Quant finance article · 6 October 2026

Quant Interview Questions and Answers (2026): 43 Real Problems Solved

Quant interview questions and answers: this guide collects 43 questions that come up again and again in quant researcher, quant trader, desk quant and market-risk interviews, each with a worked answer. They cover probability, statistics, stochastic calculus, options and Greeks, numerical methods and coding, and market risk. Read the answer, then close the page and re-derive it from scratch.

How to use it: spend about 20 minutes a day. Do one section at a time, speak your answer out loud, and always state your assumptions before you calculate. Interviewers mostly care about how you reason and how you handle a changed condition, not the final number.

1. Probability

Probability is the most reliable opener at trading firms. Expect short, fast questions with a follow-up that changes one assumption.

Q1. Expected rolls to get two consecutive sixes

Let $E$ be the expected number of rolls from scratch. You first need a six (expected 6 rolls). Then you roll once more: with probability $1/6$ you are done, otherwise you are back to the start.

$$E = 6 + 1 + \tfrac56 E \;\Rightarrow\; E = 42.$$

General form for $k$ consecutive successes with probability $p$: $E=\frac{1-p^k}{(1-p)p^k}$, which gives 42 for $p=1/6,k=2$.

Q2. Expected flips until HH vs until HT

HT takes 4 flips on average: about 2 to see the first head, then 2 to see a tail. HH takes 6, because a failure after the first head (a tail) sends you back to the start, while after HT you keep your progress.

If the two patterns race each other, each wins with probability 1/2: the flip after the first head decides it.

Q3. A test is 99% sensitive with a 5% false-positive rate. The disease affects 1% of people. You test positive. What is the chance you are ill?

Bayes: $P(D|+)=\frac{0.01\cdot0.99}{0.01\cdot0.99+0.99\cdot0.05}=\frac{0.0099}{0.0594}\approx 16.7\%$.

The point interviewers want: the prior matters. Most positives come from the large healthy group.

Q4. How many people until two share a birthday with probability above 50%?

$P(\text{no match})=\prod_{k=0}^{n-1}\left(1-\frac{k}{365}\right)$. For $n=23$ this is about 0.493, so the answer is 23. Quick estimate: $n(n-1)/2$ pairs each match with probability $1/365$, so you need about $365\ln2\approx253$ pairs.

Q5. Monty Hall: why should you switch?

Your first pick is right with probability $1/3$ and that never changes. The host, who knows where the car is and always opens a goat door, effectively hands you the other $2/3$. Switching wins with probability 2/3. The answer changes if the host picks randomly: then switching and staying are equal given that a goat was shown.

Q6. Break a stick at two random points. What is the probability the three pieces form a triangle?

Each piece must be shorter than $1/2$. Using the simplex of break points, the valid region is the middle triangle of the four, so the probability is 1/4.

Q7. Expected maximum and minimum of two independent Uniform(0,1) draws

$P(\max\le x)=x^2$, so $E[\max]=\int_0^1 x\cdot 2x\,dx=2/3$. By symmetry $E[\min]=1/3$. For $n$ draws, $E[\max]=\frac{n}{n+1}$.

Q8. Gambler's ruin: you start with $i$ units, bet one unit on a fair coin, and stop at 0 or $N$

The probability of reaching $N$ is $i/N$ (the wealth process is a martingale). The expected duration is $i(N-i)$. If the coin is biased, the formula changes to a ratio of geometric terms $\left(\frac qp\right)^i$.

Q9. Expected number of cards drawn until the first ace in a 52-card deck

The 4 aces split the other 48 cards into 5 gaps, each with the same expected length of $48/5$. The first ace therefore sits at position $48/5+1=\mathbf{53/5=10.6}$.

Q10. Two children: at least one is a boy. What is the chance both are boys?

Outcomes BB, BG, GB are equally likely, so the answer is 1/3. If you are told the older child is a boy, the answer is 1/2. The wording of the condition is the whole question.

2. Statistics and regression

Stats rounds test whether you understand estimators, not whether you can recite formulas.

Q11. Explain the bias-variance trade-off

Expected squared prediction error splits into $\text{bias}^2+\text{variance}+\text{noise}$. Flexible models reduce bias but fit noise (high variance); simple models do the opposite. Regularisation, cross-validation and more data are the levers. In finance, noise is large and the signal is small, so variance usually dominates.

Q12. What does multicollinearity do to OLS?

The estimates stay unbiased but their variance explodes, because $(X^\top X)^{-1}$ is nearly singular. Signs flip, standard errors balloon, but predictions can still be fine. Diagnose with VIF or condition number. Fix with dropping or combining variables, ridge regression, or PCA.

