Introduction

Quantitative finance relies on a broad set of mathematical tools. The most important areas are linear algebra, statistics, and calculus, but probability theory and optimization are also critical. Below are tables summarizing the key concepts in each area.

Linear Algebra

Topic	Description	Applications in Quant Finance
Vectors	Ordered lists of numbers	Portfolio weights, returns
Matrices	Rectangular arrays of numbers	Covariance matrices, transformations
Eigenvalues	Scalars associated with matrix transformations	Principal component analysis, risk factors
Eigenvectors	Vectors associated with eigenvalues	Factor models, dimensionality reduction
Matrix Multiplication	Combining matrices and vectors	Portfolio calculations, model building
Inverse/Determinant	Matrix properties	Solving systems, optimization

Statistics

Topic	Description	Applications in Quant Finance
Mean	Average value	Expected returns, forecasts
Variance	Measure of dispersion	Risk measurement, volatility
Standard Deviation	Square root of variance	Risk, portfolio theory
Covariance	Measure of joint variability	Portfolio construction, risk analysis
Correlation	Standardized covariance	Diversification, asset selection
Regression	Modeling relationships	Factor models, alpha generation
Hypothesis Testing	Statistical inference	Model validation, backtesting
Probability Distributions	Likelihood of outcomes	Option pricing, risk modeling

Calculus

Topic	Description	Applications in Quant Finance
Differentiation	Rates of change, sensitivity analysis	Greeks, hedging, risk management
Integration	Accumulating quantities, area under curves	Expected value, pricing, risk aggregation
Partial Derivatives	Multivariable sensitivity	Portfolio optimization, multi-factor models
Optimization (Calculus-based)	Finding maxima/minima	Portfolio construction, risk-return tradeoff
Stochastic Calculus	Calculus for random processes	Black-Scholes, quantitative modeling

Other Key Areas

Topic	Description	Applications in Quant Finance
Probability Theory	Modeling uncertainty	Option pricing, risk management
Optimization	Maximizing/minimizing functions	Portfolio construction, trading strategies
Numerical Methods	Computational techniques	Simulation, pricing, calibration

Conclusion

Success in quant finance requires fluency in linear algebra, statistics, calculus, probability, and optimization. Mastering these areas will empower you to build robust models, manage risk, and innovate in financial markets.