Derivatives Foundation puzzles, solved step by step

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  1. 012You are making a market on a contract that settles at the sum of two dice. During the game you sell 5 at 7.5, buy 3 at 6.5 and sell 2 at 8. The dice are rolled and total 9. What is your final position, what is your profit or loss in rupees, and what was your expected profit at the moment you finished trading?Market makingWarm upOptiverAmsterdam · 2023

    Try it first

    Before the blotter: what is the fair value of the sum of two dice?

    Show the worked solution

    You finish short 4, you lose Rs 2 on the settlement, and your expected profit when you stopped trading was Rs 6. Sold 5, bought 3, sold 2 is a net short of 4. Cash is +37.5 - 19.5 + 16 = +34. Settling at 9 costs 4 x 9 = 36, so the result is 34 - 36 = -2. Against the fair value of 7 the short would have cost 28, leaving +6: the edge of 2.5 + 1.5 + 2 captured on the three trades.

    Why keep three numbers in your head and not one?

    A shopkeeper who sells umbrellas at a markup has a profit on each sale, a stock count, and a worry about whether it rains. Three separate things. A market maker tracks the same three: edge per trade against fair value, net position, and the exposure to the final number, and the game checks that you never let one of them slip. Say the fair value, 7, first. Then say each trade's edge as you do it: +2.5 on selling 5 at 7.5, +1.5 on buying 3 at 6.5, +2 on selling 2 at 8. Then say the position: short 4.

    The blotter: running position and cash per trade, then the settlement at 9 against fair value 7TradePositionCashEdge vs fair 7Running edgesell 5 at 7.5short 5+37.5(7.5 - 7) x 5 = +2.5+2.5buy 3 at 6.5short 2+18(7 - 6.5) x 3 = +1.5+4sell 2 at 8short 4+34(8 - 7) x 2 = +2+6Position short 4 means you owe 4 x (settlement) at the end; cash of +34 is already in hand.Dice settle at 934 - 4 x 9 = 34 - 36P&L = -2At fair value 7, the expectation34 - 4 x 7 = 34 - 28expected P&L = +6Same trades, same position. The 9 is luck; the +6 was skill, locked in before the dice were thrown.
    Selling 5 at 7.5, buying 3 at 6.5 and selling 2 at 8 leaves a short of 4 and cash of +34 with an edge of 2.5, 1.5 and 2 against fair value 7, so at the settlement of 9 the position costs 36 and the result is Rs 2 lost, while at fair value the same trades were worth Rs 6.
    The relationship
    P&L=5(7.5)−3(6.5)+2(8)⏟cash=34+(−4)⏟position×S,S=9⇒−2,S=7⇒+6\text{P\&L} = \underbrace{5(7.5) - 3(6.5) + 2(8)}_{\text{cash} = 34} + \underbrace{(-4)}_{\text{position}} \times S, \qquad S = 9 \Rightarrow -2,\quad S = 7 \Rightarrow +6
    cashmoney received for sales minus money paid for purchases
    positioncontracts bought minus contracts sold, here minus 4
    Sthe settlement value of the contract, the dice total
    What it says in wordsProfit is the cash already banked plus the position times the settlement, and replacing the settlement with the fair value gives the expected profit.

    Was the loss a mistake?

    No. Every trade was done at a better price than fair value, so the trading was right; the dice came in high. Expected profit of +6 is what you controlled, and the realised minus 2 is what the dice did; an interviewer wants to hear you separate the two without being asked. The sum of two dice has a standard deviation of about 2.4, so a short of 4 carries a one-standard-deviation swing of nearly 10, far larger than the 6 of edge. The real question is whether a short of 4 was more risk than you wanted to carry against 6 of edge.

    What would you have done differently in the game?

    Skewed the quote as the short grew. After selling 5, you are short 5 and should lower both your bid and your offer so that the next trade is more likely to be a buy that cuts the position, which is exactly what buying 3 at 6.5 did. Selling 2 more at 8 added to the short again; at that point a wider or higher quote would have protected you. The limitation of the puzzle: with two interviewers trading against you, their trades carry information about nothing, because the dice are not rolled yet, so here the only reason to skew is inventory, not adverse selection.

    Where candidates lose it

    Candidates lose the position count under pressure, saying short 6 or short 2 because they forget the buy of 3. State the running position after every trade, aloud, as the sample blotter does.

    The second loss is reporting the minus 2 as if the trading was bad. The expected profit was plus 6 and the dice were unkind. Say both numbers and which one you controlled.

    What the interviewer asks next

    • The dice settle at 5 instead. What is your P&L, and does your expected P&L change?
    • What is the standard deviation of the two-dice total, and how does it size the risk of being short 4?
    • After the first sale of 5 at 7.5, what market would you show next, and why?

    Asked at Optiver, Prop Trading, Amsterdam, 2023 (Wall Street Oasis): some difficult trading games where you had to profit making a market whilst remembering your position and the position of two interviewers