Derivatives Foundation puzzles, solved step by step

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  1. 096A game flips a fair coin until the first tail and pays Rs 2^n if the first tail comes on flip n, but the house can pay at most Rs 1,024. What is the fair price, and why does the uncapped game break the idea of a fair price?Expected value and optimal stoppingCoreMarket makingProp trading firms

    Try it first

    How much does each possible flip contribute to the expected payout, below the cap?

    Show the worked solution

    Rs 11. A first tail on flip n pays 2^n with probability 1/2^n, so each of flips 1 to 10 contributes exactly Rs 1, Rs 10 in all. Reaching flip 11 or later has probability 1/1,024 and pays the cap of Rs 1,024, adding one more rupee. Without the cap every flip keeps adding Rs 1 and the expected value is infinite, yet nobody would pay much to play, which shows expected value alone cannot price a bet on outcomes the payer cannot honour.

    Why does every flip add exactly one rupee?

    Think of a raffle where each ticket in the next bundle is half as likely to win but the prize is twice as large. Every bundle is worth the same to you. Here the first tail on flip 1 pays Rs 2 half the time, on flip 2 pays Rs 4 a quarter of the time, on flip 3 pays Rs 8 an eighth of the time. The payout doubles exactly as the probability halves, so each flip contributes 2^n x 1/2^n = Rs 1 to the expected value, and the fair price is simply the number of flips before the cap bites, plus the capped tail. Ten flips reach Rs 1,024; anything later is paid at the cap, worth 1,024 x 1/1,024 = Rs 1.

    Every flip adds exactly Rs 1, so the cap is the whole priceRs 11121/21241/41381/814161/1615321/3216641/64171281/128182561/256195121/5121101,0241/1,024111+1,0241/1,02411uncapped: Rs 1 morefor every flip, foreverflippaysprobfirst tail on flip n: payout x probability = Rs 1 for every n up to the capcapped tail10 bars of Rs 1 + the capped tail Rs 1 = fair price Rs 11, though 7 times in 8 the game pays Rs 8 or less
    Each flip from 1 to 10 contributes payout times probability of exactly Rs 1, and reaching flip 11 or later adds the capped Rs 1,024 times a 1 in 1,024 chance, another Rs 1, for a fair price of Rs 11, while without the cap the Rs 1 contributions would carry on forever.

    What goes wrong without the cap?

    Remove the cap and the sum is 1 + 1 + 1 + ... with no end, an infinite expected value. Yet the game pays Rs 8 or less seven times in eight, and more than Rs 11 only one time in eight. The infinite value lives entirely in outcomes so rare and so large that no house could pay them, so the cap is not a detail; it is the price. Raising the cap barely moves it: a cap of Rs 1 crore makes the game worth only about Rs 24.19, because each doubling of the cap adds one rupee. That is the market maker's answer to the paradox: price what the counterparty can actually pay.

    The relationship
    E=∑n=1102n⋅12n  +  1024⋅P(first tail on flip 11 or later)=10+1024⋅11024=11E = \sum_{n=1}^{10} 2^n \cdot \frac{1}{2^n} \;+\; 1024 \cdot P(\text{first tail on flip } 11 \text{ or later}) = 10 + 1024 \cdot \frac{1}{1024} = 11
    2^nthe payout if the first tail comes on flip n
    1/2^nthe chance the first tail comes on flip n
    1/1024the chance the first ten flips are all heads
    What it says in wordsTen rupees from the flips the cap does not touch, and one more from the capped tail.

    Mention the other classic resolution, then put it in its place. Economists answer the paradox with diminishing utility: a doubling of wealth is worth less to you than the first rupee, so a risk-averse player pays little. True, but on a desk the binding constraint is usually the counterparty's balance sheet, not your utility. The limitation of the Rs 11 is the same as any fair value: it is an average over many plays. Played once, the price feels steep because 87.5% of the time you get Rs 8 or less.

    Where candidates lose it

    The common loss is computing an infinite expected value and stopping there, or saying the cap makes the game worth Rs 1,024. The cap removes the infinity but adds only one rupee for the capped tail.

    The second loss is getting the boundary wrong: counting 11 flips below the cap, or forgetting the capped tail entirely and saying Rs 10. Write the n = 10 payout, Rs 1,024, next to the cap and the count is obvious.

    What the interviewer asks next

    • What is the fair price if the house can pay at most Rs 1 crore?
    • What is the median payout of the uncapped game?
    • Make me a market on this capped game, and say which side you would rather be on if the house's credit were in doubt.