The trading game: quoting, requoting, and the questions at the end

MM · Chapter 513 min readAsked at Optiver, SIG, IMC, Jane Street

Assumes Inventory: carrying risk, shedding it, and the reservation price.

After this lesson you should be able to

  • Make a market on a known distribution and on an unknown quantity.
  • Requote correctly after being traded against, several rounds running.
  • Answer the closing questions: position, P&L, breakeven.

The trading game is the centrepiece of a market-making interview. You are asked to quote, you get traded against, you requote, and after five or ten rounds you are asked what you are holding and what it is worth. Almost everything being assessed is visible in how you handle rounds two through five, not round one.

Proposition 5.1

Markets on a known distribution

When the quantity is a die roll, a card draw or a sum you can compute, start from the true expectation and set the width from the standard deviation. You know the distribution exactly, so a wide market is indefensible — the interviewer can see that you could have computed it.

Holds when

  • One die: mean 3.53.5, so quote something like 33 at 44.
  • Sum of two dice: mean 77, standard deviation 2.422.42 — a market of 66 at 88 is defensible, 22 at 1212 is not.
  • Maximum of three dice: mean 4.964.96, so quote around 55, not around 3.53.5.

Proposition 5.2

Markets on an unknown quantity

Windows in the building, ping-pong balls in the room, revenue of a company. Here you cannot compute the answer, so the width is doing real work: it should be roughly the interval you would genuinely be surprised to fall outside. Estimate first, out loud, then quote around your estimate.

Holds when

  • Say the decomposition before the number, exactly as in a Fermi question.
  • A market you would not trade on both sides is not a market; if you would rather not sell at your offer, your offer is too low.
  • Expect to be traded on the side you are least comfortable with. That is the point of the exercise.

Example 5.3

Three rounds

Make a market on the sum of two dice. You quote 66 at 88; the interviewer buys from you at 88. You requote, and they buy again. What happened, and what is your third quote?

Show the worked solutionHide the worked solution

Worked solution

  1. Formula
    after each trade: update theo, then skew for inventory\text{after each trade: update theo, then skew for inventory}
  2. Substitute
    sold 1 at 8, then sold 1 more\text{sold 1 at } 8, \text{ then sold 1 more}
  3. Solve
    Round 2: short 1. Skew up: 6.5 at 8.5\text{Round 2: short 1. Skew up: } 6.5 \text{ at } 8.5
  4. Round 3: short 2, and they have bought twice\text{Round 3: short 2, and they have bought twice}
  5. Quote 7.5 at 9.5 — skewed up and wider\text{Quote } 7.5 \text{ at } 9.5 \text{ — skewed up and wider}
  6. Answer
    short 2 at an average of 8.25, breakeven 8.25\text{short 2 at an average of } 8.25, \text{ breakeven } 8.25

Sanity check. Two things changed. You are short, so you skew up to attract sellers. And they have bought twice at prices above the true mean of 7, which is evidence either that they know something or that they are testing you — so you also widen. Failing to move at all after two trades is the commonest way to lose this exercise.

Two different reasons to move your quote. Being traded against tells you two separate things, and conflating them is the central mistake. First, you now have a position, which calls for skew — a mechanical, unambiguous response. Second, somebody chose to trade at your price, which is weak evidence that your price was wrong, and calls for moving theo itself. The first is always right. The second depends entirely on who you think is on the other side: informed flow should move your theo, and a coin-flipping counterparty should not.

The rest of this lesson is in Premium

You have read the opening. 11 more sections follow, including 5 worked examples and 3 quick checks.

Start the free 7-day trialSign in

Nothing is charged for 7 days, and you can cancel before then. Or read Arithmetic that survives a clock in full, free.