Interview Market Making Strategy

This market-making question comes from a prop trading final round which I failed.

I was told to make a market on the number of prime numbers between 1-100. I was confident the number was around 20-30. (It's 25). So I made my market at 20-25. (Interviewer only allowed a spread of 5). They proceeded to buy my sell offer at 25. Then I moved from 30-35. They bought it again. I moved from 35-40. They kept buying until it was 65-70, then sold at 65.

They then stated they have profited off me and I have in fact lost money as they bought for 25,35,etc.

I'm having trouble wrapping my head around the correct way to approach this game. Am I not supposed to raise my quote when they buy? I thought I was winning out if he was buying for 50, when I knew the fair price was around 25.

Any help or direction to resources is greatly appreciated!

That is a very funny interview question, and I will keep my answer related to true quant finance questions about market making.

1. if you think about Kyle's model, that is very basic but contains the essence of market making when the MM is confronted to informed and noise traders:

• as a MM, you try to guess the true price by "following the flow of informed traders". Here you acted as if they were informed (and thus they traded the true value, ie 25) but they acted as noise traders.
• your reaction to the flow (that is a mixed of informed and noise traders) should be proportional to $$\sigma_I/\sigma_N$$ (ie the respective std of Informed and Noise traders' flows): your noise traders had a large $$\sigma_N$$, hence you should not move your price that much as a reaction to your noise traders' flows.
2. More specifically, you had information: you were an informed MM. This is not covered by Kyle's model but that is quite common nowadays (think about news in machine readable format that MM can process). Why accepting to buy at more than 30 if it was your upper bound for the true value?

To be fair you probably have been trapped by the fact that you were constrained to offer a spread of no more than 5, hence your bid followed your ask, and your ask was driven by the "belief" that they add information.

Nevertheless you had information: since you shouldn't set your bid at more than 30 (according to your information), you shouldn't let your ask go at more than 35.

3. To wrap up using Kyle's model again (it is so useful): in the original version of this model the traders sell back their inventory at the true price, not to the MM. That enforces the fact that noise traders cannot be that directional (in the model, not in true life).

I you want to go further than Kyle's model, I recommend Cetin and Danilova's excellent book: Dynamic Markov Bridges and Market Microstructure.

• I think I would have made the same mistake in this situation. The thing to remember is the bid and ask must bracket your estimate of the item's worth. You must not be tricked into focusing only on one side, the ask and not think about the bid. You never know what the next trade is going to be a buy or a sell. Apr 14 at 15:17
• Yeah I see makes sense. Thanks for your contribution guys. Apr 15 at 14:24
• If this were a game, wouldn't it depend on (1) how many turns or how long the game lasts and / or (2) whether there is settlement at the true value? If the game is endless and without settlement, then shouldn't any value suffice? If there is settlement at true value (OP assumed 25), wouldn't the market maker's best position be what the OP offered, i.e. no lower than that value (20 x 25 or higher and drop quotes fast in decline back to 20 x 25)? Apr 25 at 23:58
• interesting remark, @Catalyx: Kyle's model is a Stackelberg game with full information, there is a first mover's advantage and no need to iterate, (2) yes if the settlement is not a the true value, all changes. May 3 at 16:39

You should have forced them to agree to a settlement mechanism ie at the end of the interview, the answer is calculated (getting 25) and all trades settled. Then you made a lot of money.

If they are trying to make the point that markets can go anywhere, this is correct but not likely in the case of an underlying that settles very soon based on a non manipulable index !

Indeed, this is a rather amusing maintenance question.

In fact, in the context of this question, the market-maker is in a monopoly situation. This implies that as a market-maker, you have infinite power to manipulate the price.

Assuming a tick size of 1 and the impossibility of quoting a zero price, it seems that the best strategy is to keep quoting 1-6. This gives you the maximum bid-ask spread in relative terms: spread / mid-price = 5/3.5 = 143%.