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

Level

AnyCoreIntermediateHard

Type

AnyBrainteaserTechnicalCaseMarket viewFit

1. 011A random variable is uniform on the interval zero to ten. What are its expected value and variance?StatisticsCorephone / first roundOld Mission CapitalFinance · New York · 2018

   Say this

   Mean 5, variance 100 over 12, which is 8.33, so standard deviation about 2.89. For a uniform on a to b the mean is the midpoint and the variance is (b minus a) squared over 12.

   Then walk it

   1. Mean by symmetry: the midpoint of 0 and 10 is 5. No integration needed.
   2. Variance from the formula (b-a) squared over 12: 100 over 12 equals 8.33, standard deviation 2.887.
   3. If you want to derive it, E of X squared is the integral of x squared over 10 from 0 to 10, which is 1000/30 equals 33.33. Subtract 25 and you get 8.33. Good to be able to do it either way.
   4. The 1/12 is worth carrying in your head because it recurs: a fair n-sided die has variance (n squared minus 1)/12, and the rounding error of a value rounded to the nearest tick has variance tick squared over 12. That last one comes up in real microstructure work.
   5. Practical note: the uniform has thin support and no tails, so it is a bad default for anything financial. The moment somebody hands you a uniform in a trading context, ask what it is meant to represent.

   Where candidates lose it

   Reaching for integration under time pressure and fumbling the arithmetic. Know the (b-a) squared over 12 form cold. Also do not quote variance when they asked for standard deviation or the other way round, and say which one you are giving.

   Expect next

   • What is the expected value of the maximum of two independent draws?
   • What is the distribution of the sum of two independent uniforms?
   • What is the variance of the rounding error when you round to the nearest penny?

   Reported by candidates at Old Mission Capital (Finance, New York, 2018). Source: Wall Street Oasis.

2. 043State the central limit theorem and tell me where it fails.StatisticsCoretechnicalQuant researchQuant trading

   Say this

   For independent identically distributed variables with finite mean and finite variance, the standardised sample mean converges in distribution to a standard normal. The key conditions are finite variance and enough independence, and both fail regularly in markets.

   Then walk it

   1. Precisely: root n times (X bar minus mu) over sigma converges in distribution to N(0,1). Note it is the standardised mean that converges, and the rate is 1 over root n.
   2. Failure one, infinite variance. A Cauchy distribution has no variance and the sample mean of Cauchys is Cauchy again, no matter how large n is. Averaging buys you nothing. More generally, stable distributions with tail index alpha below 2 converge to a stable law, not a normal.
   3. Failure two, dependence. With strongly autocorrelated data the effective sample size is far below n, so you converge much more slowly and your standard errors are too small. Long-range dependence can break it entirely.
   4. Failure three, the rate in the tails. Even where the CLT holds, convergence is fastest in the middle and slowest in the tails, which is precisely where a risk manager needs accuracy. Berry-Esseen gives an error bound of order 1 over root n times the third absolute moment, so skewed data converges slowly.
   5. The practical version: daily equity returns have kurtosis of 5 to 10 and volatility clustering, so ten-day sums are much closer to normal than daily returns, but a 99.9 percent quantile computed from a normal assumption will still understate the tail badly. That is why value at risk models use empirical or extreme-value tails rather than leaning on the CLT.

   Where candidates lose it

   Stating the theorem without the finite variance condition, or claiming everything becomes normal for large n. Also do not confuse it with the law of large numbers, which is about convergence of the mean to a constant and needs only finite mean. Be ready to say what happens with infinite variance, because that is the follow-up.

   Expect next

   • What happens with a Cauchy distribution?
   • How is that different from the law of large numbers?
   • How large does n have to be in practice for returns data?
3. 044What is a p-value, and what is it not?StatisticsCoretechnicalQuant researchRisk

   Say this

   It is the probability of seeing data at least as extreme as what you saw, assuming the null hypothesis is true. It is not the probability that the null is true, and it is not the probability you are wrong.

