QuantLab

Interactive tools for hands-on probability, statistics, and financial modeling.

QuantLab
Linear Algebra
Advanced Statistics
Probability for Quants
Stochastic Calculus
Time Series Analysis
Machine Learning
The Two Views of a Vector
Vector Operations
The Dot Product, Norms, and Angles
Orthogonality
The Two Views of a Matrix
Matrix Operations
Matrix Multiplication
Special Matrices
Linear Combinations and Span
Linear Independence
Basis and Dimension
Vector Spaces and Subspaces
Framing the Problem: Ax=b
Gaussian Elimination
The Solutions to Ax=b
Reduced Row Echelon Form (RREF)
LU Decomposition
Column Space & Rank
The Null Space
Row Space & Left Null Space
The Fundamental Theorem of Linear Algebra
The Geometric Meaning of the Determinant
Calculation and Properties
Eigenvalues & Eigenvectors
The Characteristic Equation
Diagonalization (PDP⁻¹)
Applications of Eigen-analysis
The Spectral Theorem
The Cholesky Decomposition (LLᵀ)
The Inexact Problem: Why Ax=b Often Has No Solution
The Geometry of "Best Fit": Projections
The Algebraic Solution: The Normal Equations
The Problem with the Normal Equations
The Stable Solution: The Gram-Schmidt Process
The QR Decomposition
The Singular Value Decomposition (SVD)
Principal Component Analysis (PCA)
Advanced SVD Applications
The Covariance Matrix & Portfolio Risk
Portfolio Optimization & The Efficient Frontier
The Capital Asset Pricing Model (CAPM)
Arbitrage & The Fundamental Theorem of Asset Pricing
Markov Chains for State Transitions
Fixed Income (Bond) Mathematics
The Language of Possibility: Sets, Sample Spaces, and Events
The Rules of the Game: Kolmogorov's Axioms
Updating Beliefs with Conditional Probability
Law of Total Probability and Bayes' Theorem
From Outcomes to Numbers: PMF & CDF
The Center and The Spread: Expected Value & Variance
The Quant's Toolbox: Common Discrete Distributions
The Master Tool: Moment Generating Functions (MGFs)
The Continuous World: PDFs and Smooth CDFs
The Calculus of Center and Spread
The Continuous Toolbox
Masterclass on Continuous MGFs
Thinking in Multiple Dimensions: Joint Distributions
Slicing the Probability Landscape
Measuring Relationships: Covariance & Correlation
The Ultimate Separation: Statistical Independence
The King of Distributions: The Normal Curve
The Superpower of the Normal Distribution
The Multivariate Normal Distribution (MVN)
Performing Surgery on the MVN
Capstone 1: The MVN in Action: Portfolio Theory
The Law of the Average: The WLLN
The Universal Bell Curve: The Central Limit Theorem (CLT)
Capstone 2: The CLT in Action (Python Simulation)
Advanced Asymptotics: Slutsky's Theorem & The Delta Method
The Language of Inference: Parameter, Statistic, Estimator, Estimate
Judging Estimators: The Property of Unbiasedness
Efficiency and the Cramér-Rao Lower Bound (CRLB)
Consistency and Sufficiency
Method of Moments (MoM) Estimation
The Master Recipe: Maximum Likelihood Estimation (MLE)
Finding MLE Estimates via Optimization
General Construction of Confidence Intervals (CIs)
Applying the Recipe: CIs for Mean and Variance
The Verdict: p-values and Critical Regions
The Theory of the 'Best' Test: The Neyman-Pearson Lemma
The Generalized LRT and Wilks' Theorem
The Quest for the 'Best' Line: Simple Linear Regression (SLR)
The Full OLS Derivation (SLR)
The Performance Review: R-Squared and Residuals
Upgrading to a 3D World: Multiple Linear Regression (MLR)
The 'Master Formula' Derivation (MLR)
The Classical Linear Model (CLM) Assumptions
The Gauss-Markov Theorem and the BLUE Property
t-Tests for Individual Coefficients
F-Tests for Joint Hypotheses
Multicollinearity and the VIF
Heteroskedasticity: Detection and Correction
