Quantitative finance interview questions: the complete guide

In this article

1. Start here: the learning path
2. What quantitative finance interviews test
3. How do you find an expected value?
4. How do you size a bet?
5. How do you answer a market-making question?
6. The topics in detail
7. Questions interviewers ask on quantitative finance
8. All 4 guides in Quant and brainteasers
9. Follow-up questions
10. Sources

Start here: the learning path

Study the topics in the order below, because each builds on an earlier one. Statistics and random walks feed in alongside, not after.

Quantitative finance

Quantitative finance is the use of mathematics, statistics and programming to price financial instruments, manage risk and design trading strategies, and its interviews test those tools through probability puzzles, estimation, market games and coding.

Expected value

Expected value is the probability-weighted average of a random quantity's possible outcomes, which is the long-run average result if the same bet were repeated many times.

Market maker

A market maker is a firm or trader that continuously quotes a price to buy and a higher price to sell the same asset, and earns the gap between the two while managing the risk of what it holds.

Kelly criterion

The Kelly criterion is a rule for sizing repeated bets that picks the stake fraction maximising the long-run growth rate of capital, so it depends on the odds and on your edge.

Text description of the diagram

• The path has five steps from counting to market making.
• Expected value comes before bet sizing because the stake rule uses it.
• Statistics and random walks run alongside the five steps.

1. Counting and probability. Count outcomes, then divide.
2. Expected value. Weight every outcome by its probability.
3. Bayes and updating. Revise a probability when evidence arrives.
4. Bet sizing. Choose how much to risk when you have an edge.
5. Market making. Quote two prices around a fair value.

Alongside all five: statistics (what a sample tells you) and random walks (how a quantity drifts step by step).

1. How to solve Markov chain and random walk questions
2. Statistics for quant interviews: a from-zero guide
3. Probability interview questions on dice, coins, conditioning and waiting times

What quantitative finance interviews test

Quantitative finance interviews test three things in the same hour: clean probability, fast arithmetic, and judgement about risk. The first two are drilled by puzzles with one right answer. The third is tested by open questions where the interviewer pushes back and watches whether you change your mind for a good reason or a bad one.

The roles behind these interviews are numerate ones. The U.S. Bureau of Labor Statistics (BLS) says employment of mathematicians and statisticians is projected to grow 10% from 2025 to 2035, much faster than average, with about 2,000 openings a year (BLS, last modified August 2026). That is a broad occupation, not quant trading specifically, so read it as a sign of demand for the underlying skills, not a count of quant jobs. Banking roles add finance technicals on top, which the investment banking guide covers.

Two habits matter in every round. Say your assumptions before you calculate, and check your answer against a quick sanity bound, such as "a probability must lie between 0 and 1". The interview loops guide shows where such rounds sit in a hiring process.

How do you find an expected value?

Expected value is found by listing every outcome, multiplying each by its probability, and adding the results. For a fair six-sided die that pays its face value, each face has probability one sixth, so the expected value is 3.5 ¹. You cannot roll 3.5, which is the point: it is a long-run average, not a likely outcome.

How do you count outcomes first?

Probability questions start with counting. Two fair dice give 36 equally likely ordered outcomes, and exactly six of them sum to 7 (1 and 6, 2 and 5, 3 and 4, each in two orders), so the probability of a sum of 7 is 6 over 36, which is 1/6 ¹. The trap is counting 3 and 4 as one outcome when it is two: ordered pairs are equally likely, unordered ones are not.

How does Bayes' theorem change a probability?

Bayes' theorem updates a probability after evidence, and the base rate decides how much the evidence is worth. In words: the updated probability equals the chance of the evidence if the hypothesis is true, times the base rate, divided by the overall chance of seeing that evidence. In this illustrative case, 1% of people have a condition, a test catches 90% of them, and it wrongly flags 9% of everyone else. Out of 10,000 people, 100 have the condition and 90 of them test positive, while 891 of the 9,900 without it also test positive. Only 90 of the 981 positives are real, a posterior (the updated probability after seeing the evidence) of 10/109 or 9.17% ¹.