   Then walk it

   1. The conditioning runs the wrong way from what people assume. A p-value is P(data given null), and what you actually want is P(null given data). Those are different objects and Bayes tells you the second depends on your prior.
   2. Concretely: if you test a thousand strategies of which fifty genuinely work, at a five percent significance level you get roughly 47 true discoveries and 47 false ones. A p-value of 0.05 in that setting means a coin flip on whether the finding is real.
   3. It also says nothing about effect size. With a million observations a completely useless one-basis-point edge will have a p-value of 0.0001. Significance is not importance, and in high-frequency data everything is significant.
   4. And it is only valid for a pre-specified test. Choosing the test after looking at the data, or stopping data collection when the p-value crosses 0.05, invalidates it completely.
   5. What I would report instead on a desk: the effect size with a confidence interval, out-of-sample performance, and how many specifications I tried. A p-value on its own is close to useless in a research process where hundreds of hypotheses get screened.

   Where candidates lose it

   Defining it as the probability the null is true. That is the single most common statistical error in finance interviews and it is disqualifying at a research shop. Also be ready with the multiple-testing consequence, because the interviewer's real target is whether you understand why published anomalies do not replicate.

   Expect next

   • So what significance level would you use if you screened a thousand signals?
   • Explain the false discovery rate.
   • What would you report instead of a p-value?
4. 046What are the assumptions behind ordinary least squares, and which of them actually matter?RegressionCoretechnicalQuant researchRisk

   Say this

   Linearity in parameters, exogeneity meaning the error has zero mean conditional on the regressors, no perfect collinearity, homoskedasticity, and no autocorrelation. Only exogeneity is essential for unbiasedness. The last two affect efficiency and standard errors, not the coefficients.

   Then walk it

   1. Exogeneity, E of error given X equals zero, is the load-bearing assumption. Break it and every coefficient is biased and inconsistent, and no amount of data or robust standard errors saves you.
   2. Homoskedasticity and no autocorrelation give you Gauss-Markov efficiency and the usual standard error formula. Break them and OLS is still unbiased, just no longer the minimum-variance linear estimator, and your t-statistics are wrong. Robust or Newey-West errors fix the inference.
   3. Normality of errors is not needed for unbiasedness or consistency at all. It only buys exact small-sample t and F distributions. Asymptotically the CLT handles it.
   4. No perfect collinearity is a requirement for the estimator to exist, since X'X must be invertible. Near-collinearity is not a violation, it just inflates variances.
   5. On financial data the realistic picture is: heteroskedasticity almost always, autocorrelation often, and exogeneity frequently violated because everything is jointly determined. So I default to robust standard errors, and I spend my thinking time on whether my regressor is endogenous, because that is the one that actually changes the answer.

   Where candidates lose it

   Listing normality as a core assumption, or treating all five as equally important. Rank them. The interviewer wants to hear which violations bias the coefficients and which only bias the standard errors, because that distinction determines whether you patch the model or rebuild it.

   Expect next

   • Give me a concrete example of endogeneity in a returns regression.
   • Why is normality not needed?
   • What does Gauss-Markov actually claim?
5. 052Explain the bias-variance tradeoff, and where a quant strategy usually sits on it.Machine learningCoretechnicalQuant researchQuant development

   Say this

   Expected prediction error decomposes into squared bias, variance and irreducible noise. Bias is how wrong your model class is on average, variance is how much your fit moves with a different sample. In financial data the noise term dominates everything, so you sit far towards the high-bias, low-variance end.

   Then walk it

   1. The decomposition: E of (y minus f hat) squared equals bias squared plus variance plus sigma squared. Only the first two are under your control.
   2. Flexible models cut bias and raise variance. A deep tree fits any shape and moves wildly with resampling. A linear model with three factors barely moves but cannot represent an interaction.
   3. The financial context is what makes the answer different from a generic machine learning answer. Signal-to-noise on returns is tiny, sigma squared swamps the other terms, and a flexible model spends all its capacity fitting noise. So simple, heavily regularised, few-parameter models win out of sample far more often than they should on pure machine learning intuition.
   4. How you find your place on the curve: cross-validation that respects the time ordering, learning curves, and watching the gap between in-sample and out-of-sample performance. If the gap is large you are on the variance side.
   5. One honest complication: the classic U-shaped curve is not the whole story. Very overparameterised models can show double descent, where test error falls again past the interpolation threshold. That is real in vision and language. I have not seen it be useful on noisy financial data, where the tiny signal means regularisation still dominates.

