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

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?

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?

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

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?

The numeraire in the Swap Market Model (SMM) is

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?

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?

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?

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

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%.

Which of the following is not a valid property of the Heath-Jarrow-Morton (HJM) interest-rate framework?

The Libor Market Model is most often implemented by means of

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 $f ( 1,1 )$ ?

In the LMM, which of the following are martingales?

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

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%.

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

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

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

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 $f ( 1,1 )$ ?