Question

In: Finance

The price of a zero-coupon bond (ZCB) that matures at time t=10 and that has face...

The price of a zero-coupon bond (ZCB) that matures at time t=10 and that has face value 100 is $61.62 Build an n = 10 binomial model lattice model with the following parameters to compute the initial price of a futures contract on the same ZCB that has an expiration of t = 4

r0,0 = 5%

u = 1.1

d = 0.9

q = 1 - q = ½

Solutions

Expert Solution

The short rate lattice and the ZCB lattice are as shown below

12.97%
(Short rate 11.79% 10.61%
lattice) 10.72% 9.65% 8.68%
9.74% 8.77% 7.89% 7.10%
8.86% 7.97% 7.17% 6.46% 5.81%
8.05% 7.25% 6.52% 5.87% 5.28% 4.75%
7.32% 6.59% 5.93% 5.34% 4.80% 4.32% 3.89%
6.66% 5.99% 5.39% 4.85% 4.37% 3.93% 3.54% 3.18%
6.05% 5.45% 4.90% 4.41% 3.97% 3.57% 3.22% 2.89% 2.60%
5.50% 4.95% 4.46% 4.01% 3.61% 3.25% 2.92% 2.63% 2.37% 2.13%
5.00% 4.50% 4.05% 3.65% 3.28% 2.95% 2.66% 2.39% 2.15% 1.94% 1.74%
t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10
100
89.45 100
(Zero coupon 81.58 91.20 100
bond lattice) 75.68 84.53 92.69 100
71.26 79.46 87.06 93.93 100
67.97 75.62 82.74 89.22 94.98 100
65.56 72.74 79.45 85.57 91.05 95.86 100
63.84 70.62 76.95 82.76 87.97 92.58 96.58 100
62.68 69.10 75.10 80.62 85.59 90.01 93.87 97.19 100
61.97 68.07 73.78 79.03 83.78 88.01 91.72 94.94 97.69 100
61.62196 67.44 72.88 77.89 82.42 86.47 90.05 93.16 95.83 98.10 100
t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10

The futures price at t=4 is the same as that of the ZCB at t=4. Working backward without discounting, we get the the present value of the Futures contract

65.56
68.09 70.62
70.47 72.86 75.10
72.72 74.96 77.07 79.03
74.82 76.93 78.90 80.73 82.42
t=0 t=1 t=2 t=3

t=4

the initial price of a futures contract on the same ZCB that has an expiration of t = 4 is $74.82


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