Quant Finance Study Guide
114 free lessons covering the probability, statistics, pricing, and machine-learning foundations behind quantitative investing.
01
Foundations of Probability
- What is Probability? — The mathematical language for reasoning about uncertain outcomes.
- Theoretical vs Empirical Probability — Idealized models versus probabilities estimated from observed data.
- Three Views of Probability — Classical, frequentist, and Bayesian interpretations side by side.
- Sample Space and Events — All possible outcomes, and the subsets we actually care about.
- Axioms of Probability — Non-negativity, normalization, and additivity for disjoint events.
- Independence and Expectation — When events don't influence each other, and the long-run average outcome.
- Variance and Standard Deviation — How far outcomes typically fall from the mean.
- Covariance and Correlation — Whether two random variables move together, and by how much.
- Key Inequalities — Markov, Chebyshev, and Jensen — bounding probabilities without the full distribution.
02
Set Theory & Combinatorics
03
Conditional & Bayesian Probability
04
Random Variables & Distributions
- What is a Random Variable? — A function that maps outcomes to numbers we can do math with.
- Discrete vs Continuous — Countable outcomes versus a continuum of possible values.
- PDFs and CDFs — Densities and cumulative probabilities, and how to move between them.
- Expectation, Variance, and Moments — Summarizing a distribution with a few key numbers.
05
Discrete Distributions
06
Continuous Distributions
07
Limit Theorems
08
Frequentist Inference
- Confidence Intervals — A range of plausible parameter values, with a stated coverage rate.
- Hypothesis Testing — Testing claims about the world using sample data.
- p-values and Statistical Decisions — How surprising is the data under the null?
- Type I and Type II Errors — False alarms versus missed detections, and the trade-off between them.
- Power and Effect Size — Designing tests that can actually detect what you care about.
- Bootstrapping and Resampling — Approximating sampling distributions when theory is hard.
09
Advanced Probability Tools
- Law of the Unconscious Statistician — Computing expectations of functions of random variables.
- Moment Generating Functions — A transform that uniquely identifies a distribution and makes sums easy.
- Characteristic Functions — MGFs for distributions where moments may not exist.
- Markov Chains — Memoryless transitions: where you go next depends only on where you are.
- Stationary Distributions — Long-run behavior of Markov chains and how to compute it.
10
Bayesian Inference
- Bayesian Philosophy — Probability as degree of belief, not long-run frequency.
- Prior, Likelihood, Posterior — Combining initial belief with evidence to form an updated belief.
- Conjugate Priors — Pairs of distributions where the math stays in the family.
- MCMC and Modern Computation — Sampling from posteriors when closed-form solutions don't exist.
11
Regression Analysis
- Ordinary Least Squares — The line that minimizes squared residuals.
- Multiple Linear Regression — Many predictors, one response — and the interpretations get subtle.
- Regression Diagnostics — Residual plots, leverage, and the assumptions you should never trust blindly.
- Regularization — Ridge, Lasso, and Elastic Net — taming overfitting.
- Logistic and Generalized Linear Models — Regression when the response isn't normally distributed.
12
Multivariate Statistics
- Joint, Marginal, and Conditional — Three lenses on distributions over multiple variables.
- Multivariate Normal — The most useful multivariate distribution, end to end.
- Covariance Matrices — Encoding variance and pairwise relationships in one object.
- Correlation vs Causation — What correlations can and cannot tell you.
- Principal Component Analysis — Finding directions of maximum variance in high-dimensional data.
13
Stochastic Processes
- Random Walks — Discrete-time stochastic motion and its scaling limits.
- Poisson Processes — Counting random arrivals over continuous time.
- Brownian Motion — The continuous-time limit of a random walk and the workhorse of finance.
- Itô's Lemma — How to take derivatives of functions of stochastic processes.
- Martingales — Fair-game processes and the optional stopping theorem.
- Geometric Brownian Motion — The stochastic differential equation behind Black–Scholes.
14
Simulation & Approximation
- Monte Carlo Simulation — Estimating expectations by averaging random samples.
- Variance Reduction — Antithetic variates, control variates, and importance sampling.
- Bootstrapping for Finance — Resampling returns to estimate distributions and intervals.
