Quant interview preparation

Questions

100

Traced to a firm

53

Firms

15

Updated

September 2026

Asked at

All firmsOld Mission Capital12Tower Research Capital10Jump Trading7Akuna Capital5Citadel4DED.E. Shaw3Jane Street3ACAQR Capital Management2DRW2Millennium Management2Schonfeld2SCSquarepoint Capital2Susquehanna International Group2Belvedere Trading1Optiver1

Topic

All topicsProbability10Coins, cards and games6Expected value8Statistics11Market making15Estimation and mental maths4Stochastic processes4Regression5Machine learning6Time series6Programming10Options and derivatives8Fit and motivation7

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AnyBrainteaserTechnicalCaseMarket viewFit

1. 086Explain how you would price an option.Options and derivativesIntermediatetechnicalDRWQuantitative Trading · Chicago · 2025

   Say this

   The core idea is replication. If I can build a portfolio of the underlying and cash that matches the option's payoff in every state of the world, then no-arbitrage says the option must cost what that portfolio costs. Everything else, Black-Scholes included, is a way of computing that cost.

   Then walk it

   1. Start with one period and two states, because it makes the logic visible. Stock at 100 goes to 110 or 90, a call struck at 100 pays 10 or 0. Hold delta shares plus B in cash and solve two equations: delta is (10 minus 0) over (110 minus 90), which is 0.5, and then B falls out. The option price is 0.5 times 100 plus B. No probabilities were used anywhere.
   2. That is the key insight to state explicitly: the price does not depend on the real-world probability of the up move, only on the size of the moves. Rearranging gives the risk-neutral probability, which is the probability that makes the discounted stock a martingale, and pricing becomes a discounted expectation under that measure.
   3. Extend the tree to many steps and you get the binomial model, which handles American exercise naturally because you compare intrinsic against continuation at each node. Take the limit with the step size going to zero and you get Black-Scholes.
   4. Black-Scholes in words: the price is the discounted risk-neutral expectation of the payoff when the stock follows geometric Brownian motion with constant volatility. The formula's two N terms are the risk-neutral probability of finishing in the money and the delta-weighted version of it.
   5. Then the practical truth, which is the answer a trading firm actually wants: nobody uses Black-Scholes to find the price, because the price is on the screen. You use it as a translator from price to implied volatility, then you trade the volatility surface. Constant vol is false, the smile proves it, so the real work is interpolating and extrapolating the surface consistently and hedging the Greeks it implies.

   Where candidates lose it

   Reciting the Black-Scholes formula. Anyone can memorise it. The interviewer wants replication and no-arbitrage, and specifically wants to hear that the real-world probability drops out. Then close by saying the formula is used backwards, to extract implied vol from a market price. That last move is what marks a trader rather than a student.

   Expect next

   • Why does the real-world probability not appear in the price?
   • What are the assumptions, and which one fails hardest?
   • How would you price an American put?

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

2. 087You think the market is overestimating volatility. What options strategy would you use?Options and derivativesIntermediatetechnicalOld Mission CapitalProp Trading · Chicago · 2025

   Say this

   Sell volatility and hedge the direction out. The cleanest expression is a short straddle or strangle, delta-hedged so the position is a bet on volatility rather than on the underlying. If implied vol is above what I think realised vol will be, I collect the difference through the gamma-hedging P&L.

   Then walk it

   1. The mechanism: a delta-hedged short option position makes money when realised volatility comes in below the implied vol you sold. Your P&L is approximately half of gamma times the difference between implied variance and realised variance, integrated over the life of the trade.
   2. The instrument choice. A short straddle at the money has the most vega and gamma per unit of premium, so it is the purest vol expression. A short strangle has less gamma but a wider profitable range and less immediate pin risk. If I wanted a cleaner exposure with no path dependence I would sell a variance swap, where the payoff is literally implied minus realised variance.
   3. Risk management is the whole trade. Short gamma means every hedge is at a worse price than the last, so a gap move is where the loss lives. I would cap it with a long wing, turning the strangle into an iron condor, which sacrifices some premium to remove the unbounded tail.
   4. Sizing from the tail: I would set the position so the worst plausible gap, say a five percent overnight move, is a loss I can carry, not from the expected daily P&L. Short vol positions have positive expected value most days and lose several months of it in one session.
   5. And the honest caveat: implied vol trading above realised vol is the normal state of the world, not a mispricing. The variance risk premium exists because sellers are being paid to warehouse gap risk. So I need to believe implied is rich relative to that premium, not merely rich relative to realised, otherwise I am just collecting a risk premium and calling it alpha.