Q13. L1 versus L2 regularisation

L2 (ridge) shrinks coefficients smoothly and corresponds to a Gaussian prior. L1 (lasso) shrinks some coefficients exactly to zero, giving sparse models, and corresponds to a Laplace prior. Elastic net mixes the two and handles correlated predictors better than lasso alone.

Q14. What is stationarity and how do you test it?

A weakly stationary series has constant mean, constant variance and autocovariance that depends only on the lag. Prices are usually non-stationary (random-walk-like), returns are closer to stationary. Test with the Augmented Dickey-Fuller test (null: unit root) and KPSS (null: stationarity). Using regression on non-stationary series gives spurious results.

Q15. When does the Central Limit Theorem fail?

When the variance is infinite (for example a Cauchy distribution or very heavy-tailed returns), or when observations are strongly dependent. In both cases sample means do not converge to a normal at the usual $\sqrt n$ rate. Fat-tailed financial data is why you should be careful with normal-based confidence intervals.

Q16. You backtest 100 strategies and the best has a significant p-value of 0.03. Is it real?

Probably not. With 100 independent tests at the 5% level you expect about 5 false positives. Correct for multiple testing (Bonferroni, Holm, false discovery rate), or use a deflated Sharpe ratio that accounts for the number of trials. Hold out data that you never touched during research.

3. Stochastic calculus

Expect to be asked to derive, not recall. Start from the definition and show each step.

Q17. State Ito's lemma and use it to solve geometric Brownian motion

For $dX=\mu\,dt+\sigma\,dW$ and smooth $f(t,x)$: $df=\left(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\right)dt+\sigma f_x\,dW$.

For $dS=\mu S\,dt+\sigma S\,dW$ and $f=\ln S$: $d\ln S=\left(\mu-\tfrac12\sigma^2\right)dt+\sigma\,dW$, so $S_t=S_0\exp\left((\mu-\tfrac12\sigma^2)t+\sigma W_t\right)$ and $E[S_t]=S_0e^{\mu t}$. The $-\tfrac12\sigma^2$ is the Ito correction and the usual trap.

Q18. Compute $E[W_t^4]$

$W_t\sim N(0,t)$, so $E[W_t^4]=3t^2$ (the fourth moment of a normal is $3\sigma^4$). Related facts: $E[W_t^2]=t$ and $E[W_sW_t]=\min(s,t)$.

Q19. Is $W_t^3$ a martingale?

No. Ito gives $d(W^3)=3W^2\,dW+3W\,dt$, which has a drift term. The process $W_t^3-3\int_0^tW_s\,ds$ is a martingale.

Q20. What does the risk-neutral measure change, and why is it legitimate?

By Girsanov, switching measure replaces the real-world drift $\mu$ by the risk-free rate $r$ while leaving the volatility unchanged. Prices of traded assets discounted at $r$ become martingales. It works because a derivative can be replicated by dynamic hedging, so its price does not depend on investors' risk preferences.

Q21. Derive the Black-Scholes PDE

Build a delta-hedged portfolio $\Pi=V-\Delta S$ with $\Delta=V_S$. The random term cancels, so $\Pi$ earns the risk-free rate:

$$\frac{\partial V}{\partial t}+\tfrac12\sigma^2S^2\frac{\partial^2V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0.$$

Feynman-Kac says the same solution is the discounted risk-neutral expectation of the payoff.

Q22. What is the quadratic variation of Brownian motion?

$[W]_t=t$: the sum of squared increments over a partition converges to $t$ as the mesh shrinks. This is why $(dW)^2=dt$ and why Ito calculus has a second-order term, whereas smooth paths have zero quadratic variation.

Q23. What is the distribution of $\int_0^tW_s\,ds$?

It is Gaussian with mean 0 and variance $\int_0^t\int_0^t\min(s,u)\,ds\,du=\dfrac{t^3}{3}$.

4. Options, Greeks and volatility

These decide most pricing and trading interviews. Know the formulas and the intuition behind each.

Q24. State put-call parity and what breaks it

For European options without dividends, $C-P=S-Ke^{-rT}$. With a continuous dividend yield, replace $S$ by $Se^{-qT}$. It fails for American options (early exercise gives an inequality), with borrowing costs or hard-to-borrow stocks, and with transaction costs.

Q25. What is the delta of an at-the-money call?

$\Delta=N(d_1)$ with $d_1=\frac{(r+\sigma^2/2)T}{\sigma\sqrt T}$ at $S=K$. This is slightly above 0.5 (about 0.5 for short maturities, higher with longer maturities or higher rates). An at-the-money forward call has delta of about $N(\sigma\sqrt T/2)$.

Q26. How does a delta-hedged option make or lose money?

