We will first calculate the price of the bond and then recalculate the price in the two buckets when forward rates increase by 1 basis point.

The value of the bond is

$$\frac{2.5}{1.02^1} + \frac{2.5}{1.02^2} + \frac{2.5}{1.02^3} + \frac{102.5}{1.02^4} = \text{USD } 101.9039$$

When the forward rates in the 0-1 year bucket increase by one basis point, the value of the bond becomes

$$\frac{2.5}{1.02005^1} + \frac{2.5}{1.02005^2} + \frac{2.5}{1.02005^2 \times 1.02^1} + \frac{102.5}{1.02005^2 \times 1.02^2} = 101.8940$$

Thus, the value of the bond has decreased by $101.9039 - 101.8940 = 0.0099$

When the forward rates in the 1-2 year bucket increase by one basis point, the value of the bond becomes

$$\frac{2.5}{1.02^1} + \frac{2.5}{1.02^2} + \frac{2.5}{1.02005 \times 1.02^2} + \frac{102.5}{1.02005^2 \times 1.02^2} = 101.8945$$

Thus, the value of the bond has decreased by $101.9039 - 101.8945 = 0.0094$

The forward bucket 01s are therefore 0.0099 and 0.0094