Autocorrelation: Detection and Consequences
Capstone: Building and Testing a Fama-French Factor Model
The Basics: Sample Spaces & Events
Combinatorics: The Art of Counting
Conditional Probability & Independence
Bayes' Theorem
Random Variables (Discrete & Continuous)
Expectation, Variance & Moments
Common Discrete Distributions
Joint, Marginal & Conditional Distributions
Covariance & Correlation
The Law of Large Numbers (LLN)
The Central Limit Theorem (CLT)
Transformations of Random Variables
Introduction to Information Theory
Introduction to Stochastic Processes & Stationarity
Discrete-Time Markov Chains
The Poisson Process
Random Walks & Brownian Motion
Sigma-Algebras & Probability Measures
The Lebesgue Integral & Rigorous Expectation
Martingales
Introduction to Itô Calculus
The ML Landscape: Supervised, Unsupervised & Reinforcement Learning
The Core Problem: The Bias-Variance Tradeoff
The Golden Rule of ML: Train, Validate, Test
The Data Preprocessing Toolkit: Why and How to Scale Your Features
Your First Predictive Model (Intuition): K-Nearest Neighbors (KNN)
Our First Continuous Model (Intuition): Simple Linear Regression
Our First Scoring System: Accuracy, Confusion Matrix, Precision, Recall, F1-Score
Measuring Regression Error: MSE and R-Squared
From Simple to Multiple Linear Regression: The Mathematics of Fitting a Plane
The Engine of Learning: Gradient Descent and Loss Function Optimization
Taming Overfitting: The Math of Ridge (L2) and Lasso (L1)
The Geometry of Feature Selection: L1 vs. L2
From Regression to Classification: The Logic of Logistic Regression
Evaluating Classifiers: Precision-Recall vs. ROC/AUC
The Assumptions of Linear Models
Capstone Project: Building a Credit Default Predictor
The Intuition of a Decision Tree: How a Tree Learns by Asking Questions
The Mathematics of a Split: Entropy and Gini Impurity
The Achilles' Heel of Trees: Why Single Trees Overfit (High Variance)
Taming the Tree: Pruning and Setting Hyperparameters (e.g., max_depth)
Polynomial Regression: Capturing Non-Linearity within a Linear Framework
Introduction to Support Vector Machines (SVMs): Finding the Optimal Margin
The Kernel Trick: How SVMs Map Data to Higher Dimensions
Comparing Models: When to Use a Linear Model vs. a Tree vs. an SVM
The Philosophy of Ensembles: Why a "Crowd" of Models is Wiser than One
Bagging (Bootstrap Aggregating): The Intuition Behind Random Forest
Random Forest: A Deep Dive - How Adding Randomness Reduces Variance
The Philosophy of Boosting: Learning from Mistakes Sequentially
Gradient Boosting Machines (GBM): How Trees Predict the Errors of Previous Trees
The Champion Model: XGBoost - Understanding the Innovations
Comparing Bagging vs. Boosting: A Bias-Variance Perspective
Capstone Project: Forecasting Volatility with Ensembles
The Goal of Unsupervised Learning: Clustering vs. Dimensionality Reduction
Clustering with K-Means: The Algorithm (Assign & Update Steps)
Challenges with K-Means: The Initialization Problem and the K-Means++ Solution
How Many Clusters? The Elbow Method and Silhouette Score
The Intuition of PCA (Principal Component Analysis): Finding the Directions of Maximum Variance
The Mathematics of PCA: Eigenvectors, Eigenvalues, and Explained Variance
Applications of PCA in Finance: Creating Index Factors and Denoising Correlation Matrices
Other Clustering Methods: An Overview of DBSCAN and Hierarchical Clustering
The Language of Time Series: Stationarity, Autocorrelation (ACF), and Partial Autocorrelation (PACF)
Testing for Stationarity: The Augmented Dickey-Fuller (ADF) Test
Classical Models I (The "ARIMA" Family): Autoregressive, Moving Average Models