Over a small interval, the profit or loss is approximately $\tfrac12\Gamma S^2\left(\sigma_{\text{realised}}^2-\sigma_{\text{implied}}^2\right)\Delta t$. Long gamma earns when realised volatility exceeds implied and pays theta otherwise. This is the gamma-theta trade-off: $\Theta+\tfrac12\sigma^2S^2\Gamma\approx0$ for a delta-hedged position.

Q27. Where is vega highest, and how does it scale?

$\text{Vega}=S\,\phi(d_1)\sqrt T$, largest near at-the-money and growing with $\sqrt T$. Long-dated options carry far more vega than short-dated ones, while short-dated at-the-money options carry more gamma.

Q28. Why do equity indices have a volatility skew?

Investors pay up for downside protection, falling prices raise leverage and volatility (the leverage effect), and crashes are jumpy. Models that capture it: local volatility, stochastic volatility (Heston, SABR) with negative spot-vol correlation, and jump models. Skew matters for pricing barriers and digitals.

Q29. Price a digital (cash-or-nothing) call

In Black-Scholes it is $e^{-rT}N(d_2)$. More generally a digital is the limit of a tight call spread, so its price is $-\frac{\partial C}{\partial K}$ (times the payout and discounting built into $C$). With a skew this includes the slope of the smile, so using one flat volatility mis-prices it.

Q30. State Dupire's formula

$$\sigma_{loc}^2(K,T)=\frac{\partial_TC+(r-q)K\partial_KC+qC}{\tfrac12K^2\partial_{KK}C}.$$

It recovers the unique local volatility surface from a continuum of European option prices. In practice you need a smooth, arbitrage-free surface first, because $\partial_{KK}C$ in the denominator is the risk-neutral density and must be positive.

Q31. Rule of thumb: price an at-the-money option in your head

For $r=0$, an at-the-money call or put is worth about $0.4\,S\,\sigma\sqrt T$ (precisely $S\,[2N(\sigma\sqrt T/2)-1]\approx0.399\,S\sigma\sqrt T$). With $S=100$, $\sigma=20\%$, $T=1$ that is about 8.

5. Numerical methods and coding

Coding rounds test clean thinking under time pressure. Say your assumptions out loud.

Q32. How accurate is Monte Carlo and how do you improve it?

The standard error shrinks like $1/\sqrt N$, so ten times the accuracy needs a hundred times the paths. Variance reduction: antithetic variates, control variates (for example price a geometric Asian analytically), importance sampling, stratification and quasi-random sequences (Sobol).

Q33. Why can Newton-Raphson fail when computing implied volatility?

Vega is close to zero for deep in-the-money or out-of-the-money options and very short maturities, so Newton steps are huge and can overshoot into negative volatility. Use a bracketed method (Brent or bisection), or Newton with a safeguard and a good starting point. Also check that the price is above intrinsic value, otherwise no implied volatility exists.

Q34. Write a Black-Scholes call pricer in Python

from math import log, sqrt, exp
      from statistics import NormalDist

      def bs_call(S, K, T, r, sigma):
          N = NormalDist().cdf
          d1 = (log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * sqrt(T))
          d2 = d1 - sigma * sqrt(T)
          return S * N(d1) - K * exp(-r * T) * N(d2)

      print(round(bs_call(100, 100, 1.0, 0.05, 0.20), 4))  # 10.4506

Mention edge cases: $T\to0$ returns the payoff, $\sigma\to0$ returns the discounted forward intrinsic value, and validate inputs.

Q35. What is RAII and why does C++ quant code rely on it?

Resource Acquisition Is Initialisation ties a resource's lifetime to an object's scope: constructors acquire, destructors release. It makes memory and locks exception-safe without manual cleanup. Use std::unique_ptr for single ownership, std::shared_ptr only when ownership is truly shared, and avoid raw new/delete.

Q36. Why is NumPy faster than a Python loop?

Vectorised operations run in compiled C on contiguous memory, avoiding Python's per-element interpreter overhead and enabling SIMD and cache-friendly access. Pitfalls: temporary arrays use memory, and some problems with sequential dependence cannot be vectorised (use Numba, Cython or C++).

Q37. SQL: find the second-highest value in a column

SELECT MAX(price) AS second_highest
      FROM trades
      WHERE price < (SELECT MAX(price) FROM trades);

Mention the window-function alternative (DENSE_RANK() OVER (ORDER BY price DESC) = 2) and how each treats ties and an empty result.

6. Market risk

Banks and risk teams ask these to check you can connect the maths to regulation.

