Quant Academy Math · Probability · Quant Finance

Curriculum Map

The full prerequisite-ordered sequence: 124 Full Lessons across 20 phases. Each phase builds on the ones before it; the common mathematical core (Phases 0–9) is strongly recommended for every pathway.

Prerequisite spine
0 Learning science → 1 Proof → 2–3 Calculus → 4–5 Linear algebra & numerics → 6 Analysis/measure → 7 Probability → 8 Stochastic processes → 9 Stochastic calculus → 10 Optimization → 11 Statistical learning → 12 Time series → 13 Monte Carlo → 14 Math finance → 15 Vol & derivatives → 16 Algo/HFT → 17 Quant dev → 18 Capstones.

Phase 0 - Learning How to Learn Quant Material

  1. 0.1 - Two Modes of Thought: Focused, Diffuse, and Chunking (Beginner, 35 min)
  2. 0.2 - The Core Engine: Retrieval, Spacing, and Interleaving (Beginner, 35 min)
  3. 0.3 - Deliberate Practice, Mental Representations, and Error Logs (Beginner, 35 min)
  4. 0.4 - An Evidence Classification System (and the Army Human-Performance Materials) (Beginner, 40 min)
  5. 0.5 - Notation & Formula Fluency: Fluent Forever for Mathematics (Beginner, 35 min)

Phase 1 - Mathematical Language & Proof

  1. 1.1 - Sets, Functions, and Relations (Beginner, 45 min)
  2. 1.2 - Logic, Quantifiers, and Necessary vs Sufficient Conditions (Beginner, 45 min)
  3. 1.3 - Proof Techniques: Direct, Contrapositive, and Contradiction (Beginner, 45 min)
  4. 1.4 - Mathematical Induction and Recursion (Beginner, 50 min)
  5. 1.5 - Sequences, Limits, and Reading & Writing Proofs (Intermediate, 55 min)

Phase 2 - Single-Variable Calculus

  1. 2.1 - Functions, Limits, and Continuity (Intermediate, 55 min)
  2. 2.2 - The Derivative: Definition and Meaning (Intermediate, 55 min)
  3. 2.3 - Differentiation Rules, Implicit Differentiation, and Related Rates (Intermediate, 55 min)
  4. 2.4 - Taylor Series and Local Approximation (Intermediate, 60 min)
  5. 2.5 - Optimization in One Variable (Intermediate, 55 min)
  6. 2.6 - Integration and the Fundamental Theorem of Calculus (Intermediate, 60 min)
  7. 2.7 - Sequences, Series, Convergence, and Power Series (Intermediate, 60 min)

Phase 3 - Multivariable Calculus

  1. 3.1 - Vectors, Multivariable Functions, and Level Sets (Intermediate, 45 min)
  2. 3.2 - Partial Derivatives, Gradient, Jacobian, and Hessian (Intermediate, 55 min)
  3. 3.3 - Multiple Integrals and Change of Variables (Intermediate, 55 min)
  4. 3.4 - Unconstrained Optimization and the Second-Order Test (Intermediate, 50 min)
  5. 3.5 - Constrained Optimization and Lagrange Multipliers (Advanced, 55 min)

Phase 4 - Linear Algebra

  1. 4.1 - Vector Spaces, Span, Independence, Basis, Dimension (Intermediate, 50 min)
  2. 4.2 - Linear Maps, Matrices, Rank, and the Null Space (Intermediate, 55 min)
  3. 4.3 - Determinants, Eigenvalues, and Eigenvectors (Intermediate, 55 min)
  4. 4.4 - Diagonalization and Similarity (Intermediate, 50 min)
  5. 4.5 - Inner-Product Spaces, Orthogonality, and Projections (Intermediate, 55 min)
  6. 4.6 - The Spectral Theorem, Symmetric and Positive-Definite Matrices, Quadratic Forms (Advanced, 60 min)
  7. 4.7 - The Singular Value Decomposition and Applications (PCA, Covariance) (Advanced, 60 min)

Phase 5 - Numerical Linear Algebra & Computing

  1. 5.1 - Floating-Point Arithmetic, Conditioning, and Stability (Intermediate, 50 min)
  2. 5.2 - LU Factorization and Solving Linear Systems (Intermediate, 50 min)
  3. 5.3 - QR Factorization and Least Squares (Intermediate, 55 min)
  4. 5.4 - Cholesky Factorization and Positive-Definite Systems (Intermediate, 55 min)
  5. 5.5 - Eigenvalue & SVD Algorithms and Iterative Methods (Advanced, 60 min)

