After trading pokémon, what is the range of new IVs?

According to this Silphroad thread on Reddit people there are quite confident that there are IV floors that increase based on friend level. So good friends have minimum IV of 1/1/1, great friends - 2/2/2, ultra friends - 3/3/3 and best friends 5/5/5.

On IV ceiling the debate seems to be still open. Some threads suggest there is none and you could (with a low probability) get a 15/15/15 (100%) from something that was initially lower. Some threads like this suggest that the upper ceiling is the current IVs of that Pokemon.

As mentioned in the other answer, the floor depends on your friendship level. Good friends have a floor of 1/1/1, great friends have 2/2/2, ultra friends 3/3/3, and best friends 5/5/5. If a trade is lucky, the floor is 12/12/12, regardless of friendship level.

There is no ceiling to the IVs of traded Pokémon. They can be perfect even if you have a low friendship level or if the IVs are awful. Source: I have three perfects from trades, and my friends and I regularly trade to "reroll" stats of low IV Pokémon.

Stats are completely random between the floor and the ceiling, so it's often best to trade Pokémon with the worst IVs, since they have the greatest room for improvement.