The swap pays off when the interest rate exceeds 6%.

For the two scenarios that can occur in the first 6 months, the payoff of the swap will be: Payoff if rate increases to 6.5% = $1,000,000 \left(\frac{6.5% - 6%}{2}\right) = `$2`,500$ Payoff if rate decreases to 5.5% = $1,000,000 \left(\frac{5.5% - 6%}{2}\right) = -`$2`,500$

For the three scenarios that can occur after one year, the payoff of the swap will be: Payoff if rate increases to 7% = $1,000,000 \left(\frac{7% - 6%}{2}\right) = `$5`,000$ Payoff if rate remains at 6% = $1,000,000 \left(\frac{6% - 6%}{2}\right) = `$0`$ Payoff if rate decreases to 5% = $1,000,000 \left(\frac{5% - 6%}{2}\right) = -`$5`,000$

Based on the binomial tree, the probabilities are:

• At node 0 (6%), probability of an up move to 6.5% is 45%, and down move to 5.5% is 55%.
• At node U (6.5%), probability of an up move to 7% is 70%, and down move to 6% is 30%.
• At node D (5.5%), probability of an up move to 6% is 70%, and down move to 5% is 30%.

Value at Node U (6.5% at t=0.5): Expected payoff at t=1 = (0.70 \times \`5,000) + (0.30 \times \0`) = \`3,500$ Discounted back to t=0.5 at 6.5%: $\3`,500 / (1 + 0.065 / 2) = \`3,389.83$ Total value at Node U = $\2`,500 + \`3,389.83 = \$5`,889.83$

Value at Node D (5.5% at t=0.5): Expected payoff at t=1 = (0.70 \times \`0) + (0.30 \times -\5`,000) = -\`1,500$ Discounted back to t=0.5 at 5.5%: $-\1`,500 / (1 + 0.055 / 2) = -\`1,459.85$ Total value at Node D = $-\2`,500 - \`1,459.85 = -\$3`,959.85$

Value at Node 0 (6% at t=0): Expected value at t=0.5 = (0.45 \times \`5,889.83) + (0.55 \times -\3`,959.85) = \`2,650.42 - \2`,177.92 = \`472.50$ Discounted back to t=0 at 6%: $\472.50` / (1 + 0.06 / 2) = \`458.74`$

The value of the swap is $458.74.