Solution:

The octahedron can be viewed as two congruent pyramids whose shared base is the rhombus through the centers of the four side faces. This rhombus has diagonals $4$ and $5$, so its area is $\frac12\cdot4\cdot5=10$.

Each pyramid has height $\frac32$, half the prism's height. Thus the total volume is $$2\cdot\frac13\cdot10\cdot\frac32=10.$$

Thus, the correct answer is B.