Market-making interview questions

What these questions test

“Make me a market in the sum of two dice.” That single prompt exercises the whole loop of the job: estimate a fair value, quote a two-sided price, and then manage the position and information you pick up as you get filled. The interviewer is not checking arithmetic — they are checking whether you treat your own quote as a live hypothesis that every trade revises. We will run the entire page on one object, the dice sum $S$, so the mechanics stay concrete.

Start from fair value

Before you say a bid or an ask, say what you think the thing is worth. For two dice the sum is symmetric around $7$, so $\E[S] = 7$. Your spread should be governed by your uncertainty, so quantify it: a single die has variance $\tfrac{91}{6} - \tfrac{49}{4} = \tfrac{35}{12}$, and two independent dice add, so

Now a quote of 6 / 8 is not a guess — it is “fair value 7, plus or minus a bit less than half a standard deviation.” Naming the number and its dispersion out loud is worth more than any specific spread, because it shows the interviewer where every later decision comes from.

Quote two-sided, earn the spread

Against a counterparty with no edge — a noise trader who buys or sells for reasons unrelated to $S$ — your spread is pure profit. Someone lifts your 8: you are now short something worth $7$ on average, a $+1$ edge. Someone hits your 6: you are long at 6 against a fair value of 7, again $+1$. Widen to 5 / 9 and each fill is worth $+2$, but you will trade less often. That tension — edge per fill against probability of a fill — is the entire craft, and interviewers probe it by asking you to widen or tighten and justify it.

Skew for inventory

Every fill leaves you holding risk, and the next quote has to reflect it. The clean model is a reservation price: the value at which, given your current inventory, you are indifferent to holding more. With inventory $q$ and risk aversion $\gamma$, it is $7 - \gamma q$. Long five units at $\gamma = 0.1$ pulls it to $7 - 0.5 = 6.5$, and you re-centre the whole two-wide quote there:

The ask is now only $0.5$ above fair value, so it is cheap for someone to lift you and let you shed risk; the bid sits $1.5$ below fair, demanding a real edge before you add to a position you already dislike. You did not change your view of fair value — you changed the price at which you are willing to warehouse more of it.

The trap: adverse selection

Now drop the fiction that your counterparty is uninformed. Suppose whoever trades can see $S$ and will only take a price that is good for them. Quote 6 / 8 and look at who actually shows up. Nobody trades on $\{6,7,8\}$ — that middle carries $16/36$ of the mass and just sits there. Each tail carries $10/36$, and those are the only fills you get:

The conditional means are brutal: $\E[S \mid S > 8] = 10$ and $\E[S \mid S < 6] = 4$. Get lifted at 8 and you are short something worth 10 — down $2$. Get hit at 6 and you are long against a true value of 4 — down $2$ again. Same spread, opposite sign of profit, purely because the flow was informed. This is the single idea every market-making interview is built around.

The break-even spread, formally

The senior version of the question is: given informed flow, how wide must you quote to break even? Take a value that is $V_H = 10$ or $V_L = 0$, equally likely, so fair value is $5$. A fraction $\alpha$ of traders know $V$ and trade in its direction; the rest are noise, buying or selling with equal probability. Then a buy arrives with probability $\tfrac{1+\alpha}{2}$ under $V_H$ and $\tfrac{1-\alpha}{2}$ under $V_L$. Bayes gives the posterior:

Set the ask to that posterior mean and, by symmetry, the bid to $\E[V \mid \text{sell}] = 5(1-\alpha)$. The spread collapses to something you can quote from memory:

With $\alpha = 0.4$ the market is 3 / 7, spread $4$. The corners are the sanity check: $\alpha = 0$ is all noise and the spread is zero; $\alpha = 1$ is a fully informed counterparty, the spread is the entire range, and you only ever trade at the true value — exactly the dice result above, where a knowing counterparty made every fill a loss.

Variants and common mistakes

The interviewer will perturb the game to see whether your quote follows the information. Handle each by asking “what does the new fact do to fair value and to how informed my flow is?”

• “I show you one die is a 5.” Fair value jumps to $5 + 3.5 = 8.5$ and your uncertainty shrinks to a single die. Re-centre the quote and tighten it.
• “Now they can trade in size.” Size is information. A big lift is a stronger signal than a small one, so widen — and skew harder as inventory builds.
• “You get filled three times on the same side.” That is the market telling you your fair value is wrong. Move it, do not just skew for inventory.

The reflexes that cost candidates the offer:

• Quoting a spread with no fair value. A bid and ask with no stated mid is a number with no reasoning attached — the thing they are actually grading.
• Leaving the quote still after a fill. If you would quote the same market after getting lifted, you have ignored both the inventory and the information the trade carried.
• Pricing as if flow were noise. Tight spreads are free only when nobody knows more than you. Assume they might, and let adverse selection set your minimum width.