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. 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?
2. 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?
3. 016List the inputs to an option price and tell me which way each one moves a call and a put.Options basicsCorephone / first roundDerivatives operations

   Say this

   Six inputs: spot, strike, time, volatility, the risk-free rate and dividends. Spot up helps calls and hurts puts, higher strike does the reverse. More time and more volatility help both. Higher rates help calls and hurt puts. Dividends hurt calls and help puts.

   Then walk it

   1. Volatility is the only input that raises both, and that is the single most important line in the answer. Options are convex payoffs, so a wider distribution increases expected payoff without increasing the downside, which is already capped at the premium.
   2. Time works the same way for both, with one exception: a deep in-the-money European put can be worth less with more time, because the discounting of the strike dominates the extra optionality.
   3. Rates: a higher rate lowers the present value of the strike you will pay, which helps the call. For a put you are receiving the strike, so discounting it harder hurts. Another way to say it is that a call is a leveraged long, so financing cost is baked into it.
   4. Dividends reduce the forward. Lower forward means lower calls and higher puts. This is why you cannot price an equity option off spot without a dividend forecast, and why dividend risk is a real trading exposure on a long-dated book.
   5. Magnitudes matter more than signs. On a one-month ATM option, one volatility point typically moves the price far more than a 25 basis point rate change. Rates and dividends only dominate on long-dated structures.
   6. The check I would do out loud: only volatility and time raise both a call and a put, and everything else is a tug of war. If you can state that, you have not memorised a table, you have understood the shape.

   Where candidates lose it

   Reciting the table without being able to explain why volatility raises both. If you cannot say 'the payoff is convex and the downside is capped at the premium', the interviewer will assume you learned a grid rather than a mechanism.

   Expect next

   • Why does volatility raise both a call and a put?
   • When is more time worth less for a put?
   • Which input would you least trust in a real pricing run?
4. 018Give me the bounds on a European call price without using any model.Options basicsHardtechnicalProp trading firmsMarket making

   Say this

   The price sits between max of zero and spot minus the discounted strike, and spot itself. Below that lower bound or above spot, there is a static arbitrage that does not depend on any model, any distribution or any volatility assumption.

   Then walk it

   1. Upper bound: a call can never be worth more than the stock, because the most it ever delivers is the stock, and only after you pay the strike. If a call traded above spot I would sell the call, buy the stock, and be guaranteed a profit.
   2. Lower bound: the call must be worth at least spot minus the present value of the strike. If it were cheaper, I buy the call, short the stock, invest the proceeds. At expiry I exercise or buy in the market, and I have locked in the gap risk-free.
   3. Note the discounting in the lower bound. Naive intrinsic, spot minus strike, is the wrong floor for a European option, and getting that right is the point of the question.
   4. Beyond bounds there are model-free shape constraints. Call prices must be decreasing in strike, and convex in strike — the butterfly constraint. A violation means a butterfly spread with a negative cost and a non-negative payoff.
   5. And calendar monotonicity: a longer-dated call cannot be cheaper than a shorter-dated one at the same strike, for European options on a non-dividend payer.
   6. These constraints are what production systems police. An arbitrage-free volatility surface is defined by exactly these inequalities, and a fitted surface that violates butterfly convexity will let a trader book a position the risk system prices as free money. That is why the checks run before the model does.

   Where candidates lose it

   Giving spot minus strike as the lower bound. The discounting is the whole test. And the strong follow-up is the convexity-in-strike condition — if you can name the butterfly argument, you have said something most candidates cannot.

   Expect next

   • Why must call prices be convex in strike?
   • Show me the arbitrage if they are not.
   • How do those constraints get used when fitting a volatility surface?
5. 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.

6. 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.

7. 022List the Black-Scholes assumptions and tell me where each one fails.Option pricingHardtechnicalQuant tradingRisk management

   Say this

   Six that matter: lognormal returns with constant volatility, continuous trading with no transaction costs, no jumps, a constant known risk-free rate, no dividends or a known continuous yield, and European exercise. Every one of them fails, and the market patches each failure in a different way — which is why the volatility surface has the shape it does.

