Derivatives Foundation puzzles, solved step by step

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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. 065A contract pays, in rupees, the amount by which a fair die roll exceeds 4, and nothing otherwise. What is it worth? What about the matching put, which pays the amount by which the roll falls short of 4?Option payoffs and no-arbitrageWarm upBelvedere TradingChicago · 2021

    Try it first

    The die call with strike 4. What is it worth?

    Show the worked solution

    The call is worth Rs 0.50 and the put is worth Rs 1.00. The call pays 1 on a 5 and 2 on a 6, so its average payoff is 3/6. The put pays 3 on a 1, 2 on a 2 and 1 on a 3, so its average is 6/6. The put is worth more because the strike of 4 sits above the die's mean of 3.5, and call minus put equals 3.5 minus 4, which is minus 0.5.

    Why is a die option priced by averaging the payoffs?

    A school raffle with six equally likely tickets where ticket 5 pays Rs 1 and ticket 6 pays Rs 2 is worth exactly the average prize, Rs 0.50, because nothing else is uncertain and nobody can hedge a die roll. With equally likely outcomes and no hedge available, the fair price of a payoff is its expected value, so you list the payoff on each face and average. For the call with strike 4: 0, 0, 0, 0, 1, 2, which averages 0.5. For the put: 3, 2, 1, 0, 0, 0, which averages 1.0. Each payoff is floored at zero, which is what makes it an option rather than a forward.

    A call and a put on a die, strike 4: the put is worth twice the call012303roll 102roll 201roll 300roll 410roll 520roll 6payoff in rupees on each face, strike 4call: pays roll - 4 when positiveput: pays 4 - roll when positiveCall = (1 + 2) / 6 = 0.5 Put = (3 + 2 + 1) / 6 = 1.0Call - Put = 3.5 - 4 = - 0.5: parity on a die
    Face by face, the strike-4 call pays 0, 0, 0, 0, 1 and 2 for an average of 0.5, the put pays 3, 2, 1, 0, 0 and 0 for an average of 1.0, and the difference of minus 0.5 equals the expected roll of 3.5 minus the strike of 4.

    What is the parity check, and why does it work on a die?

    Add the call and subtract the put on every face. Call minus put on any single face equals the roll minus 4 exactly, because whichever side is in the money pays the gap and the other pays nothing, so the average of call minus put is the average roll minus the strike: 3.5 - 4 = - 0.5. That is put-call parity with no interest and no dividends, and it gives a one-line check: once you have the call at 0.5, the put must be 0.5 + 0.5 = 1.0. On a real option the same identity holds with the forward in place of the expected roll.

    The relationship
    C=1+26=12,P=3+2+16=1,C−P=E[roll]−K=3.5−4=−12C = \frac{1+2}{6} = \tfrac{1}{2}, \qquad P = \frac{3+2+1}{6} = 1, \qquad C - P = E[\text{roll}] - K = 3.5 - 4 = -\tfrac{1}{2}
    C, Pthe values of the die call and die put with strike 4
    E[roll]the expected face of a fair die, 3.5
    Kthe strike, 4
    What it says in wordsThe call is worth half a rupee, the put one rupee, and their difference equals the expected roll minus the strike, which is parity.

    What does the interviewer ask next, and where does the analogy stop?

    The next question is usually a different strike, or a market. Move the strike to 3 and the call pays 1, 2 and 3 on the top three faces, worth 1.0, while the put pays 2 and 1, worth 0.5, so the two swap values because the strike is now below the mean. Where the analogy stops is hedging: a real option is priced not by the expected payoff under your view but by the cost of replicating it with the underlying, which shifts the probabilities to the risk-neutral ones; on a die there is nothing to trade against, so the expectation under the real probabilities is the price.

    Where candidates lose it

    The fast wrong answer is to compute 3.5 minus 4 and say the call is worth minus 0.5, or to count two paying faces and say 1/3. A call never pays a negative amount: list the payoffs face by face and average.

    The second loss is pricing the put from scratch and getting it right while missing the parity relation. Say call minus put equals 3.5 minus 4 and the interviewer hears that you know what parity is.

    What the interviewer asks next

    • Price the call and the put with strike 3.
    • What is the value of a contract that pays the square of the roll minus 10, floored at zero?
    • Make a two-way market on the strike-4 call.
    • Why does put-call parity on a real stock use the forward rather than the expected price?

    Asked at Belvedere Trading, Generalist, Chicago, 2021 (Wall Street Oasis): Pricing an option contract on a game involving rolling a die.