Classical Models II (Volatility): The ARCH and GARCH Models for Volatility Forecasting
Feature Engineering for Time Series: Lags, Rolling Windows, and Date-Based Features
Using ML for Time Series: How to Frame a Forecasting Problem for XGBoost
The Perils of Backtesting: Look-Ahead Bias and Ensuring Walk-Forward Validation
Advanced Concept: Fractional Differentiation for Preserving Memory
The Neuron: From the Perceptron to Modern Activation Functions (ReLU, Sigmoid)
Building a Brain: The Multi-Layer Perceptron (MLP) and the Concept of Layers
How Neural Networks Learn: The Intuition of Backpropagation and Chain Rule
Optimizing the Brain: Understanding Optimizers (Adam, SGD) and Learning Rates
Fighting Overfitting in Neural Networks: Dropout and Early Stopping
The Universal Approximation Theorem: Why Neural Networks are so Powerful
Practical Session: Building Your First Neural Network in PyTorch/TensorFlow
Introduction to Convolutional Neural Networks (CNNs): For Image and Grid Data
The Problem of Memory: Why MLPs Fail on Sequence Data
Recurrent Neural Networks (RNNs): The Concept of a Hidden State
The Vanishing/Exploding Gradient Problem in RNNs
The Solution: Long Short-Term Memory (LSTM) & Gated Recurrent Units (GRU)
The Attention Mechanism: A New Way to Think About Sequence Importance
The Rise of the Transformer: The Architecture that Revolutionized NLP
Comparing Models: When to Use GARCH vs. XGBoost vs. LSTM for Forecasting
Capstone Project: Forecasting Market Volatility using an LSTM
From Text to Numbers: Classic Techniques (Bag-of-Words, TF-IDF)
Financial Sentiment Analysis: Building a Lexicon-Based Scorer
The Rise of Embeddings: The Intuition of Word2Vec
The State of the Art: Contextual Embeddings with BERT
NLP Tasks for Finance: Named Entity Recognition (NER) and Topic Modeling (LDA)
Information Extraction: Parsing Financial Reports and News Feeds
Integrating NLP Signals: How to Add Text-Based Features to a Trading Model
Capstone Project: Sentiment Analysis of Earnings Call Transcripts to Predict Stock Returns
Model Explainability & Interpretability: "Why did my model do that?" (SHAP & LIME)
Advanced Feature Engineering: De Prado's Meta-Labeling for Sizing Bets
Advanced Backtesting: The Dangers of Data Snooping and Multiple Testing
Reinforcement Learning for Trading: The Basics of Q-Learning and Policy Gradients
ML for Portfolio Optimization: Estimating Covariance Matrices with ML
Leveraging Alternative Data: An Overview of Satellite Imagery, GPS, and Web Scraped Data
AI Ethics & Regulation in Finance: Model Bias, Fairness, and Governance
Final Project: Design and Propose a Complete, Novel ML-based Trading Strategy
Key Concepts - Mean and Variance
Standard Deviation (The "Intuitive" Spread)
The Normal Distribution N(μ, σ²)
Calculus Review - Derivatives (The "Slope")
Calculus Review - Integrals (The "Sum")
The "Master Tool": The Taylor Expansion
The "Master Tool" (Part 2): The 2-Variable Taylor Expansion
The Rules of "Normal" Infinitesimals
Why Normal Calculus Fails
Brownian Motion (Wiener Process Wt)
The "Weird" Scaling Property
A Model for Stocks (Geometric Brownian Motion)
The Failure of Path Length
Quadratic Variation (The "Aha!" Moment)
The New Rules of Algebra
The Itô Integral
Itô's Lemma (Simple Case, for f(Wt))
Itô's Lemma (The Full Version, for f(t, Xt))
Understanding the Full Formula (Translating Math to Finance)
The "Magic Portfolio"
Eliminating Risk
Eliminating Drift μ
The "No-Free-Lunch" Argument & The Final PDE
The Solution (The Black-Scholes Formula)
Delta (Δ): The "Speed" of an Option
Gamma (Γ): The "Acceleration"
Vega (ν): The "Jiggle Risk"
Theta (Θ): The "Melting Ice Cube"
Rho (ρ): The "Interest Rate Risk"
