The first factor in stability is pitch inertia. Even though it is just a sheet of paper, its moment of inertia around the pitch axis is quite high, so any pitch motion is slow.
Next is aerodynamic pitch damping: As the paper starts to pitch, local forces at the edges will produce a stabilizing moment.
That the paper will slowly pitch up is due to the location of the center of lift, which will act at a quarter of the chord. The precise location is slightly ahead of the quarter chord, the more so, the more slender the paper is (ratio of span to chord length). This creates a pitch-up moment, and since the paper itself is rather floppy, will bend the whole paper into a graceful curve. This again helps to stabilize the paper, because a negatively cambered wing is naturally stable.
Your paperclip weight is needed to shift the center of gravity forward to one quarter of the paper airplane's chord. This is where the resulting lift force will act, so by shifting the cg forward, you minimize pitch moments. You could as well fold the first half of the sheet of paper into a tight roll of paper - this will produce the same effect as the paperclip. Once the pitch angle increases, so does lift and angle of attack. Beyond an angle of attack of maybe 10° or 15°, airflow on the top will separate, which will shift the center of lift backwards. Also, lift stops to increase with further angle of attack increases. Now the lift creates a pitch-down moment, which helps to regain the attitude at which the flow is still partially attached. Especially when using stiffer cardboard, this stabilizing effect of flying with a partially stalled wing is easy to reach once the center of gravity is right.
In folded paper airplanes, you will notice that a slightly bent-up trailing edge will help to stabilize the paper airplane, whereas folding it down will make sure it goes into a dive quickly. By folding the trailing edge at least partially up, you will create a local area of lower lift (or actual downforce), which will see a proportionally stronger increase of lift when the whole paper airplane pitches up. This results in a pitch-down moment, and vice versa. This is all what is needed to give a paper airplane positive stability.
A simple sheet will be too floppy to benefit from the stabilizing effect of a bent-up trailing edge, though.
Best Answer
I think I've found the answer. On the pitching moment page it says:
However the question remains, what is the position of the aerodynamic centre and centre of pressure?