The Ho-Lee model formula for interest rates in a binomial tree at a given node after $i$ steps is: $r_{i} = r_0 + \sum_{k=1}^{i} \theta_k \Delta t + (\text{net up moves}) \times \sigma \sqrt{\Delta t}$

Given:

• Initial rate $r_0 = 3.2\% = 0.032$
• Time step $\Delta t = 1 \text{ month} = 1/12 \text{ years}$
• Volatility $\sigma = 2.1\% = 0.021$
• Month 1 drift $\theta_1 = 80 \text{ bps} = 0.008$
• Month 2 drift $\theta_2 = 120 \text{ bps} = 0.012$

To find the lowest node after 2 months, we assume two consecutive "down" moves. Each down move subtracts $\sigma \sqrt{\Delta t}$. Total drift over 2 months = $(\theta_1 + \theta_2) \times \Delta t = (0.008 + 0.012) \times (1/12) = 0.020 / 12 = 0.001667$ (or $0.1667%$) Volatility term per step =$\sigma \sqrt{\Delta t} = 0.021 \times \sqrt{1/12} \approx 0.021 \times 0.288675 = 0.006062$(or `$0.60`62%$)

Since it's the lowest node after 2 steps (down, down), the net volatility impact is: $2 \times (-0.006062) = -0.012124$(or$-1.2124%$)

Now, calculate the rate at the lowest node ($r_{2, \text{min}}$): $r_{2, \text{min}} = r_0 + \text{Total Drift} + \text{Total Volatility Impact}$ $r_{2, \text{min}} = 0.032 + 0.001667 - 0.012124$ $r_{2, \text{min}} = 0.021543 \approx 2.15\%$