Market-Making stocks with random inventory exposure

Question

In the traditional case of stock market-making, a market-maker calculates optimal bid-ask quotes which are based on various variables like current inventory, spot price, volatility etc. The market-maker is in control of their inventory and will only take on inventory in they choose to.

Is it possible to develop a strategy where a market-maker receives inventory (long or short) at a random rate and random amount without their choosing.

As a literal example, a contract-for-difference (CFD) company offers quotes on the SPY for their customers at 521-522, but they also post bid-ask quotes on an open exchange which will have a tighter spread, i.. 521.60 - 521.70. The CFD broker has 0 control over how their clients choose to trade and are forced to take long/short positions. Can they effectively post bid-ask quotes in the open market for profit?

My logic is that it's possible because if their clients are collectively long at 522, therefore they have a synthetic short position at 522. They then place bid LOs at 521.60, with the advantage that they are short at 522 AND that they didn't pay any fees for that side of the trade.

An analogous situation could be that a traditional market-maker is making a market for NVIDIA stock, but they also have to factor in the positions of their equities research department. So if their ER department are suddenly long NVIDIA, they calculate their quotes on the fact that the company as a whole are currently long NVIDIA. They have 0 control over that department, the department's choices are random and they only know their position after the long NVIDIA position is placed.

Answer 1

A market is made for a particular point in time. A change in inventory is a piece of information that can be input into a pricing model to update the market that is being made.

If a pricing model has a term that is dependent on inventory or position, e.g. a pricing model that is inversely proportional to position in order to manage risk, then the prices would be updated every time there is an inventory change.

Suppose my pricing model encourages me to stay within a certain position range. What characteristics do we know about the incoming volume? If the aim is to maintain a position range, then an increased variance in the incoming volume would probably mean increased spread in my market, and any asymmetry in the net incoming volume could be multiplied by a proportional retreat in order to maintain the position.

However if a pricing model doesn't depend on inventory or position, then adding random position would not require an update to the price.

Answer 2

I'm not sure that's the correct way to view the clients of the CFD broker. You say:

The CFD broker has 0 control over how their clients choose to trade

but that's really the same as any other market maker; the market maker has 0 control over how the market chooses to trade. In fact, a CFD broker likely has no contractual obligations to their clients compared to MMs who have a contractual obligation to the exchange, so the CFD can control their position by halting trading, widening the spread etc.

As for internal clients of traditional MM's, I believe the hedge fund desk gives the Equities department a price; and the Equities department can choose to either take that price, take the market price, or just not trade.