Derivatives Foundation interview preparation

Questions

100

Traced to a firm

29

Firms

19

Updated

September 2026

Asked at

All firmsMSMorgan Stanley4Nomura4Akuna Capital2Amundi2HSBC2PIMCO2Bank of America1Barclays1Citadel1DRW1Goldman Sachs1Jane Street1Millennium Management1Mizuho1Old Mission Capital1RCRBC Capital Markets1Scotiabank1UBS1Wells Fargo Securities1

Topic

All topicsForwards and futures10Options basics8Option pricing7The Greeks10Volatility7Option strategies9Swaps and rates7Credit derivatives4Market structure and clearing6Indian derivatives8Trading and markets9Brainteasers6Fit9

Level

AnyCoreIntermediateHard

Type

AnyTechnicalCaseMarket viewBrainteaserFit

1. 086Two separate games have the same expected value. Which would you choose?BrainteasersIntermediatetechnicalAkuna CapitalSales and Trading · Chicago · 2025

   Say this

   Expected value is not enough information, so I would ask about the variance, whether I can play repeatedly, and what fraction of my capital is at stake. Same expected value, lower variance wins if I play once. If I can play many times and size small, I would take the higher variance game if it has any edge, because repetition converts edge into certainty.

   Then walk it

   1. First, name what is missing: the distributions. Two games can share an expected value and have completely different shapes — one pays 1 with certainty, the other pays 1,000 with probability one in a thousand.
   2. One-shot, meaningful fraction of capital: take the low variance game. Utility is concave and a single large loss is not recoverable, which is the whole content of the Kelly and utility arguments.
   3. Repeated play with small sizing: variance matters much less because the law of large numbers works for you. Then I would prefer whichever game has the better edge per unit of capital tied up, and the higher variance one may well be more profitable per dollar deployed.
   4. The crucial extra question is ruin. If the high variance game can take me to zero, no expected value justifies it, because zero is absorbing. So my real decision rule is: maximise growth subject to never risking ruin, which is Kelly sizing, not expected value maximisation.
   5. A concrete version: a coin flip paying 2 to 1 on heads has a positive edge, and betting your whole stack each time gives you an expected value that rises and a probability of ruin that approaches one. Expected value and survival are different objectives.
   6. So the answer I would actually give a trader: I choose neither until I know the variance and the sizing, and then I size by Kelly and prefer the game with the better ratio of edge to variance. That ratio is the Sharpe ratio, and preferring it is the same instinct as running a book rather than making a bet.

   Where candidates lose it

   Picking one and defending it. The question has no answer as stated, and the interviewer is testing whether you ask for variance, repetition and sizing. A candidate who says 'they are the same, so I am indifferent' has failed. A candidate who asks three questions before choosing has passed.

   Expect next

   • What if you could play a thousand times?
   • What if one game could take you to zero?
   • How would you size the bet?

   Reported by candidates at Akuna Capital (Sales and Trading, Chicago, 2025). Source: Wall Street Oasis.

2. 087Make me a market on something you cannot know. Now, how much would you risk to win 100 dollars if the real answer is inside your market?BrainteasersIntermediatetechnicalAkuna CapitalTrading · Chicago · 2025

   Say this

   If I believe my own market, the answer being inside it is the outcome I expect, so I should be willing to risk a meaningful amount — but the question is a test of whether my market was honest. The right response is to state my confidence as a probability, then size the bet from that probability, not from bravado.

   Then walk it

   1. First, be clear what a market means: a bid and an offer I am willing to be traded on either side of. If I quote 40 at 60, I am saying I will buy at 40 and sell at 60, and the width is my uncertainty.
   2. So the follow-up question is really 'what is your confidence that the answer lies between 40 and 60?' If I say 80 percent, the fair stake to win 100 is around 25 — because at 4 to 1 in my favour, risking 25 to win 100 is the break-even at 80 percent.
   3. That arithmetic is the answer: my willingness to bet has to be consistent with the width I quoted. If I quoted a tight market and then refuse to bet, my market was dishonest. If I quoted a wide market and bet enormously, I was sandbagging.
   4. Then apply a sizing discount for the fact that the interviewer has information I do not, or is choosing the question because it is adversarial. Betting against someone who knows the answer means adverse selection, and the correct response to adverse selection is to widen the market, not to bet bigger.
   5. So I would say: I quote 40 at 60, I am about 75 to 80 percent confident, and I would risk 20 to win 100 — and if you want me to risk more than that, I need to widen my market first. That trade-off between width and size is exactly the market maker's job.
   6. And I would say the meta-point out loud, because it is the point: the test is consistency between my quoted uncertainty and my willingness to back it. A trader who cannot price their own confidence cannot be trusted to price anything else.