   Where candidates lose it

   Answering short straddle and stopping. Two things must follow: that you delta hedge to isolate the vol view, and that short gamma means a fat left tail so you cap or size for it. Also the variance risk premium point, because saying implied is above realised therefore sell it is the reasoning that ends careers.

   Expect next

   • How do you make it a pure volatility trade?
   • What happens if the stock gaps ten percent overnight?
   • Why is implied usually above realised in the first place?

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

3. 088Walk me through the Greeks, and tell me which one a market maker actually worries about.Options and derivativesCoretechnicalProp trading firmsDerivatives

   Say this

   Delta is sensitivity to spot, gamma to how delta changes, vega to volatility, theta to time and rho to rates. A market maker hedges delta continuously and almost mechanically, so the risks they actually carry are gamma and vega.

   Then walk it

   1. Delta: first derivative of price with respect to spot, between 0 and 1 for a call, and at the money roughly 0.5. It is also approximately the risk-neutral probability of finishing in the money, which is a useful intuition.
   2. Gamma: the second derivative, highest at the money and rising sharply as expiry approaches. Gamma is why a hedge goes stale, and it is the reason a delta-hedged book still has P&L. Long gamma means you buy low and sell high while hedging; short gamma means the opposite.
   3. Vega: sensitivity to implied vol, largest for longer-dated at-the-money options. So near-dated options are a gamma trade and far-dated ones are a vega trade. That distinction drives which expiry you use to express a view.
   4. Theta: the cost of owning optionality. For a delta-hedged long option position, theta is what you pay and gamma is what you earn, and the two balance exactly when realised vol equals implied vol. That relationship is the single most useful thing in the list.
   5. So: delta gets hedged away because it is free to hedge and carries no edge. Gamma and vega are the positions a desk actually runs, and the third risk that does not appear in the standard list but dominates in practice is the correlation and skew risk across strikes, because you are never long one option, you are long a surface.

   Where candidates lose it

   Listing definitions without connecting gamma and theta. The relationship, that a delta-hedged option earns gamma and pays theta and breaks even when realised equals implied, is the answer that shows you understand what a vol trader does all day. Also be clear that delta is hedged precisely because there is no edge in it.

   Expect next

   • What is the relationship between gamma and theta?
   • Which expiry would you use to express a pure vega view?
   • What are the second-order Greeks and when do they matter?
4. 089What is put-call parity, and what would you do if you saw it violated?Options and derivativesIntermediatetechnicalProp trading firmsDerivatives

   Say this

   For European options on a non-dividend-paying stock, call minus put equals spot minus the discounted strike. It is pure arbitrage, no model, because a long call plus a short put plus the discounted strike in cash replicates the stock exactly. If it breaks, you trade both sides and lock a riskless profit.

   Then walk it

   1. The proof is a payoff table. At expiry, long call plus short put pays S minus K in every state, whether S is above or below K. Adding K in cash held to expiry gives you S. So the cost today of call minus put plus K discounted must equal S.
   2. With dividends, subtract the present value of dividends from the spot. With a cost of carry or borrow cost on the short, use the forward: C minus P equals the discounted difference between the forward and the strike.
   3. If I saw a violation, say the call is too expensive: sell the call, buy the put, buy the stock, and borrow the discounted strike. That is a conversion, and the reverse is a reversal. Lock the difference and hold to expiry.
   4. Then the reasons an apparent violation is usually not one, and this is what the question is really testing. Stale quotes on one leg. You are looking at mid prices but must trade at the bid and offer, and the parity gap is usually smaller than the combined spreads. Hard-to-borrow stock making the short leg expensive. American exercise, where early exercise of the put breaks the equality. Discrete dividends you have modelled wrong.
   5. So my actual answer: I would first check whether the apparent edge survives crossing four spreads and paying the borrow. Ninety-nine times out of a hundred it does not, and that is the point of the question. The hundredth time, borrow cost is usually the explanation, and the implied borrow rate you back out of the parity relationship is itself the useful information.

