assignment
Fixed Income Securities MGMT41250 Krannert School of Management Purdue University Problem Set 3 Lecturer: Adem Atmaz 1. Ho-Lee Model - 25 Points Suppose the short-term interest rate volatility is σ = 1.7% and the discount factors for years 1, . . . , 4 as follows: T 1 2 3 4 DT 0.954 0.902 0.851 0.800 (a) Compute the short-term discount factor tree with yearly time steps (h = 1) implied by the Ho-Lee model. That is, populate the following Binomial tree: A(j=0) D0,1 =? B(j=0) D1,2 =? C(j=1) D1,2 =? D(j=0) D2,3 =? E(j=1) D2,3 =? F(j=2) D2,3 =? G(j=0) D3,4 =? H(j=1) D3,4 =? I(j=2) D3,4 =? J(j=3) D3,4 =? t = 0 t = 1 t = 2 t = 3 (b) Compute the continuously compounded short-term interest rate tree implied by the Ho-Lee model. 1 2. Curve Fitting and Ho-Lee Model - 25 Points Consider the following data on continuously compounded interest rates: T r0,T 1 Month 5.58% 3 Month 5.72% 6 Month 5.95% 1 Year 6.11% 2 Year 6.26% 3 Year 6.50% 5 Year 7.05% 10 Year 7.80% (a) Using the Nelson-Siegel method and Excel solver determine the model implied continuously compounded interest rates for the maturity of 1 month, 2 month, ...,5 month, 6 month as well as the corresponding 6 discount factors, that is, D0, 112 , D0, 212 ,..,D0, 512 , D0, 612 . (Set the initial values of parameters as as θ0 = 5.58%, θ1 = 0, θ2 = 0, λ = 1. Subject to the constraint that λ > 0.001. Select the solving method as “GRG Nonlinear”) (b) Using Excel and the model implied 6 monthly discount factors you obtained in part (a) and the fact that the short-term interest rate volatility is σ = 1.7%, compute the short-term discount factor and the continuously compounded short-term interest rate trees with monthly time steps (h = 1/12) implied by the Ho-Lee model. 2 3. Options on Bonds - 25 Points Consider the following short-term discount factor tree implied by the Ho-Lee model: A D0,1 = 0.9540 B D1,2 = 0.9294 C D1,2 = 0.9616 D D2,3 = 0.9114 E D2,3 = 0.9429 F D2,3 = 0.9755 G D3,4 = 0.8922 H D3,4 = 0.9230 I D3,4 = 0.9549 J D3,4 = 0.9880 t = 0 t = 1 t = 2 t = 3 Consider a put option with a strike price X = 940 and maturity 2 years (T o = 2) written on a 4-year zero coupon bond (T = 4) with face value $1000. What is the no-arbitrage price of this put option at time 0? 3 4. Options on Interest Rates- 25 Points Using the annually-compounded short-term interest rates implied by the Ho-Lee model: A ra0,1 = 6.30% B ra1,2 = 8.47% C ra1,2 = 4.84% D ra2,3 = 10.0% E ra2,3 = 6.32% F ra2,3 = 2.77% t = 0 t = 1 t = 2 (a) What is the evolution of the floorlet 1 with a strike rate x = 5%, maturity T o = 1 year and the notional amount N = 1000 written on the short-term interest rate? (b) What is the evolution of the floorlet 2 with a strike rate x = 5%, maturity T o = 2 year and the notional amount N = 1000 written on the short-term interest rate? (c) What is the evolution of the floor which is the portfolio of floorlet 1 and floorlet 2? 4