All Topics
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Interactive tools for hands-on probability and statistics analysis.
Vectors, matrices, and eigenvalues. The language of data.
The science of collecting, analyzing, and interpreting data.
Master random variables, distributions, and stochastic processes.
Derive Black-Scholes from scratch and master the models that power modern finance.
Building predictive models for financial markets.
Vectors as geometric arrows vs. vectors as ordered lists of numbers (the data science view).
Addition, subtraction (tip-to-tail rule), and scalar multiplication (stretching/shrinking).
The dot product as a measure of 'projection' or 'agreement.' L1 and L2 norms as measures of length/magnitude. Cosine similarity as a practical application.
The concept of perpendicular vectors (dot product = 0) and its meaning: independence.
A matrix as a container for data (a collection of vectors) vs. a matrix as a linear transformation that moves, rotates, and scales space.
Addition, scalar multiplication, and the transpose.
Taught not just as a rule, but as the composition of linear transformations. This explains why AB ≠ BA.
Identity matrix (the 'do nothing' operation), inverse matrix (the 'undo' operation), diagonal, triangular, and symmetric matrices.
What can you build with a set of vectors?
Identifying and removing redundant vectors.
The minimal set of vectors needed to define a space and the concept of its dimension.
Formalizing these concepts. A subspace as a 'plane' or 'line' within a higher-dimensional space that passes through the origin.
Understanding Ax=b from the row picture (intersection of planes) and the column picture (linear combination of columns).
The core algorithm for solving linear systems. Row operations, row echelon form (REF).
Identifying if a system has a unique solution, no solution, or infinitely many solutions from its REF.
The ultimate, unique 'answer sheet' for a linear system, removing the need for back-substitution.
The 'matrix version' of Gaussian Elimination. Solving Ax=b becomes a fast, two-step process of forward and back substitution.
The space of all possible outputs of A. The concept of rank as the "true dimension" of the output space.
The space of all inputs that map to the zero vector. Its connection to multicollinearity in data.
Completing the picture of the four fundamental subspaces.
How the four subspaces relate to each other and partition the input and output spaces.
The determinant as the scaling factor of area/volume.
Cofactor expansion and the properties of determinants. A determinant of zero means the matrix squishes space into a lower dimension (i.e., it's not invertible).
Finding the 'special' vectors that are only scaled by a transformation, not rotated off their span (Ax = λx).
The calculation behind eigenvalues: solving det(A - λI) = 0.
Decomposing a matrix into its core components: 'changing to the eigenbasis, scaling, and changing back.'
Using eigenvalues for tasks like calculating matrix powers (e.g., for Markov chains).
For symmetric matrices (like covariance matrices), the eigendecomposition is especially beautiful and stable (A = QDQᵀ). This is the theoretical foundation of PCA.
A highly efficient specialization for symmetric, positive-definite matrices, often used in optimization and financial modeling.
Introducing the goal of minimizing the error ||Ax - b||.
Finding the closest point in a subspace (the Column Space) to an external vector.
Deriving AᵀAx̂ = Aᵀb from the projection geometry. This is the engine of Linear Regression.
Understanding why AᵀA can be ill-conditioned and lead to numerical errors.
An algorithm for creating a "nice" orthonormal basis from any starting basis.
Using Gram-Schmidt to factor A=QR. Show how this makes solving the least squares problem trivial (R = Qᵀb) and numerically robust.
The ultimate decomposition (A = UΣVᵀ) that works for any matrix and finds orthonormal bases for all four fundamental subspaces simultaneously.
A direct, powerful application of SVD on the data matrix for dimensionality reduction.
Low-rank approximation for noise reduction, and the core ideas behind recommendation systems.
Deriving portfolio variance from first principles using linear algebra.
Using linear algebra to construct optimal portfolios.
Understanding the relationship between risk and expected return.
Connecting "no free lunch" to the geometry of vector spaces.
Modeling dynamic systems like credit ratings with transition matrices.
Duration and convexity as linear algebraic concepts.
A comprehensive guide to choosing the right statistical test.
A practical guide to deciding if your results are a real breakthrough or just random noise.
Compares the means of two groups, assuming normal distribution.
Compares means of large samples (n>30) with known population variance.
Compares the averages of three or more groups.
Compares the variances (spread) of two or more groups.
Measures the linear relationship between two continuous variables.
Analyzes categorical data to find significant relationships.
Alternative to the T-Test when data is not normally distributed.
Alternative to ANOVA for comparing three or more groups.
Alternative to the paired T-Test for repeated measurements.
Measures the monotonic relationship between two ranked variables.
The non-parametric alternative to a repeated-measures ANOVA.
Tests if a sample is drawn from a specific distribution.
The detective work of data science.
Interactive guide to mean, median, skewness, and kurtosis.
Discover how order emerges from chaos.
Understanding the range where a true value likely lies.
Calculate probabilities from Z-scores and vice-versa.
The ubiquitous "bell curve."
Using random simulation to solve complex problems.
Breaking down a time series into its core components.
Measuring how a time series correlates with its past values.
Modeling the changing volatility of financial returns.
Finding the optimal portfolio for a given level of risk.
Dynamically estimating the state of a system from noisy data.
The calculus of random walks, essential for derivatives pricing.
Understanding the building blocks of probability.
The three fundamental rules that govern all of probability.
How the occurrence of one event affects another.
Updating your beliefs in the face of new evidence.
Describing the probabilities of discrete outcomes.
Calculating the center and spread of a random variable.
Exploring key models for random events.
The "fingerprint" of a distribution for deriving moments.
Describing the probabilities of continuous outcomes.
Calculating moments for continuous random variables.
Exploring Uniform, Exponential, and Gamma distributions.
Deriving moments for Normal, Exponential, and Gamma distributions.
Modeling multiple random variables simultaneously.
Extracting marginal and conditional probabilities from joint distributions.
Measuring how two random variables move together.
Defining when two variables have no influence on each other.
Mastering the bell curve and standardization.
Understanding how normal variables combine.
The cornerstone of modern portfolio theory.
Dissecting multi-asset models.
Applying MVN properties to portfolio construction and risk management.
The distribution of variances.
The backbone of hypothesis testing with small sample sizes.
Comparing variances between groups and the foundation of ANOVA.
Why casino averages are so stable.
Why the normal distribution is everywhere.
An interactive simulation to visualize the CLT with different distributions.
Tools for approximating the distribution of functions of random variables.
Distinguishing between a function of data and a guess for a parameter.
Evaluating the accuracy of estimators.
Finding the "best" possible unbiased estimator.
Ensuring estimators converge to the true value and use all available information.
An intuitive technique for finding estimators by matching sample moments to population moments.
The most important method for parameter estimation in finance.
The practical side of implementing MLE.
A framework for creating intervals for any parameter.
Using t, χ², and Z pivotal quantities to build intervals.
The fundamental setup of all hypothesis tests.
The two equivalent approaches to making a statistical decision.
Finding the most powerful test for a given significance level.
A general method for comparing nested models.
Modeling a relationship with a single predictor.
The calculus behind finding the "best fit" line.
Assessing how well your linear model fits the data.
Extending SLR to multiple predictors using linear algebra.
The matrix algebra for solving a multiple regression problem.
Defining the rules for OLS to be BLUE.
The theoretical justification for using OLS.
Testing the significance of a single predictor.
Testing the significance of a group of predictors or the entire model.
Diagnosing when predictors are too correlated with each other.
Handling non-constant variance in the error terms.
Detecting patterns in the error terms over time.
A practical application of MLR