The "Other Way" - Risk-Neutral Valuation
The "Computer Way" - Monte Carlo Methods
Fixing σ - Stochastic Volatility (Heston Model)
Fixing "No Jumps" - Jump-Diffusion (The Merton Model)
Fixing r - Stochastic Interest Rates (Vasicek & CIR)
Introduction to Time Series
Stationarity
ACF and PACF
Autoregressive (AR) Models
Moving Average (MA) Models
ARMA Models
ARIMA Models
The Box-Jenkins Methodology
Introduction to Volatility Modeling
ARCH Models
GARCH Models
GARCH Extensions (EGARCH, GJR-GARCH)
Capstone: Modeling S&P 500 Volatility
Vector Autoregression (VAR) Models
Cointegration and Error Correction Models (VECM)
State Space Models & Kalman Filters
Structural VAR (SVAR) Models
Capstone: A Pairs Trading Strategy
Feature Engineering for Time Series
Time Series Cross-Validation
Forecasting with Tree-Based Models (XGBoost)
Deep Learning for Time Series (RNNs & LSTMs)
Advanced Concepts (Meta-Labeling, Feature Importance)
Descriptive Statistics Explorer
The Central Limit Theorem (CLT)
Bayes' Theorem
Confidence Intervals
Z-Table Calculator
An Introduction to Hypothesis Testing
Hypothesis Testing Guide
Bernoulli Distribution
Binomial Distribution
Poisson Distribution
Geometric Distribution
Hypergeometric Distribution
Negative Binomial Distribution
Discrete Uniform Distribution
Multinomial Distribution
The Normal Distribution
Gamma Distribution
Beta Distribution
Exponential Distribution
The t-Distribution (Student's t)
The χ² (Chi-Squared) Distribution
The F-Distribution (Fisher-Snedecor)
Cauchy Distribution
Laplace Distribution
Weibull Distribution
Logistic Distribution
T-Test
Z-Test
ANOVA
F-Test
Pearson Correlation
Chi-Squared Test
Mann-Whitney U Test
Kruskal-Wallis Test
Wilcoxon Signed-Rank Test
Spearman's Rank Correlation
Friedman Test
Kolmogorov-Smirnov (K-S) Test
Monte Carlo Simulation
Efficient Frontier & Sharpe Ratio
Kalman Filters
Stochastic Calculus & Ito's Lemma
QuantLab
Linear Algebra
Advanced Statistics
Probability for Quants
Stochastic Calculus
Time Series Analysis
Machine Learning
The Two Views of a Vector
Vector Operations
The Dot Product, Norms, and Angles
Orthogonality
The Two Views of a Matrix
Matrix Operations
Matrix Multiplication
Special Matrices
Linear Combinations and Span
Linear Independence
Basis and Dimension
Vector Spaces and Subspaces
Framing the Problem: Ax=b
Gaussian Elimination
The Solutions to Ax=b
Reduced Row Echelon Form (RREF)
LU Decomposition
Column Space & Rank
The Null Space
Row Space & Left Null Space
The Fundamental Theorem of Linear Algebra
The Geometric Meaning of the Determinant
Calculation and Properties
Eigenvalues & Eigenvectors
The Characteristic Equation
Diagonalization (PDP⁻¹)
Applications of Eigen-analysis
The Spectral Theorem
The Cholesky Decomposition (LLᵀ)
The Inexact Problem: Why Ax=b Often Has No Solution
The Geometry of "Best Fit": Projections
The Algebraic Solution: The Normal Equations
The Problem with the Normal Equations
The Stable Solution: The Gram-Schmidt Process
The QR Decomposition
The Singular Value Decomposition (SVD)
Principal Component Analysis (PCA)
Advanced SVD Applications
The Covariance Matrix & Portfolio Risk
Portfolio Optimization & The Efficient Frontier
The Capital Asset Pricing Model (CAPM)
Arbitrage & The Fundamental Theorem of Asset Pricing
Markov Chains for State Transitions
Fixed Income (Bond) Mathematics
The Language of Possibility: Sets, Sample Spaces, and Events
The Rules of the Game: Kolmogorov's Axioms
Updating Beliefs with Conditional Probability
Law of Total Probability and Bayes' Theorem
From Outcomes to Numbers: PMF & CDF
The Center and The Spread: Expected Value & Variance
The Quant's Toolbox: Common Discrete Distributions
