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
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  1. 023A ticket pays Rs 100 if a card drawn from a well-shuffled standard deck is a picture card (jack, queen or king), and nothing otherwise. I quote the ticket at 20 bid, 22 offered. Would you trade, and on which side?Betting and sizingWarm upAkuna CapitalChicago · 2026

    Try it first

    Before you work out the fraction: the right action against a 20 bid, 22 offered quote is

    Show the worked solution

    Yes: buy at 22, because the ticket is worth 23.08. A standard deck has 12 picture cards out of 52, a 23.1% chance, so the fair value of a Rs 100 payout is 12/52 x 100 = 23.08. You buy at the offer, 22, which is below fair, and expect to make about Rs 1.08 a ticket. Selling at 20 would give away about Rs 3.08. The edge is small next to the outcome, so it pays only over many tickets.

    How do you value the ticket?

    If a friend offers you Rs 100 when a coin lands heads, you would pay up to Rs 50 and no more; the coin's odds set the price. A ticket that pays a fixed amount on an event is worth the payout times the probability of the event, before any talk of the quote. A standard deck has four suits, each with a jack, a queen and a king, so 12 picture cards in 52. The ticket is worth 12/52 x 100 = 3/13 x 100 = 23.08. Notice that the ace is not a picture card; counting it is the quickest way to get 16/52 and the wrong side.

    The relationship
    V=1252×100=313×100≈23.08V = \frac{12}{52} \times 100 = \frac{3}{13} \times 100 \approx 23.08
    12picture cards: jack, queen and king in four suits
    52cards in the deck
    100the payout in rupees if a picture card is drawn
    What it says in wordsThe ticket is worth its payout times the chance of being paid.

    Which side of the quote do you compare with?

    A quote has two prices, and you can only use one of each: you buy at the offer and sell at the bid. Trade only when fair value sits outside the quote: buy when it is above the offer, sell when it is below the bid, and pass when it falls between them. Here fair is 23.08 against a 22 offer, so you buy and expect 1.08. The figure runs two other quotes for contrast: at 22 bid, 24 offered, fair is inside and there is no trade; at 24 bid, 26 offered, you sell at 24 and expect 0.92.

    Compare fair value with each side of the quote, not with the middlefair value 12/52 x 100 = 23.0820 bid, 22 offered2022buy at 22+1.0822 bid, 24 offered2224no tradefair is inside24 bid, 26 offered2426sell at 24+0.92Buy when fair is above the offer; sell when it is below the bid; otherwise pass
    Against a fair value of 23.08, the quote of 20 bid, 22 offered lets you buy below fair for an expected 1.08 a ticket, a quote of 22 at 24 straddles fair and gives no trade, and a quote of 24 at 26 lets you sell above fair for an expected 0.92.

    How much would you buy, and what could change your mind?

    Each ticket pays 0 or 100, so its standard deviation is 100 x the square root of 0.231 x 0.769, about Rs 42, against an edge of Rs 1.08. The edge is real but small next to the noise, so it is worth taking in size over many independent draws and worth very little on a single ticket. Before trading, ask the two questions a desk would: is the deck standard and well shuffled, and does the person quoting know something you do not, such as a card already removed? The limitation of the clean answer is that it trusts the setup; in a game where the quoter controls the deck, a quote this generous is itself a warning.

    Where candidates lose it

    The common slip is to sell at 20 on the feeling that the ticket usually loses. That is true and already priced: losing most of the time is why the ticket is worth 23, not 50. Value it first, then compare.

    The second loss is comparing fair value with the middle of the quote, 21, and saying buy without naming the price. You buy at the offer. Saying buy at 20 tells the interviewer you do not know which side of a quote you can trade on.

    What the interviewer asks next

    • The ticket now pays on a picture card or an ace. Where would you trade against the same quote?
    • I draw a card, look at it without showing you, and then quote 20 at 22. What do you do now?
    • Make me your own market on the original ticket, and say why you chose that width.

    Asked at Akuna Capital, Junior Trader Interview, Chicago, 2026 (Wall Street Oasis): if you win you get 1$. how much money would be a fair bet.

