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  σ=0.30\sigma = 0.30σ=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?

Accepted Answer

A)   1.000
B)   1.026
C)   1.052
D)   1.078 E) B) and C)
F) A) and D)

Question 2

In the Vasieck (1977)  model, you are given that  drt=κ(θ−rt) dt+σdWtd r _ { t } = \kappa \left( 	heta - r _ { t } ight)  d t + \sigma d W _ { t }drt​=κ(θ−rt​) dt+σdWt​  where  k=0.5k = 0.5k=0.5  ,  θ=0.06	heta = 0.06θ=0.06  ,  σ=0.10\sigma = 0.10σ=0.10  , and the current short rate of interest is  r0=0.08r _ { 0 } = 0.08r0​=0.08  . What is the expected standard deviation of the short rate of interest one year hence?

Accepted Answer

A)   0.08
B)   0.09
C)   0.10
D)   0.11 E) A) and B)
F) All of the above

Question 3

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:  dr(T) =α(r(T) ,T) dt+σ(r(T) ,T) dW,∀Td r ( T )  = \alpha ( r ( T )  , T )  d t + \sigma ( r ( T )  , T )  d W , \quad \forall Tdr(T) =α(r(T) ,T) dt+σ(r(T) ,T) dW,∀T  where  TTT  denotes the maturity of different rates. A single-factor model implies that

Accepted Answer

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. E) B) and D)
F) All of the above C

Question 4

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

Accepted Answer

A)   The drift  μ(X) \mu ( X ) μ(X)   will be linear in  XXX  .
B)   The volatility  σ(X) \sigma ( X ) σ(X)   will be linear in  XXX  .
C)   The yield  R(X) R ( X ) R(X)   will be linear in  XXX  .
D)   The logarithm of the price scaled by maturity is the yield. E) All of the above
F) C) and D)

Question 5

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  σ=0.30\sigma = 0.30σ=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) ?

Accepted Answer

A)   1.00
B)   1.08
C)   1.16
D)   1.24 E) All of the above
F) B) and D)

Question 6

An exponential-affine short rate bond model is one

Accepted Answer

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.E) None of the above
F) A) and B)

Question 7

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

Accepted Answer

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.E) A) and D)
F) C) and D)

Question 8

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?

Accepted Answer

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.E) All of the above
F) A) and B)

Question 9

In the Ho & Lee (1986)  model, assume that the initial curve of zero-coupon discount bond prices for one and two years is   0.94340.94340.9434  and  0.87340.87340.8734  , respectively. Assume that the probability of an upshift in discount functions is equal to that of a downshift. If the parameter  δ=0.95\delta = 0.95δ=0.95  , then the price of a one-year zero-coupon bond in the up node after one year will be

Accepted Answer

A)   0.9282
B)   0.9496
C)   0.9563
D)   0.9678 E) A) and C)
F) A) and B)

Question 10

In the Cox-Ingersoll-Ross (1985)  model, interest rates are specified by the following stochastic process:   drt=k(θ−rt) dt+σrtdWtd r _ { t } = k \left( 	heta - r _ { t } ight)  d t + \sigma _ {  } \sqrt { r _ { t } } d W _ { t }drt​=k(θ−rt​) dt+σ​rt​​dWt​  The process for interest rates is mean-reverting if

Accepted Answer

A) 0">k>0k > 0k>0 B) k<0k < 0k<0 C) 0">θ>0 heta > 0θ>0 D) θ

Question 11

In the Cox-Ingersoll-Ross (1985)  model, interest rates are specified by the following stochastic process:  drt=k(θ−rt) dt+σrtdWtd r _ { t } = k \left( 	heta - r _ { t } ight)  d t + \sigma _ {  } \sqrt { r _ { t } } d W _ { t }drt​=k(θ−rt​) dt+σ​rt​​dWt​  Implementation of the model to match observed nominal rate processes generally requires of the parameters that

Accepted Answer

A)     0">k>0k > 0k>0 
B)     0">θ>0	heta > 0θ>0 
C)   Both (a)  and (b) .
D)   Neither (a)  nor (b) . E) None of the above
F) A) and B)

Question 12

In the Cox-Ingersoll-Ross (CIR 1985)  model, you are given that  drt=κ(θ−rt) dt+σrtdWtd r _ { t } = \kappa \left( 	heta - r _ { t } ight)  d t + \sigma \sqrt { r _ { t } } d W _ { t }drt​=κ(θ−rt​) dt+σrt​​dWt​  where  x=0.5x = 0.5x=0.5  ,  θ=0.06	heta = 0.06θ=0.06  ,  σ=0.10\sigma = 0.10σ=0.10  . If the yield of a five-year bond is  0.070.070.07  , then what is the price of the bond?