The Master Tool: Moment Generating Functions (MGFs)
The Continuous World: PDFs and Smooth CDFs
The Calculus of Center and Spread
The Continuous Toolbox
Masterclass on Continuous MGFs
Thinking in Multiple Dimensions: Joint Distributions
Slicing the Probability Landscape
Measuring Relationships: Covariance & Correlation
The Ultimate Separation: Statistical Independence
The King of Distributions: The Normal Curve
The Superpower of the Normal Distribution
The Multivariate Normal Distribution (MVN)
Performing Surgery on the MVN
Capstone 1: The MVN in Action: Portfolio Theory
The Law of the Average: The WLLN
The Universal Bell Curve: The Central Limit Theorem (CLT)
Capstone 2: The CLT in Action (Python Simulation)
Advanced Asymptotics: Slutsky's Theorem & The Delta Method
The Language of Inference: Parameter, Statistic, Estimator, Estimate
Judging Estimators: The Property of Unbiasedness
Efficiency and the Cramér-Rao Lower Bound (CRLB)
Consistency and Sufficiency
Method of Moments (MoM) Estimation
The Master Recipe: Maximum Likelihood Estimation (MLE)
Finding MLE Estimates via Optimization
General Construction of Confidence Intervals (CIs)
Applying the Recipe: CIs for Mean and Variance
The Verdict: p-values and Critical Regions
The Theory of the 'Best' Test: The Neyman-Pearson Lemma
The Generalized LRT and Wilks' Theorem
The Quest for the 'Best' Line: Simple Linear Regression (SLR)
The Full OLS Derivation (SLR)
The Performance Review: R-Squared and Residuals
Upgrading to a 3D World: Multiple Linear Regression (MLR)
The 'Master Formula' Derivation (MLR)
The Classical Linear Model (CLM) Assumptions
The Gauss-Markov Theorem and the BLUE Property
t-Tests for Individual Coefficients
F-Tests for Joint Hypotheses
Multicollinearity and the VIF
Heteroskedasticity: Detection and Correction
Autocorrelation: Detection and Consequences
Capstone: Building and Testing a Fama-French Factor Model
The Basics: Sample Spaces & Events
Combinatorics: The Art of Counting
Conditional Probability & Independence
Bayes' Theorem
Random Variables (Discrete & Continuous)
Expectation, Variance & Moments
Common Discrete Distributions
Joint, Marginal & Conditional Distributions
Covariance & Correlation
The Law of Large Numbers (LLN)
The Central Limit Theorem (CLT)
Transformations of Random Variables
Introduction to Information Theory
Introduction to Stochastic Processes & Stationarity
Discrete-Time Markov Chains
The Poisson Process
Random Walks & Brownian Motion
Sigma-Algebras & Probability Measures
The Lebesgue Integral & Rigorous Expectation
Martingales
Introduction to Itô Calculus
The ML Landscape: Supervised, Unsupervised & Reinforcement Learning
The Core Problem: The Bias-Variance Tradeoff
The Golden Rule of ML: Train, Validate, Test
The Data Preprocessing Toolkit: Why and How to Scale Your Features
Your First Predictive Model (Intuition): K-Nearest Neighbors (KNN)
Our First Continuous Model (Intuition): Simple Linear Regression
Our First Scoring System: Accuracy, Confusion Matrix, Precision, Recall, F1-Score
Measuring Regression Error: MSE and R-Squared
From Simple to Multiple Linear Regression: The Mathematics of Fitting a Plane
The Engine of Learning: Gradient Descent and Loss Function Optimization
Taming Overfitting: The Math of Ridge (L2) and Lasso (L1)
The Geometry of Feature Selection: L1 vs. L2
From Regression to Classification: The Logic of Logistic Regression
Evaluating Classifiers: Precision-Recall vs. ROC/AUC
The Assumptions of Linear Models
Capstone Project: Building a Credit Default Predictor
The Intuition of a Decision Tree: How a Tree Learns by Asking Questions
The Mathematics of a Split: Entropy and Gini Impurity
