Question

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

Answer 1

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