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. 005What is basis risk, and when does it hurt a hedger most?Forwards and futuresIntermediatetechnicalCorporate treasury

   Say this

   Basis is spot minus futures. Basis risk is the risk that the two do not move together, so your hedge does not offset your exposure one for one. It bites hardest when the thing you are hedging is not the thing the contract is written on, or when your exposure and the contract mature on different dates.

   Then walk it

   1. Three sources. Asset mismatch, where you hedge jet fuel with crude futures. Maturity mismatch, where your exposure runs to March and the liquid contract expires in February. And location or grade mismatch, where your physical sits in a different delivery point.
   2. At expiry basis goes to zero for the matched asset, because delivery forces convergence. Away from expiry it wanders, so a hedge you intend to lift early carries basis risk even on a perfect asset match.
   3. The cross-hedge version is the dangerous one. Jet fuel and crude are usually 90 percent correlated, which sounds fine until a refining margin shock breaks the relationship precisely during the event you were hedging.
   4. A hedge does not eliminate risk, it swaps price risk for basis risk. You do it because basis is normally an order of magnitude less volatile than outright price, not because it is zero.
   5. Numbers make it concrete. Hedging a 10 million dollar Indian mid-cap book with Nifty futures might cut your volatility from 22 percent to 12, not to zero, because the beta is unstable and the residual is idiosyncratic. You have to be honest that the hedge is partial.
   6. The rolling version compounds it. If your exposure is five years and the liquid contract is three months, you roll twenty times and each roll happens at whatever basis the market offers you that day. That is stack-and-roll risk, and it is what broke Metallgesellschaft's oil hedge.

   Where candidates lose it

   Saying basis risk means the hedge is imperfect, without naming a source. Name asset, maturity and location mismatch, and give one live example where the correlation broke during the stress you were hedging. That is the answer a desk recognises.

   Expect next

   • How would you decide between a cross-hedge and no hedge at all?
   • What is stack-and-roll risk?
   • Your hedge ratio was estimated on three years of data. Why might it be wrong tomorrow?
2. 013What does at the money mean exactly — is it the spot or the forward?Options basicsIntermediatetechnicalEquity derivativesFX derivatives

   Say this

   It depends on who is speaking, and the difference matters. Retail and screens mean at the money spot, strike equals the current price. A derivatives desk usually means at the money forward, strike equals the forward price, because that is the strike where a call and a put have the same value and the same absolute delta.

   Then walk it

   1. Moneyness in general: in the money means exercising now has value, out of the money means it does not, at the money means the strike sits at the reference price.
   2. The forward is the honest reference, because an option's fair value is driven by the forward, not the spot. Put-call parity is written against the discounted forward, so at the money forward is the strike where call and put prices coincide.
   3. At the money forward is also the delta-neutral strike in the sense that matters for a straddle: an ATM-forward straddle starts with roughly zero delta, while an ATM-spot straddle does not when rates or dividends are non-trivial.
   4. How big is the gap? On a one-month index option with rates at 5 percent and a 1.5 percent dividend yield, the forward is about 0.3 percent above spot. Trivial. On a two-year option, or a high-carry currency like the rupee where the forward points are 4 to 5 percent a year, it is a different strike entirely.
   5. FX desks quote off delta rather than strike for exactly this reason: 25-delta risk reversal, 10-delta butterfly. Delta is unambiguous across carry regimes in a way that 'at the money' is not.
   6. The practical answer on a desk: if someone says ATM I ask which one, especially on a long-dated or high-carry underlying. The confusion is a real source of trade breaks and mispriced quotes, and it is cheap to eliminate by asking.

   Where candidates lose it

   Answering just 'strike equals spot'. That answer is fine for a one-week Nifty option and wrong for a two-year USD/INR option where the forward is 10 percent away. Volunteer the forward definition and say when the two diverge enough to matter.