The Achilles' Heel of Trees: Why Single Trees Overfit (High Variance)
Taming the Tree: Pruning and Setting Hyperparameters (e.g., max_depth)
Polynomial Regression: Capturing Non-Linearity within a Linear Framework
Introduction to Support Vector Machines (SVMs): Finding the Optimal Margin
The Kernel Trick: How SVMs Map Data to Higher Dimensions
Comparing Models: When to Use a Linear Model vs. a Tree vs. an SVM
The Philosophy of Ensembles: Why a "Crowd" of Models is Wiser than One
Bagging (Bootstrap Aggregating): The Intuition Behind Random Forest
Random Forest: A Deep Dive - How Adding Randomness Reduces Variance
The Philosophy of Boosting: Learning from Mistakes Sequentially
Gradient Boosting Machines (GBM): How Trees Predict the Errors of Previous Trees
The Champion Model: XGBoost - Understanding the Innovations
Comparing Bagging vs. Boosting: A Bias-Variance Perspective
Capstone Project: Forecasting Volatility with Ensembles
The Goal of Unsupervised Learning: Clustering vs. Dimensionality Reduction
Clustering with K-Means: The Algorithm (Assign & Update Steps)
Challenges with K-Means: The Initialization Problem and the K-Means++ Solution
How Many Clusters? The Elbow Method and Silhouette Score
The Intuition of PCA (Principal Component Analysis): Finding the Directions of Maximum Variance
The Mathematics of PCA: Eigenvectors, Eigenvalues, and Explained Variance
Applications of PCA in Finance: Creating Index Factors and Denoising Correlation Matrices
Other Clustering Methods: An Overview of DBSCAN and Hierarchical Clustering
The Language of Time Series: Stationarity, Autocorrelation (ACF), and Partial Autocorrelation (PACF)
Testing for Stationarity: The Augmented Dickey-Fuller (ADF) Test
Classical Models I (The "ARIMA" Family): Autoregressive, Moving Average Models
Classical Models II (Volatility): The ARCH and GARCH Models for Volatility Forecasting
Feature Engineering for Time Series: Lags, Rolling Windows, and Date-Based Features
Using ML for Time Series: How to Frame a Forecasting Problem for XGBoost
The Perils of Backtesting: Look-Ahead Bias and Ensuring Walk-Forward Validation
Advanced Concept: Fractional Differentiation for Preserving Memory
The Neuron: From the Perceptron to Modern Activation Functions (ReLU, Sigmoid)
Building a Brain: The Multi-Layer Perceptron (MLP) and the Concept of Layers
How Neural Networks Learn: The Intuition of Backpropagation and Chain Rule
Optimizing the Brain: Understanding Optimizers (Adam, SGD) and Learning Rates
Fighting Overfitting in Neural Networks: Dropout and Early Stopping
The Universal Approximation Theorem: Why Neural Networks are so Powerful
Practical Session: Building Your First Neural Network in PyTorch/TensorFlow
Introduction to Convolutional Neural Networks (CNNs): For Image and Grid Data
The Problem of Memory: Why MLPs Fail on Sequence Data
Recurrent Neural Networks (RNNs): The Concept of a Hidden State
The Vanishing/Exploding Gradient Problem in RNNs
The Solution: Long Short-Term Memory (LSTM) & Gated Recurrent Units (GRU)
The Attention Mechanism: A New Way to Think About Sequence Importance
The Rise of the Transformer: The Architecture that Revolutionized NLP
Comparing Models: When to Use GARCH vs. XGBoost vs. LSTM for Forecasting
Capstone Project: Forecasting Market Volatility using an LSTM
From Text to Numbers: Classic Techniques (Bag-of-Words, TF-IDF)
Financial Sentiment Analysis: Building a Lexicon-Based Scorer
The Rise of Embeddings: The Intuition of Word2Vec
The State of the Art: Contextual Embeddings with BERT
NLP Tasks for Finance: Named Entity Recognition (NER) and Topic Modeling (LDA)
Information Extraction: Parsing Financial Reports and News Feeds