The natural-frequency telling, counting people, is the one to say aloud. Mandel's 2014 editorial reports that Bayesian answers to a classic mammography problem more than doubled when it was posed as natural frequencies, and that most physicians in Eddy's 1982 study gave 0.70 to 0.80 where the Bayesian answer is about 0.078 (Mandel, October 2014).

How do you size a bet?

A bet is sized by the Kelly criterion when you want the fastest long-run growth of capital. J. L. Kelly Jr. published "A New Interpretation of Information Rate" in the Bell System Technical Journal in July 1956, framing information rate as the exponential growth rate of a gambler's capital (Kelly, July 1956). For a bet that pays even money and wins with probability p, the stake fraction is 2p minus 1.

With p equal to 0.6, the Kelly fraction is 0.2: risk a fifth of capital each time. The quantity Kelly maximises is the long-run (compound) growth rate per bet, the expected log of the wealth multiple, which is 2.014% at a 20% stake, against 1.504% at a 10% stake and 1.475% at 30% ¹. At 40%, double the Kelly stake, that rate is minus 0.245%.

Average wealth still rises 8% per bet at 40% (the average multiple is 1.02, 1.04, 1.06 and 1.08 at stakes of 10, 20, 30 and 40%) ¹, but the typical path shrinks, because a loss of 40% of capital needs a gain of two thirds to repair, so capital heads toward zero over many bets while a few lucky paths carry the average. Interviewers often probe this, because it separates "I have an edge" from "I can size the edge".

Text description of the diagram

• The long-run growth rate (expected log of the wealth multiple) is 1.504% per bet at a 10% stake and 2.014% at 20%.
• It is 1.475% at a 30% stake and minus 0.245% at 40%, although average wealth still rises 8% per bet at 40%.

The inputs are illustrative, not a market. Real edges are estimated with error, so practitioners often stake a fraction of the Kelly amount. Half Kelly (a 10% stake) keeps 74.7% of the peak growth rate, with a much smaller stake at risk ¹. Over-betting is the costlier error: if the true win probability is 0.55, the true Kelly fraction is 0.1, so the 20% stake is double the true Kelly and its growth rate is minus 0.014% per bet ¹. The line chart shows the curve: it climbs to the peak and falls away faster on the right.

How do you answer a market-making question?

A market-making question asks you to quote a bid (the price you will pay) and an offer (the price you will sell at, also called the ask) on something uncertain, then trade against the interviewer. The rule is to find a fair value (your best estimate of what the thing is worth on average) first, then quote around it. For the die, the fair value is 3.5. A quote of 3 to buy and 4 to sell earns an edge (expected profit per trade) of 0.5 whichever side is hit, because you buy at 3 something worth 3.5 or sell at 4 something worth 3.5 ¹.

Two things then matter. A narrower quote wins more trades but earns less on each, and a trade tells you something: if the interviewer keeps buying from you, ask whether they know more than you do, and move your quote. This is the same updating as in the Bayes example, applied to a live conversation. Say your fair value, your width and why, out loud.

The topics in detail

Each topic below is explained in the words an interviewer would use, with the trap to avoid.

Markov and random-walk questions

A Markov process is one whose next step depends only on where it is now, not on how it got there. A random walk is a quantity that moves up or down by random steps, and the interview question is usually "what is the chance it reaches a boundary first". The classic form is the gambler's ruin: a player with a few units bets one unit at fair odds until reaching a target or going broke.

The answer is linear: starting at k units with a target of n, the chance of reaching n is k over n. The trap is applying it to a biased walk, where the answer is not linear. Describe the states, the transitions between them, and the one-line check that the probabilities in each row sum to one.

Statistics for quant interviews

Statistics questions ask what a sample tells you and how sure you can be. Expect the mean and variance, the central limit theorem (averages of many independent draws look bell-shaped), regression and its assumptions, and the trap of fitting noise.