Phase 6 - Real Analysis, Measure & Integration

  1. 6.1 - Metric Spaces, Sequences, Completeness, and Compactness (Advanced, 55 min)
  2. 6.2 - Measures and Sigma-Algebras (Advanced, 55 min)
  3. 6.3 - The Lebesgue Integral (Advanced, 55 min)
  4. 6.4 - Convergence Theorems: Monotone Convergence, Fatou, Dominated Convergence (Advanced, 55 min)
  5. 6.5 - Product Measures, Lᵖ Spaces, and the Bridge to Probability (Advanced, 60 min)

Phase 7 - Probability Theory

  1. 7.1 - Probability Spaces, Random Variables, and Distributions (Advanced, 50 min)
  2. 7.2 - Expectation, Moments, and Key Inequalities (Advanced, 50 min)
  3. 7.3 - Independence, Conditional Probability, and Conditional Expectation (Advanced, 55 min)
  4. 7.4 - Common Distributions and Transform Methods (MGF, Characteristic Functions) (Advanced, 50 min)
  5. 7.5 - The Laws of Large Numbers (Advanced, 45 min)
  6. 7.6 - The Central Limit Theorem and Modes of Convergence (Advanced, 50 min)
  7. 7.7 - Martingales and Stopping Times (Discrete Time) (Advanced, 55 min)

Phase 8 - Stochastic Processes & Brownian Motion

  1. 8.1 - Markov Chains (Advanced, 55 min)
  2. 8.2 - Random Walks, Filtrations, and Information (Advanced, 50 min)
  3. 8.3 - Brownian Motion: Construction and Defining Properties (Advanced, 55 min)
  4. 8.4 - Quadratic Variation and Path Properties of Brownian Motion (Advanced, 55 min)
  5. 8.5 - Gaussian Processes and the Foundations of Change of Measure (Research, 55 min)

Phase 9 - Stochastic Calculus

  1. 9.1 - The Itô Integral (Advanced, 55 min)
  2. 9.2 - Itô's Lemma (Advanced, 55 min)
  3. 9.3 - Stochastic Differential Equations and Geometric Brownian Motion (Advanced, 55 min)
  4. 9.4 - Girsanov's Theorem and Change of Measure (Advanced, 55 min)
  5. 9.5 - The Feynman–Kac Theorem and the PDE Connection (Advanced, 50 min)
  6. 9.6 - Risk-Neutral Pricing: Foundations (Advanced, 55 min)

Phase 10 - Convex Optimization & Numerical Methods

  1. 10.1 - Convex Sets and Convex Functions (Intermediate, 50 min)
  2. 10.2 - Convex Optimization Problems and Optimality Conditions (Intermediate, 50 min)
  3. 10.3 - Lagrangian Duality and the KKT Conditions (Advanced, 55 min)
  4. 10.4 - Algorithms: Gradient Descent and Newton's Method (Intermediate, 55 min)
  5. 10.5 - Applications: Portfolio Optimization and Regularization (Intermediate, 55 min)

Phase 11 - Statistics & Statistical Learning

  1. 11.1 - Statistical Inference, Estimation, and the Bias-Variance Decomposition (Intermediate, 50 min)
  2. 11.2 - Linear Regression: OLS, Geometry, Gauss-Markov, and Inference vs Prediction (Intermediate, 55 min)
  3. 11.3 - Model Assessment: Cross-Validation and Model Selection (Intermediate, 50 min)
  4. 11.4 - Regularization: Ridge and Lasso (Intermediate, 50 min)
  5. 11.5 - Classification: Logistic Regression and LDA (Intermediate, 50 min)
  6. 11.6 - Trees, Ensembles, Boosting, and Support Vector Machines (Intermediate, 55 min)
  7. 11.7 - Unsupervised Learning, Dimensionality Reduction, and Financial-Data Pitfalls (Advanced, 60 min)

Phase 12 - Time-Series Analysis

  1. 12.1 - Stationarity, Autocorrelation, and Partial Autocorrelation (Intermediate, 50 min)
  2. 12.2 - AR, MA, and ARMA Models (Intermediate, 55 min)
  3. 12.3 - ARIMA and Forecasting (Intermediate, 55 min)
  4. 12.4 - Volatility Models: ARCH and GARCH (Advanced, 55 min)
  5. 12.5 - State-Space Models and the Kalman Filter (Advanced, 55 min)
  6. 12.6 - Cointegration, VAR, and Backtest Pitfalls (Advanced, 60 min)

Phase 13 - Monte Carlo Methods

  1. 13.1 - Random-Number Generation and Sampling (Intermediate, 50 min)
  2. 13.2 - Monte Carlo Estimation and Error Analysis (Intermediate, 50 min)
  3. 13.3 - Variance Reduction: Antithetic, Control Variates, Stratification, Importance Sampling (Advanced, 55 min)
  4. 13.4 - Simulating Sample Paths and Discretization (Euler–Maruyama) (Advanced, 55 min)
  5. 13.5 - Monte Carlo Option Pricing, Greeks, and Quasi-Monte Carlo (Advanced, 60 min)

