How Quant Interviews Work
Quant interviews are unlike anything else in finance — or technology. You will not be asked to walk through your CV for 45 minutes. Instead, expect a series of increasingly difficult technical challenges designed to test how you think under pressure.
A typical quant interview process at a top firm looks like this:
- Online assessment — coding test, numerical reasoning, or both
- Phone screen — 30-60 minutes of probability, maths, or coding questions
- Superday / on-site — 4-6 hours of back-to-back interviews covering probability, coding, market knowledge, and fit
The questions below are drawn from real interview processes at firms including Jane Street, Citadel, Two Sigma, Optiver, IMC, Jump Trading, Goldman Sachs, and others.
Probability & Brain Teasers
These are the bread and butter of quant interviews. Every firm asks them. The goal is not just to get the right answer — interviewers want to see clear, structured thinking.
1. You roll two fair dice. What is the expected value of the maximum?
Answer: List the 36 outcomes. The maximum takes value k with probability (2k-1)/36 for k = 1,...,6.
E[max] = (1×1 + 2×3 + 3×5 + 4×7 + 5×9 + 6×11) / 36 = 161/36 ≈ 4.47
2. You flip a fair coin repeatedly. What is the expected number of flips to get two heads in a row?
Answer: Let E be the expected number from the start, and A be the expected number given the last flip was heads.
From start: E = 1 + (1/2)A + (1/2)E → after first flip, half the time we have a head (state A), half the time we restart.
From state A: A = 1 + (1/2)(0) + (1/2)E → next flip is heads (done, 0 more needed) or tails (restart).
Solving: A = 1 + E/2, and E = 1 + A/2 + E/2.
From the second equation: E/2 = 1 + A/2, so E = 2 + A = 2 + 1 + E/2 = 3 + E/2, giving E = 6.
3. Three ants are on the vertices of an equilateral triangle. Each picks a random direction and walks along an edge. What is the probability that no two ants collide?
Answer: Each ant has 2 choices (clockwise or anticlockwise), so there are 2³ = 8 equally likely outcomes. No collision occurs only if all three go clockwise or all three go anticlockwise. P(no collision) = 2/8 = 1/4.
4. You have a biased coin that lands heads with probability p. How do you use it to simulate a fair coin?
Answer: Von Neumann's trick. Flip twice: HT → call it "heads", TH → call it "tails", HH or TT → discard and repeat. P(HT) = p(1-p) = P(TH), so the outcomes are equally likely.
5. A stick is broken at two random points. What is the probability the three pieces form a triangle?
Answer: For a triangle, each piece must be less than half the total length. If breaks are at uniform points U and V on [0,1], the condition requires all three pieces < 1/2. The probability is 1/4.
6. What is the expected number of people you need to sample before finding two with the same birthday?
Answer: This is the birthday problem in expectation form. The exact answer requires summing the probability of no match up to each step. The expected number is approximately 24.6 (commonly rounded to about 25). The probability of a match exceeds 50% at 23 people.
7. You have 100 balls: 50 red and 50 blue. You distribute them between two boxes however you like. A box is chosen at random, then a ball drawn at random from that box. How do you maximise the probability of drawing a red ball?
Answer: Put 1 red ball in box A and all remaining (49 red + 50 blue) in box B. P(red) = (1/2)(1) + (1/2)(49/99) ≈ 0.747. This is optimal because box A guarantees red, and box B still has a reasonable red ratio.
8. I pick two numbers uniformly at random from [0, 1]. What is the probability their sum is greater than 1 and their product is less than 3/16?
Answer: The sum > 1 condition restricts us to the upper-right triangle of the unit square. Within that triangle, we need xy < 3/16. Parameterise and integrate. The product constraint gives a hyperbolic boundary. The answer requires computing the area of intersection — a good test of whether you can set up integrals under pressure.
9. You have a deck of 52 cards. You draw cards one at a time. I pay you £1 for each red card and you pay me £1 for each black card. You can stop at any time. What is the optimal strategy and expected value?
Answer: The expected value is 0 if you must draw all cards. But since you can stop, the value is positive. The optimal strategy uses dynamic programming. The expected value with optimal stopping is approximately £2.62.
10. In a room of n people, what is the expected number of distinct birthdays?
Answer: By linearity of expectation: E[distinct] = 365 × (1 - (364/365)ⁿ). Each day is "represented" with probability 1 - (364/365)ⁿ.
Mental Maths & Estimation
These test speed and accuracy with numbers. Practice is essential.
11. What is 17 × 23?
Answer: 17 × 23 = 17 × 20 + 17 × 3 = 340 + 51 = 391
12. Estimate √2 to two decimal places.
Answer: 1.41 (1.414... more precisely)
13. What is 1/7 as a decimal?
Answer: 0.142857 (repeating)
14. What is 85² ?
Answer: Use (80+5)² = 6400 + 800 + 25 = 7,225
15. Roughly how many piano tuners are there in London?
Answer: This is a Fermi estimation. London population ~9M, roughly 3M households, maybe 10% have pianos (300K pianos). Each piano tuned once a year, each tuning takes ~2 hours including travel. A tuner works ~250 days × 4 tunings/day = 1,000 tunings/year. So 300K / 1,000 ≈ 300 piano tuners. The exact number matters less than the structured reasoning.
