To calibrate a three-period binomial interest rate model, we can use the information provided and work backward to determine the interest rate values in each node. Here's how we can calculate the value of the interest rate in the node r_2, HH:
Start with the given information:
1-year spot rate = 3% (0.03)
A 2-year 6% annual coupon bond is trading at par
A 3-year 7.5% annual coupon bond is trading at par
Interest rate volatility (sigma) = 10% (0.10)
Determine the values for the up and down factors:
For a three-period model, the up factor (u) is calculated as u = e^(sigma * sqrt(dt)), where it is the length of each period.
Since we have an annual model, dt = 1 year.
u = e^(0.10 * sqrt(1)) = e^0.10 = 1.105171 (approximately)
The down factor (d) is the reciprocal of the up factor: d = 1/u = 1/1.105171 = 0.904554 (approximately)
Calculate the risk-neutral probability (p):
In a three-period model, p = (e^(rt) - d) / (u - d), where r is the risk-free rate and t is the length of each period.
Since we have an annual model, t = 1 year, and the risk-free rate (r) is the 1-year spot rate of 3% (0.03).
p = (e^(0.03*1) - 0.904554) / (1.105171 - 0.904554) ≈ 0.526682 (approximately)
Calculate the interest rate in the node r_2, HH:
In the binomial model, the interest rate in each node is derived based on the up and down factors and the risk-neutral probability.
To calculate r_2, HH, we multiply the previous node's interest rate by the up factor twice (HH means two upward movements).
r_2,HH = r_1 * (u^2)
Since r_1 is the 1-year spot rate of 3% (0.03), we substitute the values:
r_2,HH = 0.03 * (1.105171^2) ≈ 0.036774 (approximately)
Therefore, the value of the interest rate in the node r_2, HH is approximately 3.677% (rounded to three decimal places).