   Where candidates lose it

   Giving the textbook decomposition with no view on where financial data sits. Every candidate can recite bias plus variance. The differentiator is saying that low signal-to-noise pushes you towards simple models, and being able to say how you would diagnose which side you are on.

   Expect next

   • How would you diagnose which side of the tradeoff you are on?
   • Why do simple models often win on financial data?
   • What is double descent?
6. 054What does PCA do, how do you choose the number of components, and what are its limitations on financial data?Machine learningCoretechnicalQuant researchRisk

   Say this

   It finds the orthogonal directions of maximum variance, which are the eigenvectors of the covariance matrix, and lets you describe the data with fewer numbers. Choose the number of components by explained variance, a scree elbow, or the Marchenko-Pastur bulk edge if you want a principled cutoff.

   Then walk it

   1. Mechanically: eigendecompose the covariance or correlation matrix, or take the SVD of the centred data. Eigenvalues are the variance along each component, eigenvectors are the directions.
   2. Correlation versus covariance matters. On assets with wildly different volatilities, PCA on the covariance matrix is dominated by the most volatile names, so standardise first unless the scale is meaningful.
   3. Concrete example everyone in rates knows: PCA on the yield curve gives level, slope and curvature, explaining roughly 90, 8 and 2 percent of variance. On equities the first component is the market, explaining 25 to 40 percent depending on the regime, and it rises sharply in a crisis.
   4. Choosing k: cumulative explained variance at 90 or 95 percent, the scree elbow, or eigenvalues above the random matrix bulk edge, which is the statistically defensible version because it separates signal from estimation noise.
   5. Limitations, and these are the answer to the real question. PCA maximises variance, not predictive power, so the components need not have anything to do with your target. It is unstable: eigenvectors rotate sample to sample when eigenvalues are close, so your factor two and factor three swap places. It assumes linearity. And the components are usually uninterpretable outside a structured setting like the yield curve, which makes them awkward to risk-manage.

   Where candidates lose it

   Describing PCA as dimensionality reduction and stopping. Two things get graded: that it is unsupervised so high-variance directions are not necessarily predictive, and that you must standardise when scales differ. Also have a real example ready, because level-slope-curvature or the equity market factor proves you have used it rather than read about it.

   Expect next

   • Why is PCA not necessarily good for prediction?
   • What does the first principal component of an equity universe represent, and what happens to it in a crisis?
   • How is PCA related to a factor risk model?
7. 067What is the difference between a market order and a limit order, and who pays the spread?Market makingCorephone / first roundProp trading firmsQuant trading

   Say this

   A market order takes whatever price is available and pays the spread for certainty of execution. A limit order posts a price and waits, earning the spread if it fills, but with no guarantee it fills at all. You are choosing between price risk and execution risk.

   Then walk it

   1. The taker of liquidity pays. Buy with a market order and you pay the offer, which is above mid, so you start down by half the spread. The passive side on the other end of that trade collects it.
   2. On most exchanges the fee structure reinforces this: makers get a rebate, takers pay a fee. So the maker's economics are spread capture plus rebate minus adverse selection.
   3. The cost of a limit order is not zero, it is optionality you are giving away. A resting bid is a free put you have written to the market: it fills when the price is falling and does not fill when the price rises. That is adverse selection expressed as execution risk.
   4. So the choice is horizon-dependent. If I need to be done now because I have information or a hedge to put on, I pay the spread. If I am providing liquidity or my signal has a multi-day horizon, I post and wait.
   5. Worth adding the practical middle ground, since this is what execution desks actually do: split the order, post passively and cross only when the queue is not filling or when the signal decays. Implementation shortfall against the arrival price is how you measure whether you got that balance right.

   Where candidates lose it

   Getting the definitions right but not answering who pays the spread. The taker pays. The second miss is treating a limit order as free, when the real cost is the option you have written to anyone with better information. Say that and you are ahead of most candidates.

