To start off, I assume that Joseph's four year, $30 million deal is back-loaded at standard 4.5 percent increases. That means a starting salary of $7,025,762. It could be set up differently, the most likely alternative being a $7.5 million flat salary. I've assumed the same for Carroll. Delon Wright was signed to his contract, so his hit goes up a tiny bit. Bismack Biyombo is being signed with the Room Mid Level Exception ($5.75 million over two years), so he'll be signed after all the cap space is used up. Note that it is assumed here that Luke Ridnour is waived.
DeMarre Carroll $14,051,523
Kyle Lowry $12,000,000
DeMar DeRozan $9,500,000 (not including $600,000 unlikely incentive)
Cory Joseph $7,025,762
Patrick Patterson $6,268,675
Jonas Valanciunas $4,660,482
Terrence Ross $3,553,917
James Johnson $2,500,000
Lucas Nogueira (BeBe) $1,842,000
Bruno Caboclo $1,524,000
Delon Wright $1,509,360
Bismack Biyombo $2,814,000 (not included in cap calculation as signed with Room MLE)
Nando de Colo $1,901,900 (cap hold; $1,828,750 QO)
Above roster, if de Colo is let go, and excluding Biyombo: $64,960,811
Projected League Levels:
Cap: $67.1 million (rumours suggest it is closer to $69.1 million)
Tax: $81.6 million
Apron/Hard Cap: $85.6 million
That leaves $4,139,189 cap room. That could be enough to target a backup power forward (PF) or a backup wing if the Raptors wanted to trade some of their wing depth and picks for a starting PF.
Note that as ever, there are assumptions being made here. It's entirely possible the cap comes in at the projection of $67.1 million, which would reduce the potential cap room by half. It's also possible the Raptors structured Carroll and/or Joseph's contracts differently than I assumed. Flat deals with no increases would eat up another $1.5 million of cap room, leaving the Raptors with essentially no room at all.
One other assumption I should point out is DeRozan's contract. His cap number should drop back down to $9.5 million, due to his incentive becoming unlikely again. (This is based on him missing the All-Star game this past year.) But I've seen it suggested that it could stay at $10.1 million because he was injured. I can't find where in the CBA this would be allowed (such a consideration is only articulated at the time of signing a new contract or extension with incentives in it), but Ryan Wolstat says he has an NBA source that confirms the $10.1 million number. This would remove another $600,000 from any prospective cap room.
At this point, even if the Raptors have $4 million in cap room, any deal for a player making over $10 million is just as easy to pull off using the 150% trade rule as it is using cap space. So, any trade should follow those rules unless it is a low salary move. The upside of this is that if they are targeting a high paid PF via trade, they can make a move in free agency first to fill in another spot and it won't affect their ability to make the trade.
Unless, again, those assumptions are wrong, in which case that cap space might not exist.