- Quasi-Monte Carlo — Low-discrepancy sequences for faster integration in many dimensions.
15
Time Series
- Stationarity and Autocorrelation — When statistical properties hold across time, and when they don't.
- AR, MA, and ARIMA — Linear models for time-dependent data.
- GARCH and Volatility Clustering — Modeling the autocorrelation of squared returns.
- Cointegration and Pairs Trading — When non-stationary series move together in the long run.
- Kalman Filters — Recursive estimation in dynamic linear systems with noise.
16
Information Theory
- Shannon Entropy — Measuring the uncertainty inside a distribution.
- Kullback–Leibler Divergence — How different are two distributions, in nats?
- Mutual Information — Quantifying the information one variable carries about another.
- Maximum Entropy — Choosing the least-committal distribution consistent with constraints.
17
Linear Algebra
- Vectors, Norms, and Inner Products — The geometry of vectors and what 'distance' means.
- Matrix Operations — Multiplication, transposes, and matrix algebra essentials.
- Eigenvalues and Eigenvectors — Special directions a matrix only stretches, not rotates.
- Singular Value Decomposition — The most useful matrix factorization in applied math.
- Positive Definite Matrices — Why covariance matrices have a special structure.
- Numerical Stability — Conditioning, rank, and why naive code can silently lose precision.
18
Calculus & Optimization
- Multivariate Calculus — Gradients, Jacobians, and Hessians for functions of many variables.
- Lagrange Multipliers — Constrained optimization, the elegant way.
- Convex Optimization — Why convex problems are tractable and most others aren't.
- Gradient Descent and Variants — First-order methods for high-dimensional optimization.
- Stochastic Calculus Primer — Differentiating in the presence of Brownian noise.
19
Machine Learning Fundamentals
- Supervised vs Unsupervised — Learning from labels versus learning structure from data alone.
- Bias–Variance Trade-off — Why simple models underfit and complex models overfit.
- Cross-Validation — Estimating out-of-sample performance honestly.
- Tree-Based Methods — Decision trees, random forests, and gradient boosting.
- Support Vector Machines — Maximum-margin classifiers and the kernel trick.
- Clustering and Dimensionality Reduction — Finding groups and compact representations without labels.
- Classification Metrics — Precision, recall, ROC, and PR curves — and which to trust when.
20
Deep Learning
- Feedforward Networks — Layers, activations, and universal approximation in practice.
- Backpropagation — How gradients flow backward through composed functions.
- Optimizers and Schedules — SGD, Adam, learning-rate decay, and warmup.
- Regularization in DL — Dropout, weight decay, and early stopping.
- Architectures for Finance — MLPs for tabular data, RNNs and Transformers for sequences.
- Loss Functions — MSE, cross-entropy, Huber, and when to pick each.
21
Options Pricing
- Payoffs and Put–Call Parity — What options pay, and the no-arbitrage relationship that ties calls and puts.
- Risk-Neutral Valuation — Pricing derivatives by changing measure — and why it works.
- Binomial Trees — Discrete-time option pricing that converges to Black–Scholes.
- Black–Scholes — The closed-form European option price and the assumptions behind it.
- The Greeks — Delta, gamma, vega, theta, rho — sensitivities every trader watches.
- Volatility Smile and Surface — Why implied volatility isn't a single number.
- Exotic Options — Asian, barrier, lookback, and other path-dependent payoffs.
22
Portfolio Theory
- Mean–Variance Optimization — The Markowitz frontier and the math behind diversification.
- CAPM and Factor Models — Decomposing returns into systematic exposures.
- Sharpe, Sortino, and Information Ratio — Risk-adjusted performance measures and how they differ.
- Black–Litterman — Blending market-implied views with your own beliefs.
- Risk Parity — Allocating by risk contribution, not capital.
23
Trading & Risk Applications
- Value-at-Risk — A quantile-based summary of portfolio downside.
- Expected Shortfall — The average loss in the tail beyond VaR.
- Backtesting — Simulating a strategy on history without fooling yourself.
- Market Making Basics — Inventory, adverse selection, and quoting around fair value.
- Execution and Market Microstructure — Order books, slippage, and how prices actually form.
- Statistical Arbitrage — Mean-reverting baskets and the half-life of edge.