   Where candidates lose it

   Answering with a number to look brave, or refusing to bet at all. Both fail. The answer must tie the stake to the probability implied by the width you quoted, and it should mention adverse selection — the interviewer picked this question for a reason. Consistency is being graded, not courage.

   Expect next

   • Your market was 40 at 60. What probability does your bet imply?
   • I want to bet ten times that size. What do you do?
   • Why should you widen rather than bet bigger when I know more than you?

   Reported by candidates at Akuna Capital (Trading, Chicago, 2025). Source: Wall Street Oasis.

3. 088Compute the probability and then bet on whether you draw another black stone.BrainteasersHardtechnicalCitadelQuantitative Trading · New York · 2025

   Say this

   Two parts, and the second is the real test. Compute the conditional probability by Bayes, updating on what has already been drawn, then price a bet at odds that give you an edge and size it so a wrong answer does not end you. Most candidates get the arithmetic and then bet like the arithmetic is certain.

   Then walk it

   1. Set up the inference properly. If the composition of the bag is unknown, drawing a black stone is evidence about the composition, so you update over the possible bags rather than treating the draw as independent.
   2. The classic version: two urns, or a bag with an unknown mix, and a black draw raises the posterior weight on black-heavy compositions. With no replacement, conditioning on the draws already made is essential — a common error is to compute the unconditional probability and hand it over.
   3. Worked case to show the mechanism: three stones, one known black, one known white, one unknown with equal odds. You draw black. Posterior probability the unknown is black rises to two thirds, so the next draw being black is now more likely than the prior suggested. That is the Bayes step they are checking.
   4. Then the betting step, which is where the interview is decided. If my computed probability is 0.6, I want odds better than 3 to 2 to have an edge. I would quote a market rather than accept theirs — say I am a buyer at 55 and a seller at 65 — because that is the trading answer rather than the maths answer.
   5. Then sizing. Kelly says stake a fraction equal to the edge over the odds, and in an interview I would bet a small multiple less than Kelly, because my probability estimate is itself uncertain and Kelly assumes it is not.
   6. And I would say the honest caveat: my probability depends on my prior over the bag's composition, and if I have the prior wrong my edge is imaginary. So I would take the bet at odds that leave room for my model being wrong, which means demanding better than fair odds rather than exactly fair ones.

   Where candidates lose it

   Computing the probability and then accepting whatever odds are offered. Citadel is watching whether you distinguish your estimate from your confidence in it, quote a two-way market, and size below Kelly because the input is uncertain. The maths is the easy half.

   Expect next

   • What is your prior over the bag's composition, and how much does the answer depend on it?
   • What odds do you need to take the bet?
   • How much would you stake, and why not more?

   Reported by candidates at Citadel (Quantitative Trading, New York, 2025). Source: Wall Street Oasis.

4. 089There are n cars on a circular track and between them just enough petrol for one car to complete a lap. Show that there is a car that can complete the lap by collecting petrol from the others as it goes.BrainteasersHardsuperdayMillennium ManagementInvestments · London · 2024

   Say this

   Yes, such a car always exists. The cleanest proof: imagine a phantom car with enough fuel to complete the lap anyway, start it anywhere, and track its fuel level as it picks up each deposit. The point at which its fuel is at its minimum is a valid starting car — from there the cumulative balance never goes negative.

   Then walk it

   1. Set it up as a sequence of partial sums. Going around the circle, each car contributes a gain of its petrol and each gap costs fuel. Total gains minus total costs is exactly zero, because there is precisely one lap's worth.
   2. The argument: define the running balance starting from an arbitrary car. It ends at zero. Take the position where the running balance is at its global minimum, and start there instead. Relative to that point, every partial sum is non-negative, because you subtracted the most negative value from all of them.
   3. So the starting car is the one immediately after the minimum of the cumulative balance. That is a constructive answer, not just an existence proof, which is what makes it satisfying.
   4. There is an induction proof too: with n cars, there must exist some car that has enough petrol to reach the next one — otherwise the total would be insufficient. Merge those two into a single car and you have the same problem with n minus 1. Induct down to one car, which trivially works.
   5. One line for n equals 2 to show the mechanism: if car A has 0.7 laps of fuel and B has 0.3, and the gap from A to B is 0.4, then A cannot reach B directly if it only had 0.3 — the partial-sum argument tells you which one to pick without checking cases.
   6. Why this gets asked at a fund rather than in a maths class: it is the same structure as a cash flow or margin problem. You know the total is sufficient and you need to know whether the path ever goes negative. That is exactly a funding liquidity question, and the answer is always about the minimum of the cumulative balance, not the total.

   Where candidates lose it

   Trying small cases and asserting a pattern. The interviewer wants the partial-sum or induction argument. And the move that impresses is connecting it to cash flow timing — total sufficiency does not imply path feasibility, which is the whole of liquidity risk.