   Where candidates lose it

   Giving the formula and saying you would arbitrage it, with no mention of transaction costs, borrow or American exercise. A trading interviewer asks this specifically to see whether you treat a screen-level inefficiency as free money. Also know that parity holds for European options only, and be able to say why American puts break it.

   Expect next

   • Why does it not hold exactly for American options?
   • How would you back out the implied borrow rate from the option prices?
   • What does a persistent parity gap tell you about the stock?
5. 090What is the difference between implied and realised volatility, and what does the gap between them tell you?Options and derivativesIntermediatetechnicalDerivativesProp trading firms

   Say this

   Implied vol is the market's forward-looking price of volatility, backed out of option prices. Realised vol is a backward-looking statistic computed from returns. Implied sits above realised on average by a few points, and that gap is the variance risk premium, not a free lunch.

   Then walk it

   1. Implied comes from inverting a pricing model on a traded price, so it is a price expressed in volatility units. Realised is the annualised standard deviation of returns over a window, and how you compute it matters: close-to-close, high-low estimators like Parkinson or Garman-Klass, or sums of intraday squared returns.
   2. On the S&P, VIX has historically averaged around 19 to 20 against realised vol nearer 15 to 16. That three to four point gap is persistent and it is compensation to option sellers for taking gap risk and for providing crash insurance.
   3. So the gap does not mean options are overpriced. It means there is a premium for bearing the risk that variance spikes, and that risk is exactly the risk that hurts most when it materialises, since vol spikes coincide with equities falling.
   4. Where the gap becomes information: the term structure, which is normally upward sloping and inverts in a crisis, and the spread between implied and a good realised forecast. If implied is unusually high relative to a GARCH or HAR forecast, that is a candidate signal, but it has to clear the premium first.
   5. And the practical trap to name: implied vol from a monthly option is a forecast of realised vol over the next month, so comparing today's VIX to the last month's realised vol is comparing a forecast to the wrong period. Aligning the horizons correctly makes a lot of apparent signal disappear.

   Where candidates lose it

   Concluding that because implied exceeds realised you should always sell vol. That trade works for years and then loses everything in a week, and interviewers ask it to see whether you know the premium exists for a reason. Also mismatching horizons, which is the technical error that generates fake signals.

   Expect next

   • Why does the variance risk premium exist?
   • How would you actually forecast next month's realised vol?
   • What does an inverted vol term structure tell you?
6. 091You want to express a view that the underlying will move a lot but you have no view on direction. Straddle or strangle?Options and derivativesIntermediatetechnicalProp trading firmsDerivatives

   Say this

   Straddle if I think the move is likely but might be moderate, strangle if I think the move will be large but is less likely. The straddle costs more and starts paying sooner; the strangle is cheaper with a further breakeven and more leverage to a big move.

   Then walk it

   1. Definitions: a straddle is a call and a put at the same strike, usually at the money. A strangle is a call and a put at different out-of-the-money strikes.
   2. Put numbers on it. Stock at 100, one-month at-the-money vol 20 percent, so a monthly standard deviation of about 5.8 percent. The at-the-money straddle costs roughly 0.8 times spot times vol times root T, about 4.6, so breakevens near 95.4 and 104.6. A 95 to 105 strangle might cost 1.8, with breakevens near 93 and 107.
   3. So the straddle needs about a 4.6 percent move to break even and the strangle about 7 percent, but the strangle risks 1.8 rather than 4.6 and pays more per unit risked on a ten percent move.
   4. Greeks tell the same story. The straddle has more gamma and vega per contract and the most theta decay, concentrated near the strike. The strangle has less of everything but a flat maximum loss region, and it gains relatively more from an increase in the wings of the vol surface.
   5. What I would actually decide on, and this is the part interviewers want: whether the implied vol I am paying is cheap relative to my forecast, and where on the smile I am buying it. Out-of-the-money strikes usually carry higher implied vol because of the skew, so the strangle can be the more expensive trade in vol terms even though it is cheaper in premium. If the whole surface is cheap I buy the straddle; if only the wings are cheap I buy the strangle.

   Where candidates lose it

   Answering purely on cost, as if cheaper were better. Compare them on breakeven distance, on cost, and on where you are buying the vol surface. The skew point, that out-of-the-money options often carry higher implied vol, is the answer that shows you think in vol terms rather than premium terms.

