The Maths a Quant Interview Actually Tests

What each role actually needs

Probability, which is the centre of everything

If time is limited, spend it here. Probability questions appear in every role at every firm, and fluency shows immediately.

• Conditional probability and Bayes, to the point where you reach for them without noticing.
• Expected value and variance, including the expected value of a game you are being offered rather than one you chose.
• The standard distributions and when each arises: binomial, Poisson, normal, exponential, uniform.
• Combinatorics and counting arguments, which is where most candidates lose time rather than accuracy.
• Random walks, gambler ruin and first-passage problems, which recur constantly in interview form.
• Markov chains and stationary distributions, at least to the level of solving a small chain by hand.
• Order statistics, and the expected maximum or minimum of a sample.
• The law of large numbers and the central limit theorem, stated correctly, including what they do not say.

Statistics, where research candidates are separated

Research interviews spend most of their time here, and the questions are less about computation than about whether you can be trusted with a result.

Know linear regression thoroughly: the assumptions, what breaks when each fails, multicollinearity, heteroskedasticity and what a standard error actually is. Understand hypothesis testing well enough to explain what a p-value does and does not mean, and to discuss multiple comparisons without prompting. Know the bias-variance trade-off, cross-validation, regularisation, and the specific problem with cross-validating time series data, which is the single most tested idea in quantitative research interviews.

Linear algebra and optimisation

Linear algebra is the language of portfolio construction and of most machine learning, so research candidates need it fluently: eigenvalues and eigenvectors, positive definiteness, covariance matrices and their estimation, principal components, and why a covariance matrix estimated from short samples is unreliable.

Optimisation appears as portfolio construction and as model fitting. Convexity and why it matters, Lagrange multipliers and constrained optimisation, gradient descent and its variants, and the mean-variance problem in closed form. Being able to derive the minimum-variance portfolio is a reasonable standard.

Stochastic calculus: who genuinely needs it

This is the topic candidates most often over-prepare. Derivatives pricing desks at banks need it properly: Brownian motion, Itô lemma, the Black-Scholes derivation, risk-neutral valuation, and the relationship between a partial differential equation and an expectation.

Systematic funds trading equities at daily horizons largely do not, and market-making interviews almost never touch it. Check the role before committing months to it. If you are applying to a derivatives desk, know it to the point of being able to derive results rather than quote them, because quoting is transparent.

Mental arithmetic, which is a trainable skill

Trading interviews test arithmetic at speed and candidates dismiss it as a gimmick. It is not: the desk requires rapid estimation, and the test is also measuring composure under pressure, which is harder to assess directly.

The skill responds quickly to daily practice in short sessions and decays just as quickly, so start about six weeks out rather than six months. Learn the mechanics rather than grinding: squares to about thirty, fraction to decimal conversions, percentage shortcuts, and multiplying two-digit numbers by decomposition. Estimation to two significant figures fast is more useful than exact answers slowly.

A realistic study plan

Three months is enough to move a strong quantitative undergraduate to interview-ready, and the ordering matters more than the hours.

• Weeks one to four: probability, until the standard problem shapes are automatic. Work problems rather than reading.
• Weeks five to seven: statistics and regression for research seats, or trading games and expected value for trading seats.
• Weeks six to twelve, in parallel: coding practice in one language, aiming for fluency rather than breadth.
• Final six weeks, daily: mental arithmetic in ten-minute sessions.
• Throughout: work out loud with another person at least weekly. The interview is partly a test of whether your reasoning is audible, and that cannot be practised alone.