QUESTION #1: INTEREST RATE MODELS (15%) Setup: Suppose the risk neutral dynamics of the short rate evolve according to a binomial lattice model. Assume the following: The current short rate is ro The period length of the lattice is 1 year The rate moves up by a multiple u and down by a multiple d= 7 The risk neutral probabilities are q = 0.5 for an up move and 1-q=0.5 for a move down (a): Suppose you observe a 1-year 2% coupon bond (annual coupons) trading at $97.14 and a 2-year 3% coupon bond trading at $96.21. Calculate the prices of 1 & 2-year zero coupon bonds (ZCBs). (b): Use the prices from (a) and what you know about the short rate dynamics to determine ro, u, & d. (Hint: Derive expressions for the two ZCB prices as functions of ro, u, & d.) (c): Draw the short rate lattice for three years and label with the rates you calculate. (d): Calculate the price of a 2-year floating rate bond (annual coupons) where the interest rate in any period is capped at 5.5%. What is the value of the rate cap feature? (Hint: recall that the coupon rate for a floating rate bond resets at the short rate at the beginning of the period, but is not paid until the end of the period.) (e): Calculate the price and duration of a fixed rate bond paying annual coupons of 6% and maturing in 3 years (assume face value of $100). (*): Calculate the fair value of 2-year interest rate swap with a 4.0% fixed rate (assume annual payments and a notional of $100). What is the current swap rate? QUESTION #1: INTEREST RATE MODELS (15%) Setup: Suppose the risk neutral dynamics of the short rate evolve according to a binomial lattice model. Assume the following: The current short rate is ro The period length of the lattice is 1 year The rate moves up by a multiple u and down by a multiple d= 7 The risk neutral probabilities are q = 0.5 for an up move and 1-q=0.5 for a move down (a): Suppose you observe a 1-year 2% coupon bond (annual coupons) trading at $97.14 and a 2-year 3% coupon bond trading at $96.21. Calculate the prices of 1 & 2-year zero coupon bonds (ZCBs). (b): Use the prices from (a) and what you know about the short rate dynamics to determine ro, u, & d. (Hint: Derive expressions for the two ZCB prices as functions of ro, u, & d.) (c): Draw the short rate lattice for three years and label with the rates you calculate. (d): Calculate the price of a 2-year floating rate bond (annual coupons) where the interest rate in any period is capped at 5.5%. What is the value of the rate cap feature? (Hint: recall that the coupon rate for a floating rate bond resets at the short rate at the beginning of the period, but is not paid until the end of the period.) (e): Calculate the price and duration of a fixed rate bond paying annual coupons of 6% and maturing in 3 years (assume face value of $100). (*): Calculate the fair value of 2-year interest rate swap with a 4.0% fixed rate (assume annual payments and a notional of $100). What is the current swap rate
