Derivatives Foundation puzzles, solved step by step

Puzzles
100
Traced to a firm
66
Topics
12
Hard
29
Topic
All topicsMental maths and estimation9Random walks and Markov chains7Conditional probability and Bayes7Volatility and correlation7Option pricing intuition7Expected value and optimal stopping10Market making11Option payoffs and no-arbitrage10Probability and counting11Distributions and statistics8Games and logic8Betting and sizing5
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 11–11 of 11 · filtered from 100Clear filters
  1. 097I pick a whole number from 1 to 100 and know it; you do not. You quote 45 at 55 and I buy at 55. You requote 58 at 68 and I buy again at 68. What have you learned, and where should your next quote be?Market makingHardJane StreetNew York · 2025

    Try it first

    After the first lift at 55, where is the middle of what the number could still be?

    Show the worked solution

    The number is between 69 and 100, so centre the next quote on 84.5, say 80 at 89. Someone who knows the number buys only when your offer is too low. The lift at 55 put it in 56 to 100, midpoint 78, and your requote at 58 to 68 was far too low, so it was lifted too. Now it is in 69 to 100. You are short two at 55 and 68, worth about -46 against a fair value of 84.5, and the next quote should sit on what is left, not drift up from the last one.

    What does a trade tell you when the other side knows the answer?

    If a friend who has already peeked at the exam paper offers to bet you that the pass mark is above 55, you do not need to ask why. A counterparty who knows the number only trades when your price is wrong in their favour, so every lift of your offer is proof that the number is above it, and your fair value should jump to the middle of what is left. After the lift at 55 the number lies in 56 to 100, centred on 78. The requote at 58 to 68 treated the lift as noise and moved only 13 points. A quote centred near 78, say 73 at 83, would have used the information.

    Every lift says the number is above your offer: requote on what is leftstart: anything from 1 to 10012550751004555mid 50.5lifted at 55after one lift: 56 to 10012550751005868mid 78lifted at 68after two lifts: 69 to 10012550751008089mid 84.5next quotecentred at 63,15 below the midcentred on 84.5An informed trade is a signal, not luck: short 2 at 55 and 68, worth about -46 against fair value 84.5
    The first lift at 55 leaves 56 to 100 with a midpoint of 78, the requote of 58 at 68 sits 15 below that midpoint and is lifted again, which leaves 69 to 100 with a midpoint of 84.5, so the next quote of 80 at 89 is centred on what can still be true.

    Where should the next quote sit, and what have the trades cost?

    After the second lift the number is in 69 to 100: 32 values, mean 84.5. Centre the next quote on the middle of the remaining range, 84.5, so that whichever side is hit, the range halves, which is the fastest way to stop losing to someone who knows more than you. A quote of 80 at 89 does that. The damage so far: you sold at 55 and 68 something now worth about 84.5, an expected loss of 29.5 + 16.5 = 46. Most of it came from the timid requote; a quote centred on 78 would have been lifted only if the number was above 83.

    The relationship
    E[X∣X>55]=56+1002=78,E[X∣X>68]=69+1002=84.5E[X \mid X > 55] = \frac{56 + 100}{2} = 78, \qquad E[X \mid X > 68] = \frac{69 + 100}{2} = 84.5
    Xthe number I picked, uniform on 1 to 100 before any trade
    X > 55what my buying at 55 reveals
    84.5the fair value after both lifts, where the next quote should be centred
    What it says in wordsEach lift cuts the range to everything above your offer, and fair value moves to the middle of what is left.

    Then say how the answer changes with the counterparty, because that is the judgement being marked. If I might be guessing rather than knowing, a lift is weaker evidence and you should move less, perhaps halfway towards 78. If I know the number exactly, any tight quote inside the range loses whenever it trades, and the only quote that cannot lose is one that spans the whole range, 69 bid, 100 offered, which is no market at all. Real desks live between the two: they widen against flow they think is informed and tighten against flow they think is not.

    Where candidates lose it

    The common loss is moving the quote up a few points after each lift, as the 58 at 68 requote did. It treats an informed trade as a random one and pays for the lesson again on the next trade.

    The second loss is the opposite overreaction: refusing to quote, or quoting 1 at 100. The interviewer wants a market that uses the information and still trades, centred on the conditional mean of what is left.

    What the interviewer asks next

    • Instead I sell to you at 45 on the first quote. What is your next quote?
    • Suppose I know the number only half the time and guess otherwise. How far should the first lift move you?
    • How many lifts or hits does it take, at most, to find the number if you always quote around the middle of the range?

    Asked at Jane Street, Generalist, New York, 2025 (Wall Street Oasis): It was a probability theory based quant trading style market making questions which were intense