Assume that the interest rate starts at 4% and in each period and either increases by 2% or decreases by 2% (from 4% up to 6% or down to 2% would be the first move). The risk-neutral probabilities of ups and downs are all 1/2.
This is a one-period discount bond with face of $100 and an interest rate equal to the initial rate of 4%. Therefore, the price is 100 ---- ~ $96.15 1.04
interest tree: 8% / 6% / \ 4% 4% \ / 2% \ 0% discount bond price: 100 / 94.34 / \ 92.49 100 \ / 98.04 \ 100
cash flows: 30 / 10 / \ 0 0 \ / -10 \ -30 price (pre-cash flow) 30 / 24.151 / \ -0.267 0 \ / -24.706 \ -30 calculations: 10 + (30 + 0)/2/1.06 = 24.151 -10 + (0 - 30)/2/1.02 = -24.706 (24.151 - 24.706)/2/1.04 = -.267
Assume that the interest rate starts at 6% and in each period and either increases by 2% or decreases by 2% (from 6% up to 8% or down to 4%). The risk-neutral probabilities of ups and downs are 1/2.
$94.3 = 100/1.06
quoted spot rates: 10% / 8% / \ 6% 6% \ / 4% \ 2% discount bond prices 100 / 92.6 / \ $89.0 100 \ / 96.2 \ 100 92.6 = 100/1.08, 96.2 = 100/1.04, 89.0 = 0.5 * (92.6 + 96.2)/1.06
cash flows 0 / 0 / \ 0 1 \ / 3 \ 5 Values 0 / 0.46 / \ $2.99 1 \ / 5.88 \ 50.46 = 0.5 * 1/1.08, 5.88 = 3 + 0.5*(1 + 5)/1.04, 2.99 = 0.5*(0.46 + 5.88)/1.06