I'm having trouble grasping the operations of market makers. For example, consider Bank XYZ, which has set a bid-ask spread for T-Bond A at $90.1 (bid) - 90.2 (ask)$. Suppose a client of the bank decides to purchase T-Bond A at the quoted ask price of 90.2. My confusion lies in how the bank can fulfill this order if it doesn't currently possess T-Bond A. Additionally, for the bank to profit, it needs to acquire T-Bond A at a price lower than 90.2. I'm uncertain about how the bank manages this process?
Most market makers keep some inventory. Just like a supermarket has a stock of food ready to sell. I don't think you can call yourself a market maker and have zero inventory at all times. But it may be that the bond dealer does not have a particular bond.
Let's be perfectly clear that when you sell a bond that you don't have you are now short that bond. So you will lose money if the market price of that bond goes up.
There are several things you can do:
You can quickly go into the market and buy that bond (or a fairly similar bond). If you buy it for a higher price than you sold, you eat a loss. Presumably when you quoted 90.2 you were aware of this possibilty (maybe you should have quoted 90.3 ?). If market is very active someone else may want to sell you the bond at your bid soon, so you may be willing to wait a little.
You can buy a derivative like a bond future so you are hedged. There may be a special department whose job is to hedge the aggregate long/short position in this way, although it is seldom done just for one sale of a bond.
You can borrow the bond. Let's keep in mind that this turns the uncovered short position into a covered short position which you can continue indefinitely, but you are still short the bond and are liable for price increases until you buy it. So this does not decrease your risk and is usually done for operational and legal compliance reasons (to satisfy rules on delivery), not risk management.
Securities borrowing and lending via the repo market. If the bank doesn't own the bond, it can borrow the bond post sale and deliver it in a certain time frame to the buyer at the agreed price.
Ofcourse it will need to collateralise the borrow with say cash, for which there is some interest rate paid (the reverse repo rate for eg), and depending on the quality of the bond potentially some haircuts.
Too long for a comment, just adding a little historical color to the other good answers.
After the 2007-2008 Global Financial Crisis), many U.S. market makers became subjected to the "Volcker rule", which, greatly oversimplifying, among other things, doesn't allow market makers to keep more inventory than the RENTD. Many market makers have rather clever sophisticated RENTD models to explain and jusify their inventory. Most have been forced to transform from the market making equivalent of Wegman's - a supermarket simply trying to keep everything on its shelves - to something more resembling Aldi's, where a shopper is less likely to see something sitting on the shelf that the sellers and their models don't reasonably anticipate to move quickly.
Two tools that may help cover are worth mentioning, although the price can be anything:
Interdealer brokers help market participants trade with each other anonymously. There used to be more of them. After some merges, we are only left with BGC, ICAP Tullett Prebon, Tradition, etc.
Many traders maintain some proprietary "databases" in their heads, or in Bloomberg Notes, or on paper, saying that some buy-side friend bought a large quantity of some instrument and might be willing to sell back a small quantity as a favor.
It is good to hedge unwanted market risk - at least the interest rate sensitivity - in the inventory as well as in the short positions not yet covered.