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 1–3 of 3 · 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.

  2. 077A fund charges 2% of assets a year plus 20% of gains and expects a gross return of 10%. If it cuts the management fee to 1%, what performance fee keeps expected revenue unchanged, and why is the performance fee really a call option the investors have written?Option payoffs and no-arbitrageCoreTwo SigmaNew York · 2026

    Try it first

    At the expected return of 10%, what performance fee replaces the lost 1% of management fee?

    Show the worked solution

    A 30% performance fee holds revenue at 4% of assets at the mean, but the swap is not neutral once returns vary. Two and twenty earns 2 + 0.2 x 10 = 4; one and thirty earns 1 + 0.3 x 10 = 4. The performance fee pays 20% of max(return, 0), which is a call on the fund's return struck at zero. Investors have written it, and its value rises with volatility, so moving fee from fixed to performance raises what the manager expects to collect.

    How do you make the two schedules equal at the mean?

    A shopkeeper who swaps a fixed monthly rent for a share of sales asks one question first: at my usual sales, what share leaves the landlord no worse off? Do the same here, in percent of assets. Two and twenty collects 2 fixed plus 20% of a 10% gain, 4% of assets; cutting the fixed fee to 1 leaves 3 points to be earned from a 10% return, which needs a 30% performance fee. That is the arithmetic the interviewer wants first, and it is one line: 1 + 10x = 4, so x = 0.3.

    Both fee schedules give 4% at a 10% return, but one holds more option-20%-10%0%10%20%30%40%4%8%12%fund return for the yearfee, % of assets10% return: both pay 4%2 and 201 and 302 and 20: flat 2%1 and 30: flat 1%kink at zero: the option strikeReturns spread around 10%with 20 points of volatilityE[max(R, 0)] = 14.0%not 10%, because losses are floored2 and 20 expects4.79%1 and 30 expects5.19%the swap is not revenue neutral
    Two and twenty is a flat 2% that kinks upward at a zero return with slope 0.2, one and thirty is a flat 1% that kinks upward with slope 0.3, and the two cross at a 10% return where both pay 4%, but with returns spread around 10% with 20 points of volatility the expected fee is 4.79% under two and twenty and 5.19% under one and thirty.

    Why is the performance fee an option, and who has written it?

    The manager receives 20% of the gain when the fund is up and nothing when it is down. That is the payoff of a call on the fund's return with a strike of zero: convexA payoff that bends upward, so the average of the payoff over a spread of outcomes is higher than the payoff at the average outcome. in the return, floored at nothing. Investors are on the other side; they have granted the manager that call and are paid for it only through the management fee they do not have to pay. Because a call is worth more when the underlying is more volatile, the performance fee is worth more than its value at the mean return, and the more of the fee you move into performance, the more the schedule is worth for the same expected return.

    The relationship
    E[max⁡(R,0)]=μ Φ ⁣(μσ)+σ ϕ ⁣(μσ)=0.10 Φ(0.5)+0.20 ϕ(0.5)≈0.1396E[\max(R,0)] = \mu\,\Phi\!\left(\tfrac{\mu}{\sigma}\right) + \sigma\,\phi\!\left(\tfrac{\mu}{\sigma}\right) = 0.10\,\Phi(0.5) + 0.20\,\phi(0.5) \approx 0.1396
    muthe expected return, 10%
    sigmathe volatility of the yearly return, taken as 20 points
    Phi, phithe normal distribution and density functions
    What it says in wordsWith returns spread normally around 10% with 20 points of volatility, the average floored gain is about 14.0%, not 10%, because losses are cut off at zero but gains are not.

    Put numbers on it. With that spread the expected performance fee under two and twenty is 0.2 x 14.0% = 2.79%, so the manager expects 4.79% of assets, not 4%. Under one and thirty it is 1 + 0.3 x 14.0% = 5.19%. The swap that looked neutral at the mean adds about 0.40% of assets a year in expected revenue. Say the limitation too: real schedules carry hurdles and high-water marks, which raise the strike and cut the option's value, and the fee is charged on the net of the management fee, which shaves a little off both sides.

    Where candidates lose it

    Candidates get 30% and stop, as if the question were arithmetic. The interviewer is listening for the word option. Without it the answer is a shopkeeper's answer, not a derivatives answer.

    The second loss is saying the fee is an option and then claiming volatility makes it worth less because the fund might lose money. The fund's loss is the investor's, not the manager's; the manager's payoff is floored at zero, which is exactly why volatility helps the manager.

    What the interviewer asks next

    • Add a hurdle of 5%. Does the neutral performance fee rise or fall?
    • A high-water mark means losses must be recovered before fees resume. Which Greek of the option does that change most?
    • If the fund's volatility doubles, roughly how much does the 20% performance fee gain in expected value?