   Expect next

   • At which strike do a call and a put have the same price?
   • Why do FX desks quote by delta instead of by strike?
   • For a two-year USD/INR option, how far is ATM forward from ATM spot?
3. 014State put-call parity and tell me exactly what arbitrage enforces it.Options basicsIntermediatetechnicalProp trading firmsMarket making

   Say this

   Call minus put equals spot minus the present value of the strike, for European options on the same underlying, strike and expiry. What enforces it is the conversion and reversal trade: long call, short put, short stock is a riskless package that must be worth the discounted strike, so any deviation is free money.

   Then walk it

   1. Write it as C minus P equals S minus K times e to the minus rt, and subtract dividends from S if the stock pays them.
   2. The intuition in one line: a long call plus a short put at the same strike is a synthetic long forward. It has a delta of one and pays off spot minus strike at expiry regardless of direction, so it must cost the same as a forward.
   3. The enforcing trade is a conversion. Buy the stock, buy the put, sell the call. Whatever happens, you deliver at K, so you have bought a zero-coupon bond. If you can assemble that package for less than the discounted strike, you have picked up risk-free basis points.
   4. The reversal is the mirror: short stock, short put, long call. This is where the constraint usually breaks in practice, because it needs a stock borrow.
   5. So the deviations you see on a screen are almost never mispricings. Hard-to-borrow names show a persistent parity gap that is exactly the borrow cost, which is why the options market is where you read a stock's true short fee. Discrete dividends, early exercise on American options and financing spreads open the rest of the band.
   6. The reason to know this cold: parity is the only completely model-free relationship in options. It holds without any assumption about volatility or distribution, which makes it the one thing you can check a screen against.

   Where candidates lose it

   Reciting the formula without the replication. The question is 'what enforces it', so the answer is a trade: conversion and reversal. And do not claim a parity gap on a hard-to-borrow name is arbitrage — it is the borrow fee, and saying so is what separates someone who has traded from someone who has read.

   Expect next

   • You see a 40 cent parity violation on a small cap. Is that free money?
   • Does parity hold for American options?
   • How would you back out the implied borrow cost from option prices?
4. 015When would you exercise an American option early?Options basicsIntermediatetechnicalEquity derivativesMarket making

   Say this

   Almost never for a call on a non-dividend-paying stock, and reasonably often for a deep in-the-money put. The general rule is that you exercise early when the interest or dividend you pick up is worth more than the optionality you throw away.

   Then walk it

   1. Never for a call on a stock that pays no dividend. Exercising means paying the strike early, so you lose the interest on it, and you give up the downside protection the option gave you for free. It is always better to sell the option than exercise it.
   2. The exception is a dividend. If the stock pays 3 dollars tomorrow and your deep in-the-money call has less than 3 of remaining time value, exercise the day before the ex-date to capture the dividend. That is why open interest in deep ITM calls collapses into an ex-date.
   3. Puts are different, because exercising a put brings cash in early. A deep in-the-money put on a stock near zero is essentially a claim on the strike, and holding it means forgoing interest on that cash. With rates at 5 percent that is a real cost, so early exercise becomes optimal.
   4. There is a critical price for the put below which immediate exercise beats holding, and it rises with the interest rate and falls with volatility. Zero rates make early exercise on puts nearly irrelevant, which is why the whole topic went quiet from 2010 to 2021 and came back with rate hikes.
   5. So American calls on non-dividend stocks are worth exactly the European price; American puts are worth strictly more, and the difference is the early exercise premium.
   6. The desk-level version: assignment risk. If you are short a deep ITM call into an ex-date you can be assigned, which turns you short stock and short the dividend overnight. Managing that calendar is a daily job on an equity derivatives book.

   Where candidates lose it

   Answering 'never, because optionality has value' and stopping. That is right for the textbook call and wrong for a dividend-paying stock and wrong for puts. Split the answer into calls-with-dividends and puts-with-interest, and name assignment risk.

   Expect next

   • Where does the critical price for a put sit and what moves it?
   • If you are short a deep in-the-money call into an ex-date, what is your risk?
   • Why did early exercise stop mattering between 2010 and 2021?
5. 017An option buyer has limited loss and unlimited gain. Why does anyone take the other side?Options basicsIntermediatetechnicalProp trading firmsMarket making

   Say this

   Because the seller gets paid, and on average the premium is more than the payout. Buyers are buying insurance and insurers earn a premium for bearing tail risk. The payoff shape is asymmetric but the expected value is not the same as the shape.