   Then walk it

   1. Constant volatility fails first and worst. Volatility clusters and mean-reverts, so a single number cannot price every strike and maturity. The market's patch is to quote a different implied volatility per strike and expiry, which is the smile and the term structure.
   2. Lognormality fails in the tails. Real return distributions are fat-tailed and negatively skewed — a 5 standard deviation daily move should be a once-in-millennia event and happens every few years. The patch is the skew: out-of-the-money puts trade at higher implied volatility because the model underprices crashes.
   3. No jumps fails on any event date. A takeover or an earnings gap moves price discontinuously, so a delta hedge does not protect you through it. The patch is jump-diffusion and local-volatility models, and in practice a trader simply pays up for gamma into events.
   4. Continuous costless hedging fails always. You hedge at discrete intervals and pay bid-offer, so your realised profit and loss has a hedging error term whose size scales with your rebalance frequency. The patch is to widen the spread you quote.
   5. Constant rates and known dividends fail on long-dated equity options, where dividend uncertainty is a first-order risk and you need a stochastic rate model. For short-dated index options neither matters much.
   6. And European exercise fails for single-stock American options, where dividends make early exercise rational. That needs a tree or a numerical method, not a formula. The meta-point: Black-Scholes is used as a quoting convention rather than a belief, and the smile is the accumulated record of everywhere it is wrong.

   Where candidates lose it

   Listing assumptions without connecting each to an observable market feature. The strong version of this answer says 'fat tails plus no jumps equals the skew you see on the screen'. If you cannot link the failure to the smile, you are reciting a textbook page.

   Expect next

   • Which single assumption would you relax first if you were building a pricer?
   • So is the smile a model failure or market information?
   • How does discrete hedging show up in your profit and loss?
8. 023What do d1 and d2 actually mean in Black-Scholes?Option pricingHardsuperdayQuant tradingProp trading firms

   Say this

   N of d2 is the risk-neutral probability the option finishes in the money. N of d1 is the delta of the call, and it is also the probability of finishing in the money under a measure where the stock, rather than cash, is the numeraire. The gap between them, which is the volatility times root time, is why delta is not the same as the probability of exercise.

   Then walk it

   1. Read the formula as two pieces. The discounted strike times N of d2 is what you expect to pay, weighted by the chance you pay it. Spot times N of d1 is what you expect to receive, and it has to be weighted by the value of the stock conditional on exercise, not just the chance of it.
   2. That conditioning is the whole distinction. When the option finishes in the money, the stock is on average higher than its unconditional expectation, so the receipt leg needs a bigger weight. Hence d1 exceeds d2 by sigma root t.
   3. d2 equals the log of forward over strike, minus half the variance, all divided by sigma root t. It is a standardised distance to the strike measured in volatility units — a z-score.
   4. N of d1 being the delta is not a coincidence, it is the hedging argument: the number of shares you hold equals the probability-weighted claim on the stock.
   5. Useful desk consequence: a 25-delta option does not have a 25 percent chance of expiring in the money. For a call the probability is lower than the delta, and the gap widens with volatility and maturity. Retail traders systematically get this wrong when they size positions off delta.
   6. The caveat: all of this is under the risk-neutral measure, so N of d2 is not the real-world probability of exercise. Real-world probability needs the actual drift, which nobody knows. Quoting risk-neutral probabilities as real ones is the standard error.

   Where candidates lose it

   Calling N of d1 a probability without saying under which measure, or claiming delta is the chance of expiring in the money. The interviewer is specifically testing whether you know delta and probability of exercise are different numbers, and why.