Integrating NLP Signals: How to Add Text-Based Features to a Trading Model
Capstone Project: Sentiment Analysis of Earnings Call Transcripts to Predict Stock Returns
Model Explainability & Interpretability: "Why did my model do that?" (SHAP & LIME)
Advanced Feature Engineering: De Prado's Meta-Labeling for Sizing Bets
Advanced Backtesting: The Dangers of Data Snooping and Multiple Testing
Reinforcement Learning for Trading: The Basics of Q-Learning and Policy Gradients
ML for Portfolio Optimization: Estimating Covariance Matrices with ML
Leveraging Alternative Data: An Overview of Satellite Imagery, GPS, and Web Scraped Data
AI Ethics & Regulation in Finance: Model Bias, Fairness, and Governance
Final Project: Design and Propose a Complete, Novel ML-based Trading Strategy
Key Concepts - Mean and Variance
Standard Deviation (The "Intuitive" Spread)
The Normal Distribution N(μ, σ²)
Calculus Review - Derivatives (The "Slope")
Calculus Review - Integrals (The "Sum")
The "Master Tool": The Taylor Expansion
The "Master Tool" (Part 2): The 2-Variable Taylor Expansion
The Rules of "Normal" Infinitesimals
Why Normal Calculus Fails
Brownian Motion (Wiener Process Wt)
The "Weird" Scaling Property
A Model for Stocks (Geometric Brownian Motion)
The Failure of Path Length
Quadratic Variation (The "Aha!" Moment)
The New Rules of Algebra
The Itô Integral
Itô's Lemma (Simple Case, for f(Wt))
Itô's Lemma (The Full Version, for f(t, Xt))
Understanding the Full Formula (Translating Math to Finance)
The "Magic Portfolio"
Eliminating Risk
Eliminating Drift μ
The "No-Free-Lunch" Argument & The Final PDE
The Solution (The Black-Scholes Formula)
Delta (Δ): The "Speed" of an Option
Gamma (Γ): The "Acceleration"
Vega (ν): The "Jiggle Risk"
Theta (Θ): The "Melting Ice Cube"
Rho (ρ): The "Interest Rate Risk"
The "Other Way" - Risk-Neutral Valuation
The "Computer Way" - Monte Carlo Methods
Fixing σ - Stochastic Volatility (Heston Model)
Fixing "No Jumps" - Jump-Diffusion (The Merton Model)
Fixing r - Stochastic Interest Rates (Vasicek & CIR)
Introduction to Time Series
Stationarity
ACF and PACF
Autoregressive (AR) Models
Moving Average (MA) Models
ARMA Models
ARIMA Models
The Box-Jenkins Methodology
Introduction to Volatility Modeling
ARCH Models
GARCH Models
GARCH Extensions (EGARCH, GJR-GARCH)
Capstone: Modeling S&P 500 Volatility
Vector Autoregression (VAR) Models
Cointegration and Error Correction Models (VECM)
State Space Models & Kalman Filters
Structural VAR (SVAR) Models
Capstone: A Pairs Trading Strategy
Feature Engineering for Time Series
Time Series Cross-Validation
Forecasting with Tree-Based Models (XGBoost)
Deep Learning for Time Series (RNNs & LSTMs)
Advanced Concepts (Meta-Labeling, Feature Importance)
Descriptive Statistics Explorer
The Central Limit Theorem (CLT)
Bayes' Theorem
Confidence Intervals
Z-Table Calculator
An Introduction to Hypothesis Testing
Hypothesis Testing Guide
Bernoulli Distribution
Binomial Distribution
Poisson Distribution
Geometric Distribution
Hypergeometric Distribution
Negative Binomial Distribution
Discrete Uniform Distribution
Multinomial Distribution
The Normal Distribution
Gamma Distribution
Beta Distribution
Exponential Distribution
The t-Distribution (Student's t)
The χ² (Chi-Squared) Distribution
The F-Distribution (Fisher-Snedecor)
Cauchy Distribution
Laplace Distribution
Weibull Distribution
Logistic Distribution
T-Test
Z-Test
ANOVA
F-Test
Pearson Correlation
Chi-Squared Test
Mann-Whitney U Test
Kruskal-Wallis Test
Wilcoxon Signed-Rank Test
Spearman's Rank Correlation
Friedman Test
Kolmogorov-Smirnov (K-S) Test
Monte Carlo Simulation
Efficient Frontier & Sharpe Ratio
Kalman Filters
Stochastic Calculus & Ito's Lemma

