Hedge Funds puzzles, solved step by step

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  1. 045You make a market on the number of heads in 10 fair coin flips: 4.5 bid, 5.5 offered. A counterparty who has already seen the first three flips lifts your offer. What does the trade tell you, and where do you requote?Market making and trading gamesHardCitadelNew York · 2025

    Try it first

    Given that they bought at 5.5, the fair value is about

    Show the worked solution

    The lift says they saw at least two heads, so the fair value is now at least 5.75, not 5; requote around 5.75 bid, 6.5 offered. The other seven flips are worth 3.5 heads, so the buyer's value is heads seen plus 3.5. Paying 5.5 only makes sense with two heads (5.5) or three (6.5). Those are 3 to 1 likely, giving 5.75; a buyer who needs a strict edge saw three heads, worth 6.5.

    What is the trader's view before they trade?

    A friend offers to buy your raffle ticket after the first few numbers are drawn. The offer itself is the warning. The informed trader values the contract at heads already seen plus 3.5, the expected heads in the seven unseen flips, so their value is 3.5, 4.5, 5.5 or 6.5 with chances 1, 3, 3 and 1 in 8. Against your market of 4.5 bid and 5.5 offered, they buy only if their value is at least 5.5, and sell to you at 4.5 only if it is 4.5 or less. The flat 5 you quoted around is right only for someone who has seen nothing.

    What the buyer saw decides whether they lift: a lift means 2 or 3 heads34567your offer 5.5your bid 4.5value given a lift = 5.750 headsvalue 3.5chance 1/81 headvalue 4.5chance 3/82 headsvalue 5.5chance 3/8may lift3 headsvalue 6.5chance 1/8liftshits bidTrader's fair value after seeing the first 3 flips
    The informed trader's value is 3.5, 4.5, 5.5 or 6.5 depending on how many heads they saw, so a lift at 5.5 means two or three heads, and weighting those 3 to 1 puts the fair value given the trade at 5.75, above your 5.5 offer.

    How do you turn the trade into a new fair value?

    Condition on the fact that they traded. Only the two-head and three-head worlds produce a buy at 5.5, and they are 3/8 and 1/8 likely, so given a lift the value is (3 x 5.5 + 1 x 6.5) / 4 = 5.75. If you assume they would not bother trading at zero edge, only the three-head world is left and the value is 6.50. Either way you sold too cheaply: this is adverse selectionThe tendency of a market maker to trade most with the people who know more, so the trades that happen are the ones that lose money for the market maker., and it is the cost every market maker prices into the spread.

    Now requote. Your bid should not be below what you now believe the floor is, and your offer should sit where even the best informed buyer has no edge. Something like 5.75 bid, 6.5 offered does both: a buyer who saw three heads is indifferent at 6.5, and you are no longer selling below value. Cut your size too, because you know someone is trading with more information than you, and say you would ask whether they could see the flips before quoting again.

    Where candidates lose it

    The common loss is staying at 5 because the coin is fair. The coin is fair; the counterparty is not uninformed. The trade itself carries information and you must update on it.

    The other loss is overreacting and moving to 8 or 9, as if the trader knew all ten flips. They saw three. Condition on what could have made them trade, weight those worlds, and move by exactly that much.

    What the interviewer asks next

    • The same trader then hits your new bid. What do you conclude?
    • How wide should your first market have been if you knew one counterparty could see three flips?
    • What if the trader had seen the first three flips but traded a small size and then a large size?

    Asked at Citadel, Quantitative Trading, New York, 2025 (Wall Street Oasis): Superday was more market-making but requires very sold foundation in math and statistics.