Coding Questions
Typically asked in Python or C++. You will code on a whiteboard or shared editor.
16. Implement a function to compute the nth Fibonacci number in O(log n) time.
Answer: Use matrix exponentiation. [[F(n+1), F(n)], [F(n), F(n-1)]] = [[1,1],[1,0]]ⁿ. Compute the matrix power using repeated squaring.
17. Given a time series of stock prices, find the maximum profit from a single buy-sell pair.
Answer: Single pass: track the minimum price seen so far and the maximum profit achievable.
def max_profit(prices): min_price = float('inf') max_profit = 0 for price in prices: min_price = min(min_price, price) max_profit = max(max_profit, price - min_price) return max_profit
18. Implement a Monte Carlo simulation to estimate π.
Answer: Generate random (x, y) pairs in [0,1]². Count the fraction that fall inside the quarter circle (x² + y² ≤ 1). Multiply by 4.
import random def estimate_pi(n_samples=1000000): inside = sum(1 for _ in range(n_samples) if random.random()**2 + random.random()**2 <= 1) return 4 * inside / n_samples
19. Write a function to price a European call option using the Black-Scholes formula.
Answer:
from math import log, sqrt, exp from scipy.stats import norm def black_scholes_call(S, K, T, r, sigma): d1 = (log(S/K) + (r + sigma**2/2)*T) / (sigma*sqrt(T)) d2 = d1 - sigma*sqrt(T) return S*norm.cdf(d1) - K*exp(-r*T)*norm.cdf(d2)
20. Given an array of integers, find two numbers that sum to a target value. What is the time complexity?
Answer: Use a hash set. O(n) time, O(n) space.
def two_sum(nums, target): seen = {} for i, num in enumerate(nums): complement = target - num if complement in seen: return [seen[complement], i] seen[num] = i
Market & Finance Questions
21. Explain put-call parity.
Answer: For European options: C - P = S - K×e^(-rT). A long call and short put with the same strike and expiry replicates a forward position. This is a no-arbitrage relationship.
22. What are the Greeks and why do they matter?
Answer: Delta (sensitivity to underlying price), Gamma (rate of change of delta), Theta (time decay), Vega (sensitivity to volatility), Rho (sensitivity to interest rates). They are essential for hedging and risk management of options portfolios.
23. How would you hedge a portfolio of call options?
Answer: Delta-hedge by shorting delta × number of shares of the underlying. Re-hedge as delta changes (gamma risk). Consider vega exposure if implied volatility moves. Dynamic hedging is required because delta changes with the underlying price.
24. What happens to option prices when volatility increases?
Answer: Both call and put values increase (higher vega). This is because higher volatility increases the probability of the option finishing deep in the money, while the downside is still capped at zero (for the option holder).
25. Explain what Value at Risk (VaR) is and its limitations.
Answer: VaR is the maximum expected loss over a time horizon at a given confidence level. For example, "1-day 95% VaR of £1M" means there is a 5% chance of losing more than £1M in a day. Limitations: it says nothing about the magnitude of losses beyond the threshold (use CVaR for that), assumes historical patterns persist, and can be gamed by concentrating tail risk.
Statistics & Machine Learning
26. What is the difference between correlation and causation? Give a finance example.
Answer: Correlation measures linear association; causation implies one variable drives the other. Example: two stocks might be correlated because they are in the same sector (common cause), not because one stock's price movement causes the other's.
27. Explain overfitting in the context of backtesting a trading strategy.
Answer: Overfitting occurs when a strategy is optimised to perform well on historical data but fails on new data. In backtesting, this happens when you have too many parameters relative to the amount of data, or when you test many strategies and select the best performer (data snooping). Out-of-sample testing and cross-validation help detect it.
28. What is the bias-variance trade-off?
Answer: Bias is error from overly simplistic assumptions (underfitting). Variance is error from sensitivity to small fluctuations in training data (overfitting). The optimal model balances both. In finance, high-variance models are particularly dangerous because financial data is noisy and non-stationary.
29. How would you detect if a time series is stationary?
Answer: Use the Augmented Dickey-Fuller (ADF) test or KPSS test. Visual inspection of the ACF/PACF plots. Check if mean and variance appear constant over time. Many financial time series (stock prices) are non-stationary, but returns are typically stationary.
30. Explain the Central Limit Theorem and why it matters in finance.
Answer: The CLT states that the sum (or average) of a large number of independent, identically distributed random variables is approximately normally distributed, regardless of the underlying distribution. In finance, it justifies using normal distributions for portfolio returns (which are sums of individual asset returns), underpins VaR calculations, and is the foundation for many statistical tests.