   Expect next

   • What if the event has a known date, like earnings?
   • How would the skew change your strike choice?
   • When would you prefer a calendar spread instead?
7. 092What are the assumptions behind Black-Scholes, and which one fails hardest?Options and derivativesHardtechnicalDerivativesProp trading firms

   Say this

   Constant known volatility, geometric Brownian motion with no jumps, continuous frictionless hedging, constant rates, no dividends and European exercise. The one that fails hardest is constant volatility, and the proof that it fails is the volatility smile.

   Then walk it

   1. If the model were right, every strike and expiry on the same underlying would have the same implied vol. They do not. Equity index options show a pronounced skew, with out-of-the-money puts trading at much higher implied vol than calls, and the smile steepens for shorter expiries.
   2. Two economic reasons for the skew: returns are negatively skewed with crash risk, which a lognormal cannot represent, and there is genuine demand for downside protection that pushes puts rich. Both are real and they reinforce each other.
   3. No jumps is the second failure, and it is the same failure in a different form. Under continuous paths a delta hedge is riskless in the limit; with jumps it is not, and that unhedgeable jump risk is precisely what the skew prices.
   4. Continuous costless hedging fails too, which matters practically. You hedge discretely and pay the spread, so your realised hedging error has a variance proportional to the hedge interval, and a short-gamma book pays that cost repeatedly.
   5. But here is the thing worth saying: the model is still used everywhere despite being false, because it is a lossless translator between price and implied vol. Traders quote in vol, not in price, and Black-Scholes is the shared language. The correct summary is that it is a wrong model used as a coordinate system, with local vol, stochastic vol models like Heston, and jump models layered on top for anything path-dependent.

   Where candidates lose it

   Listing the assumptions without naming the smile as the empirical refutation. That link is the whole point. And do not conclude the model is useless, because that misses why every desk still quotes in Black-Scholes implied vol. Wrong but indispensable as a change of variables is the answer.

   Expect next

   • If you know the model is wrong, why still use it?
   • What is local volatility, and what does it fix?
   • How would you price a barrier option given a smile?
8. 093You are long a delta-hedged call. Where does your profit and loss actually come from?Options and derivativesHardsuperdayProp trading firmsDerivatives

   Say this

   From the difference between realised and implied volatility. You earn gamma by rehedging, buying the underlying when it falls and selling when it rises, and you pay theta for the privilege. If realised vol beats the implied vol you paid, the gamma earnings exceed the theta bill.

   Then walk it

   1. The mechanics of gamma scalping: long a call means delta rises as spot rises. To stay hedged you sell into rallies and buy into dips, which is systematically buying low and selling high. Each round trip banks money proportional to the square of the move.
   2. The algebra: daily P&L is approximately half gamma times (change in S) squared minus theta times the time step. Substituting the Black-Scholes relationship between gamma and theta gives P&L proportional to half gamma S squared times (realised variance minus implied variance) times dt.
   3. So the position is a bet on variance, not on direction, and it settles continuously rather than at expiry. Put a number on it: a one percent daily move against a 20 percent annual implied vol, which implies about 1.26 percent daily, means you lose on that day because the move was smaller than what you paid for.
   4. Real-world frictions that eat the theory: you hedge discretely, so you capture only part of the gamma and the hedging error has variance proportional to the hedge interval. You pay the spread on every rehedge, so hedging too often costs more than the gamma it captures. There is an optimal hedge frequency that trades hedging error against transaction cost, and it scales with gamma and the spread.
   5. And the residual you cannot get rid of: vega. If implied vol falls while realised vol is fine, you lose on the mark even if your gamma P&L is positive. A long-dated option is mostly a vega position where the gamma story barely matters, which is why the expiry you choose determines which of these two effects dominates.

   Where candidates lose it

   Saying you profit if the stock goes up. You are delta hedged, so direction is neutralised by construction. The answer must be realised versus implied volatility, with gamma earned against theta paid. Then volunteer the discrete hedging cost, because in practice it is what determines whether a theoretically profitable long gamma position makes money.

   Expect next

   • How often would you rehedge, and what determines the optimal frequency?
   • What if implied vol collapses but realised vol is high?
   • How does this change for a one-week option versus a one-year option?

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.