There isn't magic here. Your transactions are presumably recorded in your books and that of your counterparty. There are billions of books...
This is misunderstanding BIS's point. The BIS is talking about the size of the market - ie they are concerned about liquidity and system wide risk. They are not saying the pension funds are short trillions via FX swaps.
Off balance sheet is a term of art in this space that means it’s not listed as an asset or liability for the purposes of say fx exposure.
The reason for this is that fx swaps dont create exposure to fx or credit swings the way that fx trading or borrowing against future cash flows do because they are fixed rate contracts.
There is counterparty risk but a) that is typically mitigated much more cheaply and us generally less risky and b) you’d move the loss “on book” if the counterparty risk shapes up.
Imagine you go to the bank to get a loan for a new tractor for your business. They ask to see a list of your assets and liabilities (your balance sheet).
You own a house worth $1 million with no mortgage.
You have an insurance policy.
You have nothing else.
If bushfires start right next to your house as you are meeting the bank manager, you don't say to him/her: I have a house that is worth 900k marked to market because of the bushfire risk, and I marked my insurance policy to be worth $100k.
You have a house worth $1 million. The insurance policy is "off balance sheet" despite the fact it could be worth a lot of money.
In this thread I give an example of the Australian Future Fund and Coke bonds. It shows what is meant by off balance sheet. It's a matter of netting things v not, it's not a matter of just leaving things out.
This is misunderstanding BIS's point. The BIS is talking about the size of the market - ie they are concerned about liquidity and system wide risk. They are not saying the pension funds are short trillions via FX swaps.