Question

The price of a zero-coupon bond (ZCB) that matures at time t=10 and that has face value 100 is $61.62

An n = 10 binomial model lattice model has the following parameters: r0,0 = 5% u = 1.1 d = 0.9 q = 1 - q = ½

Compute the price of an American call option on the above ZCB. The option has expiration t = 6 and strike = 80.

Answer 1

The short rate lattice and the Zero Coupon Bond with parameters given is as given below

Short Rate Lattice :

										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

Zero Coupon Bond

										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

The Value of American Call option at Maturity (t=6) is given by max(St-K, 0)

and then moving backward , the Value of American option at every node is given by

= max ((q*value of option in upside+(1-q)*value of option in downside)/(1+shortrate at the node), St-K)

Using this formula the American option Lattice is as given below

						0.00
					0.00	0.00
				0.00	0.00	0.00
			0.29	0.62	1.31	2.76
		0.84	1.49	2.53	4.00	5.59
	1.56	2.45	3.64	5.08	6.56	8.01
2.36	3.39	4.65	6.03	7.42	8.77	10.05
t=0	t=1	t=2	t=3	t=4	t=5	t=6

So, the Value of the American Call option on the ZCB is $2.36