Probability
Core topics include counting, conditional probability, Bayes’ rule, expectation, variance, covariance, common distributions, concentration, laws of large numbers, central-limit behavior, stochastic processes, and martingales. Trading interviews often emphasize intuitive probability and expected value.
Statistics
Estimation, sampling, regression, regularization, hypothesis testing, confidence intervals, resampling, cross-validation, time series, multiple testing, and causal limitations support research. Understand both formulas and assumptions.
Linear algebra
Vectors, matrices, projections, eigenvalues, singular-value decomposition, covariance matrices, and numerical conditioning appear in regression, factor models, dimensionality reduction, optimization, and machine learning.
Optimization
Unconstrained and constrained optimization, convexity, Lagrange multipliers, regularization, and numerical methods are relevant to portfolio construction, calibration, and model fitting. Real problems include unstable inputs and transaction costs.
Calculus and differential equations
Single- and multivariable calculus, Taylor expansion, integration, ordinary and partial differential equations, and numerical approximation are important for derivatives, optimization, and continuous models.
Stochastic calculus
Brownian motion, Itô’s lemma, stochastic differential equations, risk-neutral valuation, and change of measure are central to many derivatives-quant roles but less central to some trading or data roles.
Discrete mathematics and algorithms
Combinatorics, graphs, recursion, dynamic programming, data structures, complexity, and randomized algorithms support interviews and implementation.
Numerical methods
Floating-point behavior, root finding, interpolation, Monte Carlo, optimization, matrix methods, and error analysis determine whether a theoretical model can be computed reliably.
Role-weighted depth
A trader may need very strong probability and fast reasoning. A statistical researcher needs deeper inference and validation. A derivatives quant needs stochastic calculus and numerical methods. An engineer needs algorithms, systems, and performance. Fundamentals should be strong before advanced specialization.
Recommended learning order
Begin with algebra, functions, counting, probability, expectation, variance, and basic statistics. Add linear algebra and calculus, then regression, optimization, time series, algorithms, and numerical methods. Role-specific advanced subjects should sit on top of these foundations.
Application examples
Covariance and eigenvectors appear in factor and risk models. Optimization appears in portfolio weights and model calibration. Dynamic programming appears in algorithmic problems and some pricing methods. Monte Carlo appears in simulation, risk, and derivative valuation. Time-series analysis appears in forecasting and signal research.
Mathematical maturity
The important skill isn't only remembering a theorem. It is recognizing assumptions, constructing a derivation, producing a counterexample, and deciding whether an approximation is appropriate. Interview preparation should therefore mix proofs or derivations with calculations and empirical application.