We need $0 < st < 1$, which means $st$ is positive and less than 1.

Check each statement:

I. $s < -1$ and $t > 0$: If $s < -1$ (negative) and $t > 0$ (positive), then $st < 0$ (negative). This violates $st > 0$. ✗

II. $s < -1$ and $t < -1$: Both negative means $st > 0$ ✓. If $s = -2$ and $t = -0.4$, then $st = 0.8$, which satisfies $0 < st < 1$ ✓

Actually, wait. Let's reconsider: if $s < -1$ and $t < -1$, then both are less than $-1$, so both magnitudes are greater than 1. The product would be greater than 1, violating $st < 1$. Let me recalculate: if $s = -1.5$ and $t = -0.5$, then $st = 0.75$, which works. But $t = -0.5$ does not satisfy $t < -1$. If both $s < -1$ and $t < -1$, then $|s| > 1$ and $|t| > 1$, so $|st| > 1$, meaning $st > 1$ since both are negative (making product positive). This violates $st < 1$. ✗

III. $s > -1$ and $t < -1$: This gives us $-1 < s$ and $t < -1$. If $s$ is positive (satisfies $s > -1$) and $t$ is negative with $|t| > 1$, then $st < 0$. ✗ If $s$ is negative with $|s| < 1$ (like $s = -0.5$) and $t < -1$ (like $t = -1.5$), then $st = (-0.5)(-1.5) = 0.75 > 0$ and $< 1$ ✓

Actually, I need to reconsider option I more carefully...