   Expect next

   • So is a 25-delta call's probability of exercise higher or lower than 25 percent?
   • Why is d1 greater than d2 by exactly sigma root t?
   • Is N of d2 the real-world probability of exercise?
9. 026What is delta, and give me three different ways to think about it.The GreeksCorephone / first roundMarket making

   Say this

   Delta is the change in the option's value for a one-unit change in the underlying. Three readings: it is the sensitivity, it is the hedge ratio — the number of shares you hold to be flat — and it is approximately the risk-neutral probability of finishing in the money, though only approximately.

   Then walk it

   1. As a sensitivity: a 0.4 delta call gains 40 paise for a 1 rupee move in the stock. Calls run from 0 to 1, puts from 0 to minus 1, and at the money sits near 0.5 in absolute terms.
   2. As a hedge ratio: short one 0.4 delta call and buy 40 shares and you are locally flat. This is the reading that matters on a desk, because it is the trade you actually put on.
   3. As a probability proxy: it is close to N of d2 but not equal to it, so treating a 25-delta option as a 25 percent chance of exercise is a real error that widens with volatility and time.
   4. As exposure: delta times the number of contracts times the contract multiplier gives you delta-equivalent notional, which is how the whole book gets aggregated. A book of 400 different options collapses to one number you can hedge with futures.
   5. The thing that makes delta interesting is that it is not constant. It moves with spot, which is gamma, and it moves with time and volatility, which are charm and vanna. A delta hedge is a snapshot that starts going stale the moment you put it on.
   6. The limitation to state: delta is a first-order local approximation. In a gap move it is nearly useless — a 0.4 delta call in a 15 percent overnight drop does not lose 0.4 times the move, it loses far less, because gamma works in the buyer's favour. Anyone who managed risk through March 2020 with delta alone learned this.

   Where candidates lose it

   Giving one definition and stopping. Junior interviews ask this to see whether you connect the maths sensitivity to the actual hedging trade. If you cannot say 'so I buy 40 shares', you have described a partial derivative rather than a job.

   Expect next

   • What is the delta of an at-the-money option, exactly?
   • How does delta change as expiry approaches?
   • Is delta the probability of expiring in the money?
10. 027What is gamma and why do traders care about it more than delta?The GreeksCoretechnicalMarket makingProp trading firms

    Say this

    Gamma is the rate of change of delta — the curvature of the option's value. Traders care more because delta can be hedged away in one trade, while gamma is what determines whether that hedge keeps working. Gamma is the risk you actually carry between rebalances.

    Then walk it

    1. Long gamma means your delta moves in your favour: you get longer as the market rises and shorter as it falls, so mechanical rebalancing sells high and buys low. Every rebalance banks a small profit.
    2. Short gamma is the reverse and it is vicious. You get shorter into a rally and longer into a selloff, so hedging forces you to buy high and sell low. The losses compound with the size of the move because the payoff is concave.
    3. Gamma is largest at the money and increases sharply as expiry approaches. A one-day ATM option has enormous gamma and almost no vega, which is why expiry-day books are managed completely differently from long-dated ones.
    4. You pay for gamma with theta. A long gamma position bleeds every day it does not move, and the break-even is roughly whether realised volatility exceeds the implied volatility you paid. That trade-off is the core of a market maker's daily profit and loss.
    5. Concrete version: long a 100-strike straddle at 20 implied on a stock that then realises 30 percent volatility. You lose theta every quiet day and make it back on the days that move, and the sum over the life is positive because realised beat implied.
    6. Market-wide, gamma positioning explains a lot of intraday behaviour. When dealers are short gamma they must hedge in the direction of the move, which amplifies it; when they are long they dampen it. That is the mechanism behind the gamma-squeeze stories, and it is real, though usually overstated in the press.

    Where candidates lose it

    Defining gamma as the second derivative and stopping. The interviewer wants the trading consequence: long gamma means your hedges make money, short gamma means they lose money, and you are paying or receiving theta for the privilege. Say what you do on the rebalance.

    Expect next

    • So how do you make money from being long gamma?
    • Where is gamma largest, and what does that do to your hedging on expiry day?
    • What does it mean for the market when dealers are collectively short gamma?

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.