year maturity call and put options on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon) have equal prices?

A) $98.32
B) $99.52
C) $100.12
D) $101.42

Question 16

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What are the one-year rates (up and down) after one year?

A) 9.2% and 6.1%
B) 9.6% and 5.8%
C) 10.0% and 4.0%
D) 10.4% and 5.7%

Question 17

In the Vasieck (1977) model, you are given that $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ where $k = 0.5$ , $\theta = 0.06$ , $\sigma = 0.10$ , and the current short rate of interest is $r _ { 0 } = 0.08$ . What is the expected standard deviation of the short rate of interest one year hence?

A) 0.08
B) 0.09
C) 0.10
D) 0.11

Question 18

In the Ho & Lee (1986) model, the parameter $\delta$ plays a crucial role. Which of the following statements best describes this parameter?

A) $\delta > 1$ .
B) As $\delta$ increases the volatility of interest rates increases.
C) As $\delta$ increases the volatility of interest rates decreases.
D) $\delta < 0$ .

Question 19

In the Cox-Ingersoll-Ross (1985) model, interest rates are specified by the following stochastic process: $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma _ { } \sqrt { r _ { t } } d W _ { t }$ The process for interest rates is mean-reverting if

A) $k > 0$
B) $k < 0$
C) $\theta > 0$
D) $\theta < r _ { t }$

Question 20

In the Black-Derman-Toy (BDT) model, short rates are distributed as
(a) Normal
(b) Lognormal
(c) Exponential
(d) None of the above

Question 21

In the Cox-Ingersoll-Ross (CIR 1985) model, you are given that $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ where $x = 0.5$ , $\theta = 0.06$ , $\sigma = 0.10$ . If the yield of a five-year bond is $0.07$ , then what is the price of the bond?

A) 0.65
B) 0.70
C) 0.75
D) 0.80

Question 22

An exponential-affine short rate bond model is one

A) That most bond traders have an affinity for.
B) Where the bond prices are linear in the short-rate.
C) Where the logarithm of bond prices is linear in the short rate.
D) Where the bond price is based on discrete compounding.

Question 23

In the CIR (1985) model, which of the following statements is true? The price of the bond increases when

A) The short rate $r _ { t }$ increases.
B) The rate of mean reversion ${ K }$ rises.
C) The long-run mean rate $\theta$ increases.
D) The volatility $\sigma$ increases.

Question 24

An affine factor model is one in which multiple factors $X$ may be present. Which of the following is not true of an affine factor model.

A) The drift $\mu ( X )$ will be linear in $X$ .
B) The volatility $\sigma ( X )$ will be linear in $X$ .
C) The yield $R ( X )$ will be linear in $X$ .
D) The logarithm of the price scaled by maturity is the yield.