Question

n = 10 r0,0 = 5% u = 1.1 d = 0.9 q = 1 - q = ½

Compute the price of a zero-coupon bond (ZCB) that matures at time t=10 and that has face value 100.

Answer 1

The short rate r0,0 = 5% . From the values of u and d , the two possibilities of interest rates in the next period (t=1) are  r0,0*u and r0,0*d . Similarly for t=2 and so on till t=10. The short rate lattice for 10 periods is as shown below :

										12.97%
									11.79%	10.61%
								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

Now, a Zero coupon bond maturing at t=10 will only pay face value ie. $100 at t=10.

So, at t=10, no matter what the interest rates be, the ZCB will pay $100

So, at all previous periods , the Expected value of the ZCB will be the present value of the expected value of the ZCB in the two possible situations next period., calculated by discounting at the short rate prevailing at the beginning of the period

For ex- at t=9, when r9,9 =11.79%, the two possible values of ZCB at t=10 are $100 and $100

So, present value of expected value of ZCB in next period

= (q*$100+(1-q)*$100)/1.1179

= (0.05*$100+0.5*$100)/1.1179 = $89.45

Similarly one can calculate values of ZCB for all possible values of short rate at t=9. Next , we move backwards in time and calculate values of ZCB for all possible values of short rate at t=8 and so on,

The ZCB price lattice is as shown below

										100
									89.45	100
								81.58	91.20	100
							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

From the lattice, we can say that the ZCB has a price of $61.62 at t=0.

So, the price of the Zero coupon bond today is $61.62