Q38. VaR versus expected shortfall

VaR is a quantile: the loss not exceeded with a given confidence. Expected shortfall (ES) is the average loss beyond that quantile. ES is a coherent risk measure (it is subadditive) and sees the tail shape, while VaR can penalise diversification and ignores losses beyond the threshold. That is why the Basel FRTB framework moved the internal model from 99% VaR to 97.5% ES. See our deeper VaR vs ES walkthrough.

Q39. How is a VaR model backtested?

Count exceptions (days where the loss exceeds VaR). For 99% one-day VaR over 250 days you expect about 2.5. Under the Basel traffic-light approach, 0-4 exceptions is green, 5-9 yellow and 10 or more red. Statistical tests: Kupiec (unconditional coverage) and Christoffersen (independence of exceptions). Details in our backtesting guide.

Q40. Can you scale one-day VaR to ten days with the square root of time?

Only under strong assumptions: independent, identically distributed, normal returns with zero drift. Volatility clustering, autocorrelation, fat tails and non-linear positions break it. Regulators moved to liquidity horizons inside FRTB precisely because a single scaling factor is too crude.

7. Brainteasers and mental maths

These are quick checks on structure and speed. Talk through your reasoning.

Q41. 100 doors, 100 passes: which doors are open at the end?

Door $n$ is toggled once per divisor of $n$. It ends open only if it has an odd number of divisors, which happens only for perfect squares. So 10 doors are open: 1, 4, 9, ..., 100.

Q42. Two eggs, a 100-storey building: minimise worst-case drops to find the critical floor

Drop the first egg from floors 14, 27, 39, ... with gaps shrinking by one each time, so that the worst case is always 14 drops. It solves $n(n+1)/2\ge100$, giving $n=14$.

Q43. Compute $37\times43$ in your head

Use $(a-b)(a+b)=a^2-b^2$ with $a=40,b=3$: $1600-9=\mathbf{1591}$. Look for symmetry around a round number before multiplying digit by digit.

Cheat sheet: 15 results to memorise

Result	Value
Expected rolls to two sixes in a row	42
Expected flips for HH / HT	6 / 4
People for a birthday match above 50%	23
Triangle from a broken stick	1/4
Expected max / min of two uniforms	2/3 and 1/3
Gambler's ruin (fair, start $i$, target $N$)	$i/N$ and duration $i(N-i)$
Cards until first ace	53/5 = 10.6
$E[W_t^4]$	$3t^2$
Variance of $\int_0^tW_s\,ds$	$t^3/3$
GBM log-drift	$\mu-\tfrac12\sigma^2$
ATM option value (r = 0)	$\approx0.4\,S\sigma\sqrt T$
Put-call parity	$C-P=S-Ke^{-rT}$
Digital call	$e^{-rT}N(d_2)$
Expected 99% VaR exceptions in 250 days	2.5
Two eggs, 100 floors	14 drops

How to answer in the room

Restate the problem in your own words and confirm any ambiguity (fair coin? independent draws? continuous or discrete?).
State your approach before computing: conditioning, symmetry, a martingale, a known distribution.
Compute step by step and say what you are doing. Silence reads as being stuck.
Sanity-check the result: limits, extreme cases, units, whether a probability lies between 0 and 1.
Offer the extension: what changes if the coin is biased, volatility is stochastic, or the sample is dependent. This is where strong candidates separate themselves.

When you do not know, say so, then show how you would find out. Interviewers can teach knowledge but they cannot teach honesty about the limits of it.

FAQ

How many quant interview questions should I practise?

Depth beats volume. Solving 150 to 200 problems properly, with derivations and re-derivation a week later, is more useful than skimming a thousand. Group them by topic and revisit the ones you missed.

Which topics matter most?

Probability and options theory appear in almost every process. Stochastic calculus matters for pricing and research roles, statistics for research and systematic trading, coding for desk and developer roles, and market-risk questions for risk and model-validation roles.

Do I need a PhD?

Not always. Many desk and risk quants come from master's programmes or engineering. What you do need is fluency in the topics above and the ability to explain them clearly.

How long should preparation take?

Most people need eight to twelve focused weeks. Use our free career diagnostic to find the gaps that matter for your target role and build a study order from them.

Where can I practise coding questions?

Use the free Python and C++ playground to run pricing and Monte Carlo code in the browser with no installation.

Keep practising with worked solutions

If you want a larger problem set with derivations, the Quant Interview Problem Book collects 1,000+ problems with full solutions across probability, statistics, derivatives, coding and risk. For role-specific preparation, browse the interview playbooks in the resource library, check market pay in the salary explorer, or read our complete quant career guide.

Want feedback on how you explain these out loud? Book a 1-on-1 session and practise with a working quant.

Continue with the quant interview guides or browse the Desk2Quant resource catalog.