I've been a lurker on this forum for a long time now but answering this question has caused me to create an account.
The main derivative that US banks are exposed to are interest rate swaps so I will only talk about those.
An interest rate swap is an agreement between two parties to swap the interest/coupon payments on a
notional amount, but this notional amount is never exchanged.
For example, if a European corporation with a substantial (fixed-rate) debt load who thinks USD interest rates are going to be lower and an American corporation with a substantial floating-rate (variable rate) debt load who thinks EUR interest rates are going to be lower; they can agree to swap the interest rates: for example swapping 5% for the Secured Overnight Financing Rate + 3%, etc.
When reporting derivative exposures, it is only the notional amount that is reported in financial statements even though the notional values are never exchanged.
So why do banks hold these interest rate swaps? 2 main reasons
1)
Hedging interest rate risk: the value of bonds and interest rates are negatively correlated, thus for banks that hold substantial amounts of mortgages, treasuries and other kinds of debt, if interest rates are going to increase, to mitigate this risk, they would hold something on their balance sheet whose price moves up when the price of previously issued debt goes down. Banks do this both with each other and with speculative institutions such as hedge funds and asset managers. These are mainly structured in terms of a fixed-rate being swapped for a floating rate based on the interest rate on a US Treasury security. Generally, either the banks swap their floating deposit rates with a fixed-rate (so banks don't have to pay more interest to depositors when interest rates increase) and/or swapping their fixed-rate loans with a floating rate swap (so banks don't take book value loses on previously issued loans when interest rate increase). This is mainly done to avoid a Silicon Valley Bank type situation where increases in interest rate risk force a financial institution to sell underlying bonds at a loss (SVB did not have interest rate swaps).
2)
Intermediation: private non-financial corporations will often want to hedge interest rate risk with each other but use a bank such as BNY Mellon, State Street, or Goldman Sachs to write up the agreement and collect and disperse payments between them. Corporations of course, wish to lower their interest rate expenses on previously issued debt and will agree with each other to (if they take different opinions on the path of interest rates) to swap payments in the future. This generally involves no bank money but because the bank is exposed, the notional value is added to a bank balance sheet.
The other big reason why issuers will engage in derivatives are for
speculation, but these are going to be done by financial institutions that are not systematically important such as hedge funds and asset managers that take the opposite side of the trade that banks take; due to the post-2008 regulations that require banks to hold substantial amounts of high-quality liquid assets and retained capital, banks rarely speculate on their own accounts (but often, they will speculate for other individuals on their private accounts held in trust at the bank)