   Expect next

   • What is the hidden cost of a resting limit order?
   • How would you decide between posting and crossing?
   • What is implementation shortfall?
8. 080Can you implement a linked list, and when would you actually use one on a trading system?ProgrammingCorephone / first roundJump TradingProp Trading · Remote · 2022

   Say this

   Yes, a node with a value and a next pointer, plus a head, and the usual care about the empty list and about updating head when you insert or delete at the front. But the honest answer to the second half is: rarely, because pointer chasing destroys cache performance.

   Then walk it

   1. The implementation: struct with value and next, insert at head in O(1), search in O(n), delete given the previous node in O(1). Use a dummy head node and most of the edge cases disappear, which is the trick worth knowing for interviews.
   2. The standard edge cases they will check: empty list, single element, deleting the head, and not leaking the node you unlinked. In C++ that means being explicit about ownership, and in a real codebase it means a unique pointer or an arena.
   3. What a linked list genuinely buys you: O(1) splice of a node from the middle if you already hold a pointer to it, and stable addresses so a pointer stays valid across insertions. That is exactly the requirement in a limit order book, where you need to cancel an arbitrary resting order in constant time, so orders at a price level are typically an intrusive doubly linked list with a hash from order id to node.
   4. What it costs: every traversal is a potential cache miss, and a vector beats a list for iteration by an order of magnitude even when the asymptotics say otherwise.
   5. So the real-world answer is an intrusive list over a pre-allocated node pool, not a textbook list with individual heap allocations. Saying that is the difference between having done the exercise and having written low-latency code.

   Where candidates lose it

   Writing the code correctly and having nothing to say about why you would use one. At a trading firm the interesting half is the memory and cache discussion, and the order book cancel case is the one concrete example where a linked list is genuinely the right structure. Also do not forget the dummy head trick, it removes most of the bugs.

   Expect next

   • Reverse it in place.
   • Detect a cycle in constant space.
   • Why would a vector usually beat a list even when the complexity says otherwise?

   Reported by candidates at Jump Trading (Prop Trading, Remote, 2022). Source: Wall Street Oasis.

9. 085C++ or Python? Where does each belong in a quant stack?ProgrammingCorephone / first roundQuant researchQuant development

   Say this

   Both, in different places. Python for research, where iteration speed and the data-science ecosystem dominate. C++ for anything on the critical path, where you need deterministic microsecond latency and control over memory. The split is a question of which cost dominates, developer time or machine time.

   Then walk it

   1. Python's real advantage is not the language, it is pandas, numpy, scipy, statsmodels and scikit-learn plus notebooks. A research idea gets tested in an afternoon. The performance is acceptable because the heavy loops sit in vectorised C underneath.
   2. Python's disqualifying weakness for execution is non-determinism: garbage collection pauses, the global interpreter lock, and unpredictable allocation. A tail latency you cannot control is worse than a mean latency that is higher.
   3. C++ gives you no garbage collector, control of memory layout and cache behaviour, zero-cost abstractions, and access to kernel bypass networking. The cost is development speed and a large surface for undefined behaviour.
   4. How real stacks resolve it: C++ or Rust for the gateway, book building and order entry, Python for research, signal development and analysis, with the shared logic compiled once and bound into Python through pybind11 so research and production use the same code. That last point matters, because a research-production mismatch is a reliable source of live losses.
   5. And I would name the middle ground rather than pretend the choice is binary. Numba, Cython, JAX and polars cover a lot of ground where Python is too slow but full C++ is unjustified, and Rust is genuinely taking share on the systems side. The judgement I would offer is: write it in Python until you have measured that it is too slow, then move only the measured hot spot.

   Where candidates lose it

   Picking a side as a matter of taste. It is a judgement question about where each tool fits, and a candidate who says C++ is better shows they have only worked on one side of the stack. Mention the research-to-production consistency problem, because it is the practical issue this split creates and few candidates raise it.

   Expect next

   • How would you keep research and production code consistent?
   • What specifically makes Python unsuitable for the critical path?
   • Where would you use Rust?
10. 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?

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.