   Then walk it

   1. The empirical fact first: implied volatility exceeds subsequent realised volatility most of the time, in most markets. That wedge — the variance risk premium — is the seller's edge, and it is typically a couple of volatility points on index options.
   2. Why it exists: investors are structurally short the market and want protection, and protection pays off precisely when the rest of the portfolio is losing. That correlation makes it worth paying above fair value for, exactly like fire insurance.
   3. So the seller is not stupid, they are an insurer. The trade works the way insurance works: small steady income, occasional large loss, positive expectancy if priced right and sized right.
   4. The real risk is not the expectancy, it is the path. A short option book has negative skew, so it grinds up and then gives back years of premium in a week. February 2018 wiped out short-vol products in a single session on a move that was not even a large one by historical standards.
   5. Which is why sellers hedge. A market maker is not taking a directional view; they sell the option, delta hedge it, and try to earn the spread between implied and realised volatility. The naked seller and the hedged seller are completely different businesses.
   6. The honest framing: buyers pay for convexity and certainty of maximum loss, sellers earn a premium for supplying it. Neither is a free lunch, and the seller's version has a fatter left tail than a Sharpe ratio will show you.

   Where candidates lose it

   Answering 'because most options expire worthless'. That is a statistic about frequency, not about expected value, and an interviewer will immediately ask whether you would sell 1-in-1000 lottery tickets at any price. Name the variance risk premium and the negative skew of the seller's return.

   Expect next

   • So would you rather be systematically long or short volatility?
   • What happened to short-volatility products in February 2018?
   • How does a market maker sell options without taking a directional view?
6. 019Explain how you would price an option.Option pricingIntermediatetechnicalDRWQuantitative Trading · Chicago · 2025

   Say this

   By replication, not by forecasting. I build a portfolio of the underlying and cash that reproduces the option's payoff in every state, and the option has to cost what that portfolio costs, or there is an arbitrage. Everything else — binomial trees, Black-Scholes, Monte Carlo — is just machinery for doing that in different settings.

   Then walk it

   1. Start with the simplest case, one period, two states. If the stock goes to 110 or 90 and I hold a 100-strike call, I can find a number of shares and a cash amount that pays exactly 10 in the up state and 0 in the down state. That portfolio's cost today is the option price.
   2. The striking thing is that the real probabilities never appear. They cancel, because I am hedging rather than betting. That is why the answer is the same whether you think the stock is going up or down, and it is the single most important idea in the subject.
   3. Equivalently, discount the expected payoff under the risk-neutral measure, where the underlying is assumed to drift at the risk-free rate. Same number, and easier to compute.
   4. Take that to many small steps and it becomes the binomial tree; take the limit and you get Black-Scholes, a closed form for a European option on a lognormal underlying. For a path-dependent payoff you simulate instead, and for an American feature you need a tree or a backward induction so you can test early exercise at every node.
   5. Then the practical part, which is where the real work is. The only unobservable input is volatility, so in practice the model is run backwards: I take the market price and solve for implied volatility, then trade the volatility rather than the price.
   6. And the limitation up front: replication assumes continuous hedging with no transaction costs and no gaps. In the real world I hedge discretely and pay spread, so my realised profit and loss is implied minus realised volatility, less the cost of hedging — which is why a theoretically fair option can still lose money.

   Where candidates lose it

   Leading with the Black-Scholes formula. DRW and every prop shop are asking whether you understand replication and risk-neutral valuation, not whether you can recall a closed form. If you cannot explain why the real-world probability drops out, you have not answered the question.

   Expect next

   • Why do the real-world probabilities disappear?
   • Now price it in a two-step tree and tell me what changes.
   • What is the one input you cannot observe, and what do you do about it?

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

7. 020Price a one-period call with a binomial tree. Stock at 100, up to 120 or down to 80, strike 100, rate zero.Option pricingIntermediatetechnicalProp trading firmsQuant trading

   Say this

   Ten. The risk-neutral up probability is 0.5 because the up and down moves are symmetric around 100 with a zero rate, so the option is worth 0.5 times 20 plus 0.5 times 0, undiscounted. And I can prove it with a hedge rather than a probability.