Phase 14 - Mathematical Finance

  1. 14.1 - No-Arbitrage and One-Period Pricing (Intermediate, 50 min)
  2. 14.2 - Replication and Risk-Neutral Valuation (Intermediate, 55 min)
  3. 14.3 - The Fundamental Theorems of Asset Pricing (Advanced, 55 min)
  4. 14.4 - The Black–Scholes PDE and Formula (Advanced, 60 min)
  5. 14.5 - Portfolio Theory, Utility, and Optimal Investment (Advanced, 60 min)

Phase 15 - Derivatives & Volatility

  1. 15.1 - Black–Scholes Assumptions, the Greeks, and Hedging (Advanced, 55 min)
  2. 15.2 - Implied Volatility and the Volatility Smile/Skew (Advanced, 50 min)
  3. 15.3 - Local Volatility and the Dupire Equation (Advanced, 55 min)
  4. 15.4 - Stochastic Volatility and the Heston Model (Advanced, 55 min)
  5. 15.5 - The Volatility Surface: Dynamics, Arbitrage Constraints, and Calibration (Advanced, 55 min)

Phase 16 - Algorithmic & High-Frequency Trading

  1. 16.1 - Market Microstructure and the Limit Order Book (Advanced, 50 min)
  2. 16.2 - Price Formation, the Bid–Ask Spread, and Liquidity (Advanced, 50 min)
  3. 16.3 - Market Impact and Transaction Costs (Advanced, 50 min)
  4. 16.4 - Optimal Execution (the Almgren–Chriss Framework) (Advanced, 60 min)
  5. 16.5 - Market Making, Inventory Risk, and Adverse Selection (Advanced, 55 min)
  6. 16.6 - Backtesting, Risk Controls, and Regulatory/Ethical Considerations (Advanced, 60 min)

Phase 17 - Quant Programming & Research Engineering

  1. 17.1 - Python for Quants: NumPy, pandas, and Vectorization (Intermediate, 55 min)
  2. 17.2 - Numerical Correctness and Testing (Intermediate, 55 min)
  3. 17.3 - Reproducibility, Version Control, and Experiment Tracking (Intermediate, 50 min)
  4. 17.4 - Software Design: DRY, Orthogonality, and Modularity (Intermediate, 55 min)
  5. 17.5 - Performance: Profiling, Vectorization, and Introductory C++ Concepts (Advanced, 55 min)
  6. 17.6 - Data Pipelines, SQL, and Research-to-Production (Intermediate, 55 min)

Phase 18 - Integrated Quant Capstones

  1. 18.1 - Capstone 1 - Mathematical and Numerical Foundations (Advanced, 120 min)
  2. 18.2 - Capstone 2 - Probability and Brownian Motion (Advanced, 120 min)
  3. 18.3 - Capstone 3 - Stochastic Calculus and Derivative Pricing (Advanced, 130 min)
  4. 18.4 - Capstone 4 - Volatility Modeling (Advanced, 130 min)
  5. 18.5 - Capstone 5 - Statistical-Learning Research Pipeline (Research, 140 min)
  6. 18.6 - Capstone 6 - Algorithmic Execution (Research, 130 min)
  7. 18.7 - Capstone 7 - Market-Making / HFT Simulation (Research, 140 min)
  8. 18.8 - Capstone 8 - Final Quant Research Project (Research, 8+ hrs)

Phase 19 - Computational Mathematics & Problem Solving

  1. 19.1 - Divisibility, GCD, and the Euclidean Algorithm (Beginner, 50 min)
  2. 19.2 - Primes, Sieves, and Integer Factorization (Intermediate, 60 min)
  3. 19.3 - Modular Arithmetic, Inverses, and Fast Exponentiation (Intermediate, 60 min)
  4. 19.4 - Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps (Intermediate, 55 min)
  5. 19.5 - Combinatorics: Counting, Binomials, and Inclusion–Exclusion (Intermediate, 60 min)
  6. 19.6 - Recurrence Relations and Generating Functions (Advanced, 65 min)
  7. 19.7 - Dynamic Programming: Memoization and Tabulation (Intermediate, 70 min)
  8. 19.8 - Bit Manipulation and State Compression (Advanced, 60 min)
  9. 19.9 - Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees (Intermediate, 70 min)
  10. 19.10 - Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle (Advanced, 70 min)
  11. 19.11 - Integer Partitions and Counting Structures (Advanced, 65 min)
  12. 19.12 - Continued Fractions, Pell Equations, and Diophantine Approximation (Advanced, 70 min)
  13. 19.13 - Matrix Exponentiation and Linear Recurrence Acceleration (Advanced, 60 min)
  14. 19.14 - Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct (Intermediate, 65 min)