Question

V.

Compute the initial value of a forward-starting swap that begins t=1, with maturity t=10 and a fixed rate of 4.5%. The first payment then takes place at t=2 and the final payment takes place at t=11 as we are assuming, as usual, that payments take place in arrears. You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.

n = 10
r0,0 = 5%
u = 1.1
d = 0.9
q = 1 - q = 1/2

Answer 1

From the u and d given , the short rate lattice is as given 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

The swap pays the interest rate at any node - 4.5% in the next period. So, by discounting the cashflow receivable in the next period, the cashflow in the present period can be determined

So, if the notional principal is $1 ,

the payoff from the swap at t=10 ,   when short rate is 7.10% is determined as

=(0.071-0.045)/1.071 = 0.024276 or 0.0243

Similarly, the cash flows from the swap (with notional of $1) at different values of short rate is determined at t=10

Next, at t=9, we discount the cashflows due next period along with the cashflows already given by the swap

For example, when short rate is 7.89% at t=9

Payoff from the swap = (0.0789-0.045)/1.0789 + (q* payoff fom swap in the next upmove+ (1-q)* payoff from the swap in the next downmove)/1.0789

= (0.0789-0.045)/1.0789 + (0.5* 0.0385+ 0.5* 0.0243)/1.0789 = 0.0605

Similarly , we keep moving backwards till t=8, 7, 6 and so on till t=1

Since this is a forward swap, at t=0, we determine the payoff from the swap as

= (q* payoff fom swap in the next upmove+ (1-q)* payoff from the swap in the next downmove)/1.05

The complete Swap lattice is determined accordingly and is shown as 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

As can be seen that the swap has a value of $0.03337424 for a notional principal of $1

So, for a notional principal of $1 million, value of forward swap = 0.03337424 *1000000= $33374.24