Exam 27: Factor Models of the Term Structure

In the Vasieck (1977)model,you are given that $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ 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 standard deviation of the short rate of interest one year hence?

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

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

In the Cox-Ingersoll-Ross (CIR 1985)model,you are given that $d r _ { t } = x \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ 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?

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

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

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.

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

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

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

An exponential-affine short rate bond model is one

In the Cox-Ingersoll-Ross (CIR 1985)model,you are given that $d r _ { t } = x \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ 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?

Vasicek (1977)posits a general mean-reverting form for the short-rate: $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ 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

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

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

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)?

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)?

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?

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?

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?