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
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  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.