Question

The initial value of a forward-starting swap that begins at t=1, with maturity t = 10, and a fixed rate of 4.5% is $33,374.

Build an n = 10 binomial model lattice model with the following parameters to compute the initial price of a swaption that matures at time t=5 and has a strike of 0. The underlying swap is the same swap as described in the previous question with a notional of 1 million. To be clear, you should assume that if the swaption is exercised at tt=5 then the owner of the swaption will receive all cash-flows from the underlying swap from times t=6 to t = 11 inclusive. (The swaption strike of 0 should also not be confused with the fixed rate of 4.5% on the underlying swap.)

r_0,0 = 5%

u = 1.1

d = 0.9

q = 1 - q = ½

Answer 1

From the values of r0,0 , u and d as given, the short rate lattice is as shown

										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

The forward swap lattice is as shown below :

										0.0750
									0.1234	0.0552
								0.1524	0.0897	0.0385
							0.1666	0.1083	0.0605	0.0243
						0.1692	0.1146	0.0698	0.0356	0.0124
					0.1626	0.1112	0.0689	0.0366	0.0145	0.0024
				0.1486	0.0998	0.0598	0.0292	0.0082	-0.0033	-0.0059
			0.1282	0.0819	0.0439	0.0149	-0.0050	-0.0159	-0.0183	-0.0128
		0.1026	0.0583	0.0223	-0.0052	-0.0239	-0.0341	-0.0362	-0.0308	-0.0185
	0.0724	0.0301	-0.0042	-0.0301	-0.0477	-0.0571	-0.0588	-0.0533	-0.0412	-0.0232
0.03337424	-0.0023	-0.0348	-0.0593	-0.0757	-0.0842	-0.0854	-0.0796	-0.0675	-0.0498	-0.0271
t=0	t=1	t=2	t=3	t=4	t=5	t=6	t=7	t=8	t=9	t=10

(Please note that the swap shown above has a notional principal of $1. So the total swap value = $0.03337424*1000000 = $33374.24)

Now, a swaption expiring at t=5 will have the maximum of either the swap value or 0 at t=5 . So working backwards by discounting the expected two values by the current short rate at the node, the value of the swaption can be found at t=0.  It is calculated as shown in the lattice

					0.1626273
				0.1222739	0.0998227
			0.0891090	0.0678044	0.0439084
		0.0618503	0.0420755	0.0209286	0.0000000
	0.0410751	0.0248182	0.0100180	0.0000000	0.0000000
0.0263111	0.0141781	0.0048140	0.0000000	0.0000000	0.0000000
t=0	t=1	t=2	t=3	t=4	t=5

(Notional principal is $1)

So, the value of the swaption = $0.0263111 * 1000000 = $26311.08