Fundamental Tools

Descriptive Statistics Explorer

Interactive guide to mean, median, skewness, and kurtosis.

The Central Limit Theorem (CLT)

Discover how order emerges from chaos.

Bayes' Theorem

A framework for updating beliefs with new evidence.

Confidence Intervals

Understanding the range where a true value likely lies.

Z-Table Calculator

Calculate probabilities from Z-scores and vice-versa.

An Introduction to Hypothesis Testing

A practical guide to deciding if your results are a real breakthrough or just random noise.

Hypothesis Testing Guide

A comprehensive guide to choosing the right statistical test.

Discrete Distributions

Bernoulli Distribution

Modeling a single trial with two outcomes.

Binomial Distribution

Modeling a series of success/fail trials.

Poisson Distribution

Modeling the frequency of rare events.

Geometric Distribution

Modeling trials until the first success.

Hypergeometric Distribution

Modeling sampling without replacement.

Negative Binomial Distribution

Modeling trials until a set number of successes.

Discrete Uniform Distribution

Modeling where all outcomes are equally likely.

Multinomial Distribution

Generalizing the Binomial for multiple outcomes.

Continuous Distributions

The Normal Distribution

The ubiquitous "bell curve."

Gamma Distribution

Modeling waiting times and skewed data.

Beta Distribution

Modeling probabilities, percentages, and proportions.

Exponential Distribution

Modeling the time between events in a Poisson process.

The t-Distribution (Student's t)

The backbone of hypothesis testing with small sample sizes.

The χ² (Chi-Squared) Distribution

The distribution of variances.

The F-Distribution (Fisher-Snedecor)

Comparing variances between groups and the foundation of ANOVA.

Cauchy Distribution

Modeling extreme events and 'fat-tailed' phenomena.

Laplace Distribution

Modeling with a sharp peak and 'fat tails'.

Weibull Distribution

Modeling time-to-failure and event durations.

Logistic Distribution

A key distribution in machine learning and growth modeling.

Statistics Tools

T-Test

Compares the means of two groups, assuming normal distribution.

Z-Test

Compares means of large samples (n>30) with known population variance.

ANOVA

Compares the averages of three or more groups.

F-Test

Compares the variances (spread) of two or more groups.

Pearson Correlation

Measures the linear relationship between two continuous variables.

Chi-Squared Test

Analyzes categorical data to find significant relationships.

Mann-Whitney U Test

Alternative to the T-Test when data is not normally distributed.

Kruskal-Wallis Test

Alternative to ANOVA for comparing three or more groups.

Wilcoxon Signed-Rank Test

Alternative to the paired T-Test for repeated measurements.

Spearman's Rank Correlation

Measures the monotonic relationship between two ranked variables.

Friedman Test

The non-parametric alternative to a repeated-measures ANOVA.

Kolmogorov-Smirnov (K-S) Test

Tests if a sample is drawn from a specific distribution.

Monte Carlo Simulation

Using random simulation to solve complex problems.

Efficient Frontier & Sharpe Ratio

Finding the optimal portfolio for a given level of risk.

Kalman Filters

Dynamically estimating the state of a system from noisy data.

Stochastic Calculus & Ito's Lemma

The calculus of random walks, essential for derivatives pricing.