   Then walk it

   1. Risk-neutral probability: p equals one plus r minus d over u minus d. With u of 1.2, d of 0.8 and r of zero, p is 0.2 over 0.4, so 0.5. Payoffs are 20 up and 0 down, so the value is 10.
   2. Now the replication, which is the answer they actually want. Delta is the payoff spread over the price spread: 20 minus 0 over 120 minus 80, so 0.5. Hold half a share, which costs 50, and borrow 40. In the up state the half share is worth 60, repay 40, net 20. In the down state 40 minus 40 is zero. Both match, and the portfolio cost 10.
   3. So the hedge ratio and the price come out of the same arithmetic, and the 0.5 that appears twice is a coincidence of the symmetric tree — one is a delta, the other a probability.
   4. Note that p is not a forecast. If I told you the stock has a 90 percent chance of going up, the option is still 10, because I can hedge it. The real probability affects whether you want the trade, not what it costs.
   5. Add a rate and both pieces move: p shifts up because the risk-neutral drift is higher, and you discount the expectation. With r at 5 percent p becomes 0.625 and the value rises to about 11.9.
   6. The limitation worth saying: a one-period tree is a cartoon. It only works because two states and two instruments make the market complete. Add a third state and I can no longer hedge exactly, and the price becomes a range rather than a number — which is the real world with jumps in it.

   Where candidates lose it

   Using the real-world probability, or averaging 120 and 80 to get an expected stock price and working from there. Also, do the replication: the interviewer wants to see you derive delta as the ratio of payoff spread to price spread, not quote a formula.

   Expect next

   • Redo it with the rate at 5 percent.
   • What if I tell you the real probability of the up move is 90 percent?
   • Now make it two periods and tell me what changes about the hedge.
8. 021Explain Black-Scholes.Option pricingIntermediatetechnicalGoldman SachsWealth Management · Zurich · 2025

   Say this

   It is a closed-form price for a European option, derived from the insight that an option can be perfectly hedged with the underlying. If you can hedge it, its price cannot depend on your view or on risk appetite — only on volatility, time, rates and the distance to the strike. The formula is the answer to that hedging argument, not a forecasting model.

   Then walk it

   1. The derivation in one line: build a portfolio that is long the option and short delta of the stock, and the random term cancels. What is left grows at the risk-free rate, and imposing that gives a differential equation whose solution is the formula.
   2. In words, the call price is the discounted expected payoff under a lognormal distribution: spot times N of d1, less the discounted strike times N of d2. N of d2 is roughly the risk-neutral probability of finishing in the money; spot times N of d1 is the expected value of the stock you receive if you do.
   3. So the inputs are spot, strike, time, rate, dividend and volatility. Five are observable. Volatility is not, which means in practice the formula is used inverted — you put the market price in and read the implied volatility out.
   4. That is its real job on a desk. Nobody believes the assumptions. It is a translation device that turns option prices in dollars into a single comparable number in volatility terms, so you can compare a one-month Nifty option with a two-year S&P option on the same axis.
   5. For a private client I would put it plainly: the formula prices the insurance. The further out of the money, the shorter the time and the calmer the market, the cheaper the insurance — and the fair price of that insurance is what the model gives you.
   6. The limitation, said before being asked: it assumes constant volatility and continuous, costless hedging, and it assumes prices do not jump. All three are false, which is why the market charges a different implied volatility for every strike. That pattern is the volatility smile, and it is the market's way of correcting the model.

   Where candidates lose it

   Writing out the formula and naming the terms without ever saying why the hedging argument removes the drift. And on a wealth management desk specifically: if you cannot restate it in one plain sentence about the price of insurance, you have failed the actual test, which was whether you can explain it to a client.

   Expect next

   • Which assumption fails worst in practice?
   • What does N of d2 actually represent?
   • If nobody believes the assumptions, why is it still on every screen?

