Explanation

In the Ho-Lee model, the interest rate tree follows a binomial structure where:

• The up and down movements have equal probabilities (0.5 each)
• The volatility σ is constant
• The drift λₜ is time-dependent

From the given tree:

• Initial rate r(0) = 4%
• Month 1: Up state = 4.661%, Down state = 3.506%
• Month 2: From up state: Up-up = 5.305%, Up-down = 4.15%

We need to find the missing node [2,0] (the down-down state).

Step 1: Calculate the volatility σ

The difference between up and down states at time 1: 4.661% - 3.506% = 1.155%

Since this is a one-period difference, and in Ho-Lee model the spacing between nodes is σ√Δt: σ√Δt = 1.155%

Assuming Δt = 1 month = 1/12 year: σ√(1/12) = 1.155% σ = 1.155% × √12 ≈ 1.155% × 3.464 = 4.0%

Step 2: Find the missing node [2,0]

From the down state at time 1 (3.506%), the down movement should be: r[2,0] = r[1,1] - σ√Δt = 3.506% - 1.155% = 2.351%

However, this doesn't match any options exactly. Let's verify using the tree structure:

From the given nodes:

• The spacing between [1,1] and [1,0] is 1.155%
• The spacing between [2,2] and [2,1] is 1.155%
• Therefore, the spacing between [2,1] and [2,0] should also be 1.155%

So: r[2,0] = r[2,1] - 1.155% = 4.15% - 1.155% = 2.995%

This matches option C.

Verification: The tree maintains constant volatility with equal spacing between adjacent nodes at each time step, confirming that 2.995% is the correct value for the missing node [2,0].