Behavioural & Fit Questions
31. Why do you want to work in quantitative finance?
Tip: Be specific. Connect your quantitative interests to the firm's work. Avoid generic answers about "loving maths." Talk about specific problems that excite you — signal research, pricing model complexity, low-latency optimisation.
32. Tell me about a time you found an error in your own analysis.
Tip: Show intellectual honesty and rigour. Describe the error, how you discovered it, what you did to fix it, and what process you put in place to prevent similar errors.
33. How do you handle ambiguity?
Tip: Give a concrete example where you made progress on a problem without clear direction. Quant work is inherently ambiguous — firms want people who can make structured progress without being told exactly what to do.
34. Describe a complex technical concept to me as if I were non-technical.
Tip: This tests communication skills. Pick something from your experience and explain it with analogies, not jargon. Interviewers assess whether you can collaborate with traders, PMs, and other non-quant stakeholders.
35. What is your biggest weakness?
Tip: Be genuine but strategic. A good answer: "I sometimes spend too long optimising code before validating the approach. I have learned to prototype quickly and iterate." Avoid cliché non-answers.
Advanced Questions (Senior Roles)
36. Derive the Black-Scholes equation from first principles.
Answer: Start with geometric Brownian motion for the asset: dS = μS dt + σS dW. Construct a self-financing portfolio of the option and the underlying that eliminates the stochastic term (delta-hedging). Apply Itô's lemma. The resulting PDE is the Black-Scholes equation.
37. How would you model the volatility smile?
Answer: The vanilla Black-Scholes model produces flat implied volatility across strikes. The observed smile/skew can be modelled with local volatility (Dupire), stochastic volatility (Heston, SABR), or jump-diffusion models (Merton). Each has trade-offs between calibration quality, tractability, and dynamic hedging behaviour.
38. Explain the difference between risk-neutral and real-world probability measures.
Answer: The real-world (physical) measure P describes actual market dynamics. The risk-neutral measure Q is a mathematical construction where all assets earn the risk-free rate in expectation. We price derivatives under Q because it gives the unique no-arbitrage price. The change of measure is captured by the Radon-Nikodym derivative (Girsanov's theorem).
39. How would you build a pairs trading strategy?
Answer: Identify co-integrated pairs (not just correlated). Estimate the spread and its mean-reverting dynamics. Enter positions when the spread deviates beyond a threshold and exit when it reverts. Key challenges: pairs can break down (structural breaks), transaction costs erode profits, and the strategy has become crowded.
40. Design a system for real-time risk monitoring of a trading book.
Answer: Key components: real-time position feed, market data streaming, risk engine (computing Greeks, VaR, stress scenarios), alerting system for limit breaches, and a dashboard. Latency requirements depend on the trading style. Need to handle instrument pricing across multiple asset classes, correlation matrices, and scenario generation.
Quick-Fire Practice (Try These Timed)
Give yourself 30 seconds per question:
- What is 13 × 19? (247)
- If you roll 3 dice, what is the probability they all show different numbers? (120/216 = 5/9)
- What is e^0.1 approximately? (≈ 1.105)
- A stock is at 100. It goes up 10% then down 10%. What is the price? (99)
- What is ln(2) approximately? (≈ 0.693)
- If a fair die is rolled, what is the expected number of rolls to see all 6 faces? (14.7)
- What is 23/7 as a decimal to 2 places? (3.29)
- A jar has 3 red and 5 blue balls. You draw 2 without replacement. P(both red)? (3/28)
- What is the derivative of x^x? (x^x (ln(x) + 1))
- In a 5-card poker hand, how many ways can you get a flush? (5,148)
How to Prepare
- Practice daily — spend 30 minutes on brain teasers and mental maths every day for at least 4 weeks before interviews
- Time yourself — interviewers expect quick responses, not leisurely derivations
- Think out loud — the process matters as much as the answer
- Study the fundamentals — our courses on probability and statistics cover the mathematical foundations you will be tested on
- Do mock interviews — practise with a friend or use our interview question generator for unlimited practice questions
- Download our cheatsheet — the quant interview prep cheatsheet is a concentrated reference for last-minute review
Frequently Asked Questions
How long should I prepare for quant interviews?
Minimum 4-6 weeks of daily practice if you already have a strong quantitative background. Career changers should allow 3-6 months of combined skill-building and interview preparation.
Do all quant firms ask brain teasers?
Most do, especially prop trading firms (Jane Street, Optiver, IMC, Citadel). Hedge funds and banks may lean more toward technical and market knowledge. But probability questions appear almost universally.
Should I memorise answers?
Memorise key formulas and results (like the birthday problem or coupon collector), but interviewers can tell if you are reciting rather than reasoning. Focus on understanding the solution approach so you can adapt to variations.
What if I get stuck in an interview?
Say so. "I am not sure of the answer, but here is how I would approach it..." then lay out your thinking. Partial credit is real in quant interviews. Silence is the worst possible response.
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