    Asked at Two Sigma, Equity Capital Markets, New York, 2026 (Wall Street Oasis): the 2/20 rule, and if one part of this equation changed, how would the other variable make up for it

  3. 092Screen quotes: the 95 call is 7.00 bid, 7.40 offered, and the 100 call is 4.10 bid, 4.50 offered. You buy the 95 call and sell the 100 call. What do you pay, what is the most you can make and lose, and where do you break even at expiry?Option payoffs and no-arbitrageCoreWolverine Trading, Chicago, ILUSA · 2019

    Try it first

    What does the call spread cost you to put on?

    Show the worked solution

    You pay 3.30; the most you can make is 1.70, the most you can lose is 3.30, and you break even at 98.30. You buy the 95 call at the offer, 7.40, and sell the 100 call at the bid, 4.10. At expiry the spread is worth nothing below 95, the stock minus 95 between the strikes, and 5 above 100. So the profit runs from minus 3.30 to 5 - 3.30 = 1.70, and is zero when the stock is 95 + 3.30 = 98.30.

    Which side of each quote do you deal on?

    At a currency counter at the airport there are two rates on the board: the one they buy at and the one they sell at, and you always get the worse one for you. Option screens are the same. If you want to trade now, you buy at the offer and sell at the bid, so a two-leg trade pays the spread on both legs. Buy the 95 call at 7.40, sell the 100 call at 4.10, net debit 3.30. At the mid prices, 7.20 and 4.30, the same trade would cost 2.90. The 0.40 difference is the cost of crossing two bid-offer spreads of 0.40 each, half of each.

    You pay the offer on what you buy and receive the bid on what you sell859095100105110-4-20+2stock price at expiryprofit per share after the premiummax loss 3.30, the premiummax profit 1.70 = 5 - 3.30break-even 98.30at mids: pay 2.90The screenbidoffer95 call7.007.40100 call4.104.50you buy hereyou sell here7.40 - 4.10 = 3.30 to pay0.40 more than at mids
    Bought across the bid-offer for 3.30, the 95/100 call spread loses 3.30 below 95, gains one for one between the strikes, makes 1.70 above 100 and breaks even at 98.30, while the same spread at mid prices would cost 2.90 and break even at 97.90.

    How do you read off the maximum profit, loss and break-even?

    Walk the stock price up. Below 95 both calls expire worthless, so you lose the 3.30 you paid. Between 95 and 100 only the long call pays, one for one, so the profit rises from minus 3.30. Above 100 the short call pays out as fast as the long call pays in, so the value is capped at the strike gap of 5. A call spread can never be worth more than the gap between its strikes, so the most you can make is 5 minus what you paid, 1.70, and you break even where the stock has risen 3.30 above the lower strike, at 98.30.

    The relationship
    cost=7.40−4.10=3.30,max profit=(100−95)−3.30=1.70,S∗=95+3.30=98.30\text{cost} = 7.40 - 4.10 = 3.30, \qquad \text{max profit} = (100 - 95) - 3.30 = 1.70, \qquad S^{*} = 95 + 3.30 = 98.30
    7.40the offer on the 95 call, what you pay to buy it
    4.10the bid on the 100 call, what you receive to sell it
    S*the break-even stock price at expiry
    What it says in wordsPay the offer, receive the bid, and the spread's payoff is boxed between losing the premium and making the strike gap less the premium.

    Two checks to say aloud. First, the price passes the no-arbitrage bounds: a 95/100 call spread must cost between 0 and 5, and 3.30 does. Second, the risk-reward: you risk 3.30 to make 1.70, which only makes sense if you think the stock finishes above 98.30 more than 3.30/5 = 66% of the time. The limitation: this is the payoff at expiry. Before expiry the spread's value moves with volatility and time, and you could unwind it, but again only by selling the 95 at its bid and buying the 100 at its offer.

    Where candidates lose it

    The common loss is using mid prices and answering 2.90, as if the screen would trade with you at the middle. The interviewer gave you two-sided quotes precisely to see whether you know which side you hit.

    The second loss is getting the maximum profit wrong by forgetting the cap, and saying it is unlimited because you own a call. You also sold one, and above 100 the two cancel; the most a 5-wide spread can ever be worth is 5.

    What the interviewer asks next

    • What would you pay if you could work both orders at mid?
    • At what price would you buy the 95 call and sell the 100 call so that you risk exactly as much as you can make?
    • Build the same view with puts. What are the costs on this screen if the 95 put is 1.80 at 2.10 and the 100 put is 3.70 at 4.00?

    Asked at Wolverine Trading, Prop Trading, Chicago, IL, USA, 2019 (Wall Street Oasis): pricing options given an ask and a bid price for options with different strikes if you were to short one and long another