Accepted Answer

A)   0.65
B)   0.70
C)   0.75
D)   0.80 E) A) and B)
F) A) and C)

Question 13

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

Accepted Answer

A)   The short rate  rtr _ { t }rt​  increases.
B)   The rate of mean reversion  K { K }K  rises.
C)   The long-run mean rate  θ	hetaθ  increases.
D)   The volatility  σ\sigmaσ  increases. E) All of the above
F) None of the above

Question 14

In the Cox-Ingersoll-Ross or CIR model, interest rates are specified by the following stochastic process:  drt=k(θ−rt) dt+σrtdWtd r _ { t } = k \left( 	heta - r _ { t } ight)  d t + \sigma \sqrt { r _ { t } } d W _ { t }drt​=k(θ−rt​) dt+σrt​​dWt​  One attractive feature of this process relative to the Vasicek interest rate process  drt=k(θ−rt) dt+σdWtd r _ { t } = k \left( 	heta - r _ { t } ight)  d t + \sigma d W _ { t }drt​=k(θ−rt​) dt+σdWt​  is that

Accepted Answer

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. E) All of the above
F) None of the above

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  σ=0.30\sigma = 0.30σ=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) ?

Accepted Answer

A)   0.80
B)   0.90
C)   1.00
D)   1.10 E) B) and C)
F) None of the above

Question 16

In the Black-Derman-Toy (BDT)  model, short rates are distributed as

Accepted Answer

A)   Normal
B)   Lognormal
C)   Exponential
D)   None of the aboveE) A) and C)
F) All of the above B

Question 17

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  σ=0.30\sigma = 0.30σ=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?

Accepted Answer

A)   $98.32
B)   $99.52
C)   $100.12
D)   $101.42 E) B) and C)
F) A) and D)

Question 18

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

Accepted Answer

A) 1">δ>1\delta > 1δ>1 . B) As δ\deltaδ increases the volatility of interest rates increases. C) As δ\deltaδ increases the volatility of interest rates decreases. D) δ<0\delta < 0δ<0 . E) All of the above F) C) and D)

Question 19

Vasicek (1977)  posits a general mean-reverting form for the short-rate:   drt=κ(θ−rt) dt+σdWtd r _ { t } = \kappa \left( 	heta - r _ { t } ight)  d t + \sigma d W _ { t }drt​=κ(θ−rt​) dt+σdWt​  He then derives, in the absence of arbitrage, a restriction on the market price of risk  λ\lambdaλ  of any bond, where  (μ−r) /η=λ( \mu - r )  / \eta = \lambda(μ−r) /η=λ  of any bond, with  μ\muμ  being the instantaneous return on the bond, and  η\etaη  being the bond's instantaneous volatility. The derived restriction is that

Accepted Answer

A)    λ\lambdaλ  is a constant.
B)    λ\lambdaλ  may be a function of time  ttt  , but not of any other time-  ttt  information or of the maturity  TTT  of the bond.
C)    λ\lambdaλ  may be a function of the time-  ttt  short rate  rtr _ { t }rt​  , but not of current time  ttt  or of the bond maturity  TTT  .
D)    λ\lambdaλ  may be a function of time  ttt  and the time-  ttt  short rate  rtr _ { t }rt​  , but not of the bond maturity  TTT  . E) A) and B)
F) A) and C)

Question 20

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  σ=0.30\sigma = 0.30σ=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?

Accepted Answer

A)   1.000
B)   1.025
C)   1.050
D)   1.075 E) None of the above
F) C) and D) B

Exam 27: Factor Models of the Term Structure

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?

Correct Answer
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C

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.

Correct Answer
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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) ?

Correct Answer
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An exponential-affine short rate bond model is one

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

Correct Answer
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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?

Correct Answer
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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

Correct Answer
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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?

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

Correct Answer
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Correct Answer
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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) ?

Correct Answer
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In the Black-Derman-Toy (BDT) model, short rates are distributed as

Correct Answer
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B

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

Correct Answer
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Correct Answer
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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?

Correct Answer
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B

Exam 27: Factor Models of the Term Structure

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 σ=0.30\sigma = 0.30σ=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?

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedC

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

Correct AnswerverifiedShow Answer

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 σ=0.30\sigma = 0.30σ=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) ?

Correct AnswerverifiedShow Answer

An exponential-affine short rate bond model is one

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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?

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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 σ=0.30\sigma = 0.30σ=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) ?

Correct AnswerverifiedShow Answer

In the Black-Derman-Toy (BDT) model, short rates are distributed as

Correct AnswerverifiedB

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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 σ=0.30\sigma = 0.30σ=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?

Correct AnswerverifiedB

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?

Correct Answer
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C

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.

Correct Answer
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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) ?

Correct Answer
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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) ?

Correct Answer
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B

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

Correct Answer
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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?

Correct Answer
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B