  2. 072A stock at 100 will be at 80, 100 or 130 at expiry, each equally likely in your view. What is the expected payoff of a 100-strike call and of a 100-strike put, and why is the expected payoff not what a market maker would charge?Option payoffs and no-arbitrageWarm upSell-side sales and trading

    Try it first

    Three equally likely outcomes. Expected call payoff and expected put payoff?

    Show the worked solution

    Expected payoffs are 10 for the call and 6.67 for the put, and neither is a price. The call pays 0, 0 and 30 across the three outcomes, averaging 10; the put pays 20, 0 and 0, averaging 6.67. A market maker charges the cost of hedging, and with zero rates call minus put must equal stock minus strike, which is 0, so prices of 10 and 6.67 would be an arbitrage against the maker. Hedge-implied odds that keep the stock worth 100 give both options the same price.

    Why is the expected payoff under your odds not the price?

    A shopkeeper who believes the monsoon will be good does not price umbrellas off that belief; the price is set by what the umbrellas cost to stock and what the shop next door charges. An option maker does not hold the option to expiry hoping for the payoff; they hedge it with the stock, so the price is the cost of that hedge, and the cost of the hedge depends on the stock's current price of 100, not on anyone's forecast of where it goes. Your equal odds imply the stock is expected to be worth 103.33, above today's 100; that optimism is yours to trade, not something the maker will pay you for inside an option price.

    Expected payoff under your odds: call 10, put 6.67. But a price must obey parity801001300102030stock at expiry, each outcome 1/3payoffcall 30put 20E[call] = 10E[put] = 6.67Parity, zero ratescall - put = stock - strike= 100 - 100 = 0but 10 - 6.67 = 3.33: not pricesHedge-implied oddsmust make the stock worth 100:e.g. 0.40, 0.33, 0.27 on 80, 100, 130call = 0.267 x 30 = 8put = 0.40 x 20 = 8equal, as parity demands
    Under equal odds on 80, 100 and 130 the call's expected payoff is 10 and the put's is 6.67, but with zero rates call minus put must equal stock minus strike, which is zero, so those two numbers cannot both be prices; hedge-implied odds that keep the stock worth 100, such as 0.40, 0.33 and 0.27, price both options at 8.

    What is the one-line test that catches the mistake?

    Put-call parity. Buying the call and selling the put with the same strike gives you the stock minus 100 in every outcome, which with zero rates is worth 100 - 100 = 0 today, so the call and the put must have the same price. Expected payoffs of 10 and 6.67 fail that test by 3.33, and a maker who quoted them would be lifted on the put and hit on the call until the prices met. The test needs no probabilities at all, which is the point: parity is enforced by hedging, not by views.

    The relationship
    C−P=S−K e−rT=100−100=0but E[C]−E[P]=10−6.67=3.33=E[S]−KC - P = S - K\,e^{-rT} = 100 - 100 = 0 \qquad \text{but } E[C] - E[P] = 10 - 6.67 = 3.33 = E[S] - K
    C, Pthe prices of the 100-strike call and put
    S - Kstock minus strike, the value of a long call and short put in every state
    E[S] - Kthe drift your odds put on the stock, 103.33 - 100
    What it says in wordsPrices must satisfy parity, and the gap between the two expected payoffs is exactly the drift your personal odds assign to the stock.

    What odds would a maker use, and are they unique here?

    Odds that make the stock worth its forward, 100 with zero rates. Any set of probabilities on 80, 100 and 130 with an average of 100 prices the two options consistently; one such set is 0.40, 0.33 and 0.27, which gives the call 0.267 x 30 = 8 and the put 0.40 x 20 = 8, equal as parity demands. The limitation is that with three outcomes and only the stock to hedge with, the set is not unique: the common level of the call and put price is pinned down by parity only up to a range, and in practice it is the market's volatility quote that chooses the point inside it.

    Where candidates lose it

    The arithmetic is easy and candidates get 10 and 6.67 quickly; the loss comes in the second half, where they say a market maker would add a spread to the expected payoff. The real answer is that the probabilities themselves are wrong for pricing, because the price is the cost of a hedge.

    Say parity: call minus put equals stock minus strike, zero here, so the two prices must be equal. That one sentence shows you know why risk-neutral pricing exists.

    What the interviewer asks next

    • Find a set of probabilities on 80, 100 and 130 under which the stock is worth 100, and price both options.
    • Interest is 5% for the period. What does parity say the call minus the put is worth now?
    • Why is the set of hedge-implied probabilities not unique with three outcomes and one stock?
    • The stock will be at 80 or 130 only. Price the call by replication.