The common probe is a backtest that looks great: try many strategies and some will look good by luck alone, so the honest follow-up is what you tested out of sample, on data the strategy never saw, and how many variants you tried before this one. A good answer names the number of trials, states the uncertainty as a range and says what result would change your mind. The Gaussian distribution guide explains the bell curve behind the central limit theorem.

Probability, expected-value and Bayes questions

Probability questions ask for the chance of an event or the average result of a game, and most come from a few classic families: coin flips, dice, cards, conditional probability and independence, plus expected-value games and the base-rate trap from the Bayes example above. Each answer starts with the count, then the fraction, then a sanity check. The common error is treating dependent events as independent, for example drawing cards without replacement. In games where you choose whether to roll again, which is expected value with a stopping rule, continue only if the average of what you could get beats what you hold.

Questions interviewers ask on quantitative finance

Questions fall into four families: probability, expected value and games, statistics, and markets. These are common ones, each with a one-line answer in our own words. For the arithmetic speed these rounds also test, the case math drills practise percentages and growth rates against the clock.

Probability

1. What is the chance two dice sum to 7? Six of 36 ordered outcomes, so 1/6.
2. A test is 90% accurate and a condition is rare; what does a positive mean? It depends on the base rate: count people, not percentages.
3. What does independent mean? One event's outcome does not change the other's probability.

Expected value and games

1. What is the expected value of a die roll? Add each face times one sixth, which gives 3.5.
2. Would you pay 4 to play a game that pays the face of a die? No: the average payout is 3.5, so you expect to lose 0.5 per play.
3. How much should you bet with an edge? Use the Kelly fraction as a ceiling and say that estimation error argues for less.

Statistics

1. What does a standard deviation measure? The typical distance of values from their mean.
2. Why can a backtest mislead? Trying many strategies makes it likely that some look good by chance.

Markets

1. What is a bid-ask spread? The gap between the price to sell to a market maker and the price to buy from one.
2. How do you quote a market? Estimate fair value, add a width for risk and information, and move after each trade.
3. Why might you widen a quote? More uncertainty, or a counterparty who may know more.
4. Tell me about a time you were wrong with numbers. Use a real, bounded example: what you assumed, how you found out, what changed. The behavioural interview guide shows the structure.

For estimation-style prompts, break the quantity into factors you can estimate, state each assumption, and multiply.

Follow-up questions

How would you check that answer?

Compare it with a quick bound. A probability must lie between 0 and 1, an expected value must lie between the smallest and largest outcome, and a die average of 3.5 sits between 1 and 6. Then redo the count a second way, for example by counting the outcomes you do not want and subtracting their probability from one.

What if the die were loaded?

Then the fair value changes, because expected value uses the true probabilities. If high faces come up more often than one time in six, the average rises above 3.5: selling at 4 earns less on each fill, and loses money once the average passes 4. Estimate the new probabilities, recompute the weighted average, and move the quote.

What is a market-making question in an interview?

A market-making question asks you to quote a buy price and a sell price on an uncertain quantity, then trade with the interviewer. They are checking that you set a fair value, choose a width, and update after each trade.

Why does betting more than the Kelly fraction hurt?

Average wealth can keep rising while the typical path falls. In the illustrative bet, a 40% stake still raises average wealth by 8% per bet, yet the long-run (compound) growth rate is minus 0.245% per bet, because a loss on a large stake costs more than an equal-sized win adds back. Over many bets capital heads toward zero on the typical path.

Sources

1. A New Interpretation of Information Rate — Bell System Technical Journal (J. L. Kelly Jr.), 1 Jul 1956
2. Mathematicians and Statisticians, Occupational Outlook Handbook — U.S. Bureau of Labor Statistics, 27 Aug 2026
3. The psychology of Bayesian reasoning — Frontiers in Psychology (D. R. Mandel), 9 Oct 2014