Quoting and Managing Information Risk in a Card Sum Market-Making Game

# Market Making Card Sum Game


# Market Making Card Sum Game












I am preparing for an interview with a prop trading firm and wanted to discuss potential strategies for the classic market making games. I have seen similar posts on the forum, but a lot of the answers do not feel complete, so wanted to discuss on here. Here is a proposed game.

- Game #1 : Make a market on the sum of 3 cards. Each round, a card will be revealed.

I have a statistical value for the expectation, which is 21 (each card has an expected value of 7). Hence, I can make my spread pretty small and make a market of 20 @ 22. Assuming that the interviewer knows the cards and knows that the sum is 25, he will take my offer. Hence, I am now long 1 stock at 22. In the next round, the revealed card is 1, so my hypothetical expected value is now decreased at 15. This is where I would appreciate some input. These are the two choices that I am thinking about:

- Since I know that the interviewer is an informed trader, I can make the assumption that the next two cards must be very high despite this initial small one, as he had bought at 22. Hence, despite the fact that my theoretical expected value went down, I still make higher prices for the market as I follow the informed trader's direction. Hence, I now quote 22 @ 24.

- I can treat the interviewer as a noise trader, and hence do not let his trades influence my decisions too much. Hence, as my new expected value is 15, I lower the market to say 14 @ 16 despite the fact that he just took my offer.

In what event do I let the interviewer's trades take me outside of my expected value? I know that I am the market maker, so I ultimately decide on the prices. Is this simply dependent on whether the interviewer is an informed or noise trader? If he is an informed trader, do I follow his trades with no limit, even if they far exceed my personal expected value?

Also, as an additional question, do I try and stay flat and keep a neutral position?

I am a little confused to the available approaches, and would love to hear some input!

## Answer by Mats Lind (score 1)

https://quant.stackexchange.com/a/81317

Background

The less informed market maker, MM, cannot profit from trading with the informed trader. If the value from informed expectations is inside the bid/ask, the trader does not trade. Otherwise the trader does, earning an expected profit from the MM while giving away part of the information.

More precisely the trader tells the MM and the rest of the market that the "fundamental value" is likely outside the spread, and on which side at that. Now in order to value that information, MM will use additional info about the trader, is the trader really informed, is there information out there for instance near a quarterly report or CB decision, or is it maybe an uninformed trader making it likely after all that the fundamental value could anway be inside the spread.

Answers

The best MM can to to avoid losses to the informed trader is to quote prices whith a huge spread; at or outside the minimum and maximum of possible outcomes (payouts in the end).

"In what event do I let the interviewer's trades... " -You should not let the informed trader trade you, go out of the market by quoting a crazy wide spread or withdraw your quotes altogether.

"Is this simply dependent on whether the interviewer is an informed or noise trader?" -Exactly, that's the whole thing!

"If he is an informed trader, do I follow his trades with no limit, even if they far exceed my personal expected value?" -You cannot use your uninformed expectations when quoting prices.

"do I try and stay flat and keep a neutral position?" - yes

"so I ultimately decide on the prices" - You decide on your quotes, but then you get hit on them and while your inventory nears your limit the decision goes over more to your risk officer. The prices are decided by the players expectations on the soon to come terminal payout.

Discussion

In one version of the game, there has to be a trade before the next card is drawn. And in a multi-player setting there is a good chance of a trade to occur. But in the two player setting, with private information, I think the idea is to illustrate that there are situations in which the MM has to avoid trades to avoid losing money, even when risking its compliance with the MM agreement.

## Answer by quantinho (score 0)

https://quant.stackexchange.com/a/78228

Few corrections:

- The expected value is not 21 since all the cards are from the same deck (it's something between 19.5-19.6).

- If he hits your offer at 22, you will be short 1 share.

- Knowing the expected value is not enough to make such a tight spread unless you are asked to make bid/offer within N% confidence interval.

You have to stay flat for this game to make sense, meaning when the true sum is revealed if you are short/long you will have to close that position at the value of the sum.

Since it is an interview question, your reasoning is more important than winning. You need to make assumptions such as bid/ask price is a whole number and size is always one, the other side is playing optimal etc. Only then having a strategy makes sense.

To answer your questions, if you assume the interviewer knows the sum, he is already an informed trader and the only way for you to make money is when he is noise trading. So you have to make market based on his moves to minimize your loss (choice 1). On the other hand if you are both uninformed you can follow expected value and ignore his moves (choice 2).