Debt Capital Markets puzzles, solved step by step

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  1. 065In a trading game you are asked to make a two-way market on the sum of three fair dice. Quote a bid and an offer. The first die is then shown to be a 6. Where do you move your market, and why?Probability and expected valueHardBelvedere TradingChicago · 2022

    Try it first

    After the 6 is shown, where should the middle of your market be?

    Show the worked solution

    Start around 10.5, say 9.5 bid and 11.5 offered, then move to about 12.2 bid, 13.8 offered once the 6 is shown. Each fair die is worth 3.5 on average, so three are worth 10.5. After the reveal the value is 6 plus 3.5 plus 3.5, which is 13.0. The market should also narrow, because one die's uncertainty has gone: the standard deviation falls from 2.96 to 2.42, so the width shrinks by about a fifth, not a third.

    Where does the first quote come from?

    A shop that buys and sells used phones offers to buy below what it thinks a phone is worth and to sell above it, and earns the gap. A market maker quotes around the expected value, with a bid below and an offer above, and the width reflects how uncertain that value is. Each fair die averages 3.5, so three dice average 10.5. A quote of 9.5 bid, 11.5 offered straddles that value; the sum can still land anywhere from 3 to 18, so the market cannot be tight.

    Quote around the expected value; move and narrow as information arrives3456789101112131415161718Before: three hidden dicefair 10.5, quote 9.5 bid / 11.5 offered10.53456789101112131415161718After the first die shows 6fair 13.0, quote 12.2 bid / 13.8 offered13.0spread of outcomes: sd 2.96sd 2.42: about 18% narrower
    Before any die is shown the sum centres on 10.5 with a standard deviation of 2.96; once a 6 is shown it centres on 13.0 with a standard deviation of 2.42, so the quote moves up 2.5 and narrows from 9.5 to 11.5 to about 12.2 to 13.8.

    What changes when the first die is shown?

    Two things, and candidates usually say only one. The expected value jumps to 6 plus 3.5 plus 3.5, which is 13.0, and the uncertainty shrinks because only two dice are still hidden. Variance adds across independent dice, 35/12 for each, so it falls from 8.75 to 5.83, a third lower; the standard deviation falls from 2.96 to 2.42, about 18% lower. Scale the width by that: 2 points becomes about 1.6, so quote roughly 12.2 bid, 13.8 offered.

    The relationship
    E=6+2×3.5=13.0σ=2×3512=2.42 against 3×3512=2.96E = 6 + 2 \times 3.5 = 13.0 \qquad \sigma = \sqrt{2 \times \tfrac{35}{12}} = 2.42 \ \text{against} \ \sqrt{3 \times \tfrac{35}{12}} = 2.96
    3.5the expected value of one fair die
    35/12the variance of one fair die
    sigmathe standard deviation of the part of the sum still unknown
    What it says in wordsThe expected value adds the known die to the average of the hidden ones; the uncertainty comes only from the dice still hidden.

    What does the interviewer want to hear beyond the numbers?

    Move the market the moment information arrives, because a stale quote is a free option for everyone else at the table. If you stay at 9.5 to 11.5 after a 6 shows, every player buys your offer at 11.5 against a fair value of 13.0. Then say what you would do if you suspected the other side knew more than you, for example had seen a second die: widen, or lean your quote towards the risk. The limit: the width here is a choice; a real desk sets it by competition and by how much risk it can hold, not by a formula.

    Where candidates lose it

    The common loss is leaving the market where it was, or moving the middle and forgetting the width. Fair value jumps to 13.0 the moment the 6 is shown and the uncertainty is smaller, so both the level and the width should change.

    The second is narrowing the width by a third because one of three dice is known. Variance falls by a third, but the standard deviation falls by only about 18%, so the market narrows by about a fifth.

    What the interviewer asks next

    • Someone lifts your 13.8 offer three times in a row. What do you do next?
    • Before any die is shown, what is the probability that the sum is 13 or more?
    • How does your market change if the second die is also shown to be a 6?

    Asked at Belvedere Trading, Equity Capital Markets, Chicago, 2022 (Wall Street Oasis): superday with two 1-1s and a group trading game