   Reported by candidates at Goldman Sachs (Wealth Management, Zurich, 2025). Source: Wall Street Oasis.

9. 029What is vega, and where on the surface is it biggest?The GreeksIntermediatetechnicalEquity derivativesVolatility trading

   Say this

   Vega is the change in option value for a one-point move in implied volatility. It is biggest for long-dated at-the-money options, and it scales roughly with the square root of time. That is the key structural fact: gamma lives at the front of the curve, vega lives at the back.

   Then walk it

   1. Both calls and puts have positive vega, because higher volatility widens the distribution and both are convex payoffs. Being long options is being long volatility, full stop.
   2. Magnitude: vega is proportional to spot times root time times the standard normal density at d1. Root time means a two-year option has about five times the vega of a one-month option on the same notional.
   3. So the division of labour on a desk: if you want to trade the level of volatility you use long-dated options or variance swaps. If you want to trade the movement of the underlying you use short-dated options, where gamma dominates and vega is almost irrelevant.
   4. Vega is not one risk, it is a surface of risks. A book can be vega-flat in total and still be badly exposed if it is long front-month and short back-month volatility — that is vega term structure risk, and desks bucket vega by expiry rather than summing it.
   5. Numbers: an at-the-money one-year index option with 10 million of notional might carry 40,000 of vega, so a three-point volatility spike is 120,000. In March 2020 index implied volatility went from 15 to 80 in three weeks. That is the scale of the risk.
   6. The limitation: vega assumes a parallel shift in implied volatility, and real surfaces do not shift in parallel. Front-month volatility moves far more than back-month, and the skew steepens as the level rises. So you need vanna and volga — the cross-sensitivities to spot and to volatility itself — before your vega number means anything on a skewed book.

   Where candidates lose it

   Saying vega is biggest at the money and stopping, without the root-time scaling. The distinction that matters is that gamma is a short-dated risk and vega a long-dated one; if you cannot say which instrument to use for which view, you have not answered a trading question.

   Expect next

   • So which option would you buy to express a pure view on the level of volatility?
   • Why is a vega-neutral book still exposed to volatility?
   • What are vanna and volga, and when do they matter?
10. 030Rho gets ignored. When does it actually matter?The GreeksIntermediatetechnicalRates derivativesEquity derivatives

    Say this

    Rho is the sensitivity of an option's value to the interest rate. It is negligible on short-dated equity options, which is why nobody talks about it, and it is first-order on long-dated options, on FX, and on anything with a large strike relative to spot. It came back into focus when rates went from zero to five percent.

    Then walk it

    1. Sign: calls have positive rho, puts negative. A higher rate lowers the present value of the strike you will pay, which helps the call and hurts the put.
    2. Size: rho scales with time and with the discounted strike. On a one-month at-the-money option a 25 basis point rate move is a rounding error. On a five-year option it can be worth more than a volatility point.
    3. So where it bites: long-dated structured products, LEAPS, and the embedded options in insurance and pension liabilities. Anyone running a long-dated book in 2022 saw their option values move on rates as much as on volatility.
    4. In FX, rho is not one number but two, because you have the rate on each currency. The forward points are the rate differential, so an FX option's exposure to rates is really an exposure to the carry, and on a high-differential pair like USD/INR that dominates.
    5. Rates also change behaviour, not just value. Early exercise on American puts becomes optimal at higher rates, and the cost of carrying a delta hedge is a financing cost that grows with the rate. At five percent, financing a one-million-share hedge is a real line item in the profit and loss.
    6. The honest caveat: for the typical short-dated index option trade, rho is genuinely ignorable and pretending otherwise is false precision. The judgement being tested is whether you know which Greeks to care about for which instrument, rather than whether you can list five of them.

    Where candidates lose it

    Dismissing rho entirely, or over-claiming its importance. The right answer is a judgement about maturity and instrument: irrelevant for weekly index options, first-order for a five-year structured note and for FX carry. And name the financing cost of the hedge, which most candidates miss.

    Expect next

    • How does rho work differently in FX?
    • What did the 2022 rate cycle do to long-dated option books?
    • How does a higher rate change your delta hedging cost?

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.