The model given follows the Ho-Lee framework, which can be discretized over a monthly time step ($\Delta t = 1/12 \approx 0.08333$). The change in the short-term rate for an upward move at a specific step $t$ is calculated as: $\Delta r_t = \lambda(t) \Delta t + \sigma(t) \sqrt{\Delta t}$

For the first month (date 0 to date 1), the change for an upward move is: $\Delta r_1 = 0.0018 \times \left(\frac{1}{12}\right) + 0.0050 \times \sqrt{\frac{1}{12}}$ $\Delta r_1 = 0.00015 + 0.0050 \times 0.288675 = 0.00015 + 0.001443 = 0.001593 \text{ (or } 15.93 \text{ bps)}$

For the second month (date 1 to date 2), the change for another upward move is: $\Delta r_2 = 0.0030 \times \left(\frac{1}{12}\right) + 0.0080 \times \sqrt{\frac{1}{12}}$ $\Delta r_2 = 0.00025 + 0.0080 \times 0.288675 = 0.00025 + 0.002309 = 0.002559 \text{ (or } 25.59 \text{ bps)}$

To find the change from the current level to the upper node at date 2 (two consecutive up moves), we sum the two changes: $\text{Total Change} = \Delta r_1 + \Delta r_2 = 15.93 \text{ bps} + 25.59 \text{ bps} = 41.52 \text{ bps}$

The closest estimate is 42 bps.