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 XXXXXXXXXX
DT XXXXXXXXXX.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.
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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 co
esponding 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 > XXXXXXXXXXSelect 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.
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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-a
itrage price of this put
option at time 0?
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4. Options on Interest Rates- 25 Points
Using the annually-compounded short-term interest rates implied by the Ho-Lee model:
A
a0,1 = 6.30%
B
a1,2 = 8.47%
C
a1,2 = 4.84%
D
a2,3 = 10.0%
E
a2,3 = 6.32%
F
a2,3 = 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?
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