Paracord is usually cut cleanly with a hot knife. Short of that, you can melt it in two over a flame or cut with a knife and then melt the ends. I'll show a simple and very clean alternative. This can be used on paracord or any other small, synthetic cord with an outer sheath and inner fibers.
Step 1: Heat Carefully Over a Flame
At the point where you'll make your cut, slowly heat the cord over a flame until the outer sheath gets a slight sheen. This tells you that the sheath is melting. If you heat slowly, the inner fibers will also begin to melt.
Note that if the cord begins to melt too much, the diameter of the cord will start to increase (bubble). This may not be a problem but if you need a very clean end at the same nominal size of the cord, you want to be conservative here and not heat too much.
Step 2: Check for Adequate Heating
The outer sheath will look as shown in the photo and will feel hard. The point of the cut should be slightly shinier than the rest of the cord.
Step 3: Cut the Cord
With scissors or a sharp knife, cut the cord in the center of the heated area.
Step 4: Melt a Little More If Needed
If you heated the cord just enough, you'll already have a clean and fray-free end. To ensure your ends are robust, you may want to heat the tip of each end just a little more.
Step 5: Nice, Clean, and Small Cord End
You should now have a nice, clean end. The end is much closer to the original cord diameter which is hard to accomplish when cutting the cord prior to heating.