  2. 058We roll two fair dice and you win if the total is exactly 10. How much would you be willing to risk to win Rs 100?Betting and sizingWarm upAkuna CapitalChicago · 2025

    Try it first

    First, how many of the 36 outcomes total 10?

    Show the worked solution

    At most Rs 100/11, about Rs 9.09. A total of 10 comes from (4, 6), (5, 5) and (6, 4), three of the 36 equally likely outcomes, so the chance is 1/12 and the odds are 11 to 1 against. A fair bet matches the odds: risk 1 to win 11, so to win Rs 100 the fair stake is 100/11. Risk Rs 10 and you lose about Rs 0.83 per game on average.

    Why is the answer odds, not a probability?

    A friend offers to pay you Rs 100 if your cricket team wins a match it wins one time in twelve. Paying Rs 20 to enter is a bad deal, and the way you know is by comparing what you risk to what you get: you lose Rs 20 eleven times and win Rs 100 once. A bet is fair when the stake and the prize are in the same ratio as the losing and winning chances, which is what odds express directly. One in twelve means eleven losses per win, so a fair stake is one eleventh of the prize: Rs 100/11, which is Rs 9.09. Anything above that is paying for the privilege of playing.

    Three winning cells out of 36, so the fair stake is 100 divided by 11234567345678456789567891067891011789101112112233445566second die across, first die down: 36 equally likely cells3 cells total 10Chance of a 103 / 36 = 1 / 12Odds against11 losing to 1 winningFair betrisk 1 to win 11To win Rs 100stake Rs 100 / 11 = Rs 9.09Stake Rs 10 instead and you lose Rs 0.83 per game on average
    Of the 36 ordered outcomes of two dice, exactly three total 10, so the chance is 1/12, the odds are 11 to 1 against, and the fair stake to win Rs 100 is Rs 100/11, about Rs 9.09; staking Rs 10 loses about Rs 0.83 per game on average.

    How do you check the stake with expected value?

    Set the expected profit to zero. With stake s, you win Rs 100 with chance 1/12 and lose s with chance 11/12, so the expectation is 100/12 - 11s/12, which is zero when s = 100/11. The expected value route and the odds route must agree, and saying both in the room shows the interviewer you can move between them. Then say the direction: a lower stake is a profitable bet for you, a higher one is profitable for the house, and at Rs 10 the house edge is 8.3% of your stake, which is roughly the edge a casino takes on its better games.

    The relationship
    E[profit]=112×100−1112×s=0  ⇒  s=10011≈9.09E[\text{profit}] = \tfrac{1}{12}\times 100 - \tfrac{11}{12}\times s = 0 \;\Rightarrow\; s = \tfrac{100}{11} \approx 9.09
    1/12the chance of a total of 10, three cells out of 36
    sthe stake you risk
    100/11the fair stake, where the bet has zero expected value
    What it says in wordsThe fair stake is the prize divided by the odds against, which makes the expected profit zero.

    What would a trader add after the number?

    That the fair stake is a ceiling, not an offer. A trader asked to risk money quotes below fair, say Rs 8, and explains the gap as the edge for taking the other side, because a bet at exactly fair value makes nothing over many plays. The limitation is that 1/12 is the chance only if the dice are fair and the counting is right; if the interviewer changes the target to 7, the chance rises to 6/36 and the fair stake jumps to Rs 20, so the first thing to re-check whenever the rules change is the count of winning cells.

    Where candidates lose it

    The common error is in the count: giving 2 winning outcomes by treating 4-6 and 6-4 as the same, or 4 by counting 5-5 twice. Lay out the ordered pairs and say three of 36.

    The second loss is quoting Rs 8.33, which is 100/12. That is the expected prize, not the fair stake. Odds of 11 to 1 mean the stake is 100/11.

    What the interviewer asks next

    • Same game but you win on a total of 7. What is the fair stake now?
    • You are offered the bet at a stake of Rs 8. What is your expected profit per game?
    • How would you change the stake if the dice might be loaded?
    • You can play 100 times at Rs 8. What is the chance you end up behind?

    Asked at Akuna Capital, Trading, Chicago, 2025 (Wall Street Oasis): We are playing a game where we roll two dice. You win if you roll a 10, how much would you risk to win $100?