   Expect next

   • Give me the induction version of the proof.
   • Is the starting car unique?
   • What financial problem has exactly this structure?

   Reported by candidates at Millennium Management (Investments, London, 2024). Source: Wall Street Oasis.

5. 090How many taxis are there in Hong Kong Central?BrainteasersIntermediatetechnicalHSBCSales and Trading · Hong Kong · 2026

   Say this

   I would build it from demand rather than guess at a fleet. Central has roughly 250,000 to 300,000 daytime workers and visitors, say 10 percent take a taxi on a given day, so about 25,000 to 30,000 trips. A taxi does maybe 25 trips a day, so around 1,000 to 1,200 taxis serving Central at any time. As a sanity check, Hong Kong licenses about 18,000 taxis in total, so Central holding 5 to 7 percent of them is plausible.

   Then walk it

   1. State the approach before the arithmetic: demand side, because I can estimate people and trips more reliably than I can estimate a fleet directly.
   2. Population of Central during the working day: it is a dense financial district, so a few hundred thousand is the right order. I would say 250,000 and flag that as my biggest uncertainty.
   3. Trips per person per day: most people walk or take the MTR in Central because it is compact and well-served, so I would use 10 percent rather than something higher. That gives 25,000 trips.
   4. Trips per taxi per day: a 10-hour shift with an average trip and repositioning taking 20 to 25 minutes gives roughly 25 trips. So 25,000 over 25 is about 1,000 taxis.
   5. Then the cross-check from the supply side, which is what makes the answer credible: Hong Kong's total licensed fleet is around 18,000, a number that is publicly known and stable because licences are capped. My 1,000 to 1,200 for Central is a believable share of it.
   6. And I would name the sensitivity: the answer is most sensitive to the trips-per-person assumption. Double it to 20 percent and I get 2,000 taxis. So my honest answer is a range of roughly 1,000 to 2,000 with a central case around 1,200, and I would want the actual taxi stand data to tighten it.

   Where candidates lose it

   Guessing a number and then rationalising it. And in Hong Kong specifically, ignoring that Central is compact and MTR-dense — a candidate who uses a New York taxi propensity will be out by a factor of three. Cross-check from the licensed fleet, and name which assumption drives the answer.

   Expect next

   • Which assumption is your answer most sensitive to?
   • Now do it as a fleet-size estimate and see if the two agree.
   • How many beds are there in a New York hotel?

   Reported by candidates at HSBC (Sales and Trading, Hong Kong, 2026). Source: Wall Street Oasis.

6. 091How many beds are there in a New York hotel?BrainteasersCoretechnicalHSBCSales and Trading · Hong Kong · 2026

   Say this

   Around 250 to 350 beds for a typical mid-size Manhattan hotel. Build it from the building: about 20 floors, roughly 12 rooms per floor, so around 240 rooms, and about 1.3 beds per room once you account for doubles and suites. That gives roughly 300 beds.

   Then walk it

   1. Clarify the question first — one hotel, not the city, and a typical hotel rather than a specific one. Asking that takes five seconds and prevents answering the wrong question.
   2. Build from the physical structure: a Manhattan hotel on a standard lot, 20 floors, a floor plate that fits about 12 rooms around a central corridor. That is roughly 240 rooms.
   3. Beds per room: most rooms have either one king or two queens. Call it 60 percent single-bed and 40 percent double, so about 1.4 beds a room, less a few floors for lobby, function space and back of house. Round to 300 beds.
   4. Sanity check against known anchors: New York has roughly 120,000 to 140,000 hotel rooms across about 700 hotels, which is an average of about 180 to 200 rooms per property. So my 240 is on the larger side of average, which is fine if I say I am describing a mid-to-large hotel.
   5. That cross-check is the part that earns credit, because it shows I hold one or two real anchor numbers and use them to test a bottom-up estimate rather than relying on either alone.
   6. I would close with the range and the driver: 250 to 350 for a typical property, and the answer scales almost entirely with floors times rooms per floor, so if you tell me it is a 40-storey tower I would double it. Naming what the answer is a function of matters more than the point estimate.

   Where candidates lose it

   Answering a different question — beds in all New York hotels — or giving a number with no structure. Estimation questions are graded on the decomposition and the cross-check, not on accuracy. Build it, then test it against one anchor you actually know.

   Expect next

   • Now estimate the total number of hotel rooms in New York.
   • What is the occupancy rate, and what would that imply for revenue?
   • Which of your assumptions would you check first?

   Reported by candidates at HSBC (Sales and Trading, Hong Kong, 2026). Source: Wall Street Oasis.

Firm tags come from public, anonymous candidate reports on Wall Street Oasis: strong signal, not sworn testimony. Firms are named as the places a question was reported, not as partners of Fin Maverick. Answers are written for this page to show how to think out loud; they are not scripts to recite.