Foundations
Quantitative finance is not a set of plug-and-play formulas. It is the applied intersection of rigorous mathematics, computational systems, and empirical market microstructure. When a trading strategy incurs catastrophic drawdowns, a derivative pricing engine misquotes, or an optimizer produces singular portfolios, the root cause is almost always a violation of fundamental mathematical or physical assumptions.
This Foundations provides the rigorous ground truth underlying all 8 operational quantitative disciplines. Use these nodes to master the prerequisite mathematics and diagnose whether a mathematical model has structural integrity or is fundamentally flawed.
Core Theoretical Pillars
- Linear Algebra & Matrices: Vector spaces, spectral theory, positive semi-definiteness, singular value decomposition (SVD), PCA, and random matrix theory.
- Calculus & Optimization: Single & multivariable calculus, gradients, Hessians, Taylor series (the foundation of Greeks), Lagrange multipliers, and KKT conditions.
- Probability & Measure Theory: Probability spaces , filtrations, conditional expectations, martingales, and the Radon–Nikodym derivative.
- Stochastic Calculus & Itô’s Lemma: Quadratic variation, continuous Brownian paths, Itô integration, Itô-Doeblin formula, and Girsanov change of measure.
- Statistics & Inference: Point estimation (MLE), the CLT, confidence intervals & hypothesis testing, bias-variance, bootstrap.
- Econometrics & Time Series: Weak vs strict stationarity, unit root tests (ADF), Engle–Granger and Johansen cointegration, and ARCH/GARCH volatility modeling.
- Bayesian Statistics: Bayes’ theorem & conjugate priors, posterior inference, regularization-as-MAP, and MCMC (Metropolis, Gibbs).
- Numerical Methods: Finite-difference methods, Monte Carlo, numerical optimization, numerical linear algebra - the general computational toolbox.
- Ergodicity & Statistical Mechanics: Ensemble averages vs time averages, non-ergodic multiplicative wealth dynamics, the Kelly criterion, and ruin probability.
Reading Path - the consumption order (read this before a pillar)
This is not a menu; it is a contract. Each pillar’s 01-from-zero-intuition pages assume the foundations listed here, and the pillar hubs name them in their “Before this pillar (foundations)” line. Read the core trio first, then the pillar-specific foundations before you enter that pillar.
Step 1 - the core trio (everyone, before any pillar): 1 · Linear Algebra → 2 · Calculus & Optimization → 3 · Probability & Measure Theory. These unlock everything else.
Step 2 - the pillar-specific foundations (read before entering that pillar):
| Before this pillar | Read first |
|---|---|
| 1 · Quantitative Research | Econometrics & Time Series (stationarity, unit roots, cointegration, GARCH) + Probability |
| 2 · Algorithmic & HFT | Probability + Numerical Methods (execution/impact numerics) |
| 3 · Derivative Pricing | Stochastic Calculus (Itô, Girsanov) + Probability |
| 4 · Quantitative Risk | Probability + Statistics & Inference (quantiles, estimation) |
| 5 · Portfolio Optimization | Linear Algebra (spectral, PSD) + Calculus (KKT) + Statistics; Ergodicity for Kelly |
| 6 · Market Making | Probability + Stochastic Calculus (SDEs for optimal quoting) + Econometrics |
| 7 · ML & Alt-Data | Statistics & Inference + Linear Algebra (PCA, regularization) |
| 8 · Quantitative Development | Numerical Methods + basic Python/C++ |
Step 3 - deeper / optional: Bayesian Statistics (the bridge into Pillar 5 and ML), and Ergodicity & Statistical Mechanics (multiplicative growth, Kelly sizing, ruin).
Exit ramp - now enter a pillar
You have the foundations; pick a pillar and walk its Reading Path:
- Build strategies? → Pillar 1 · Quantitative Research, start at Statistical Arbitrage & Pairs.
- The mathematics of pricing? → Pillar 3 · Derivative Pricing, start at Options Fundamentals & Markets.
- Understand risk? → Pillar 4 · Quantitative Risk, start at VaR & Expected Shortfall.
- Portfolio construction? → Pillar 5 · Portfolio Optimization, start at MPT & Mean–Variance.
When a topic in any pillar relies on linear projections or martingales, return here to rebuild the immutable mathematical intuition before touching production code.