The Iron Bank has the right to claim Dragonstone.
Mortgages do not exist for the Iron Bank (not in this sense, at least).
In real life, mortgages are a way to settle a debt that the debtor is refusing to pay, without having to resort to violence. In essence, you sign a contract promising to give away your property should you fail to pay the debt.
However, the Iron Bank can (and will) use violence. Violence is not only still actively used, but it is also accepted more (e.g. flogging, torture, armed combat), it still exists even in legislature (trial by combat).
Tommen tries to remove the legal option of trial by combat, and while Tommen honestly believed that it was done because it's needlessly barbaric, in reality it was a ploy by the High Sparrow to prevent Cersei from escaping justice by using the Mountain to fight for her and basically acquit her from her crimes.
Mortgages are not needed if you're allowe to use violence against debtors who refuse to pay you back. So let's discuss the Iron Bank's strategy to ensure that debtors will pay back their debt.
1. Are you likely to be able to pay back your debt?
Are you asking for a loan that you will never be able to pay back? Are you considered capable enough to actually achieve what you want and be able to pay the Iron Bank (including interest)?
If not, you won't even get the loan to start with. But let's assume that the Iron Bank gives you the loan because you are a wise investment.
2. You stop paying your debt.
If this is a temporary issue, and you will soon be able to resume the payments, the Iron Bank may grant you an extension (more interest for them to collect).
If this seems to be a permanent issue, then the Iron Bank takes action. Their favorite way of doing so, especially for loans extending to (would-be) kings, is to start financially supporting your opponent.
The main goal of the Iron Bank here is not to collect your outstanding debt. But rather, the Iron Bank uses revenge, thus ensuring your quick demise.
They do manage to recover some money, from the interest that the opponent pays them.
This approach is intended to breed fear into the next people who ask the Iron Bank for a loan, who will be less likely to enter into a loan agreement if they might not be able to pay you back.
This is corporal punishment. Teachers and parents use violence against you when you misbehaved, so that you wouldn't want to misbehave again (fearing the pain). This would also work by proxy. If your sibling gets a beating, then you will be afraid too. And that's what the Iron Bank does. It punishes one, so that the others are now afraid and more likely to fall in line.
It's not impossible for the Iron Bank to lose money in the short term. This was the case for Stannis. However, you should consider the Iron Bank as "playing the stock market", backing claimants to the throne (as opposed to companies).
While the Iron Bank can potentially lose money on a debtor here and there (Stannis), they will always come out ahead in the bigger picture.
Tycho Nestoris often speaks of "wise investments", and he means "giving money to people who are likely to pay it back with interest". There's no guarantee, but they back the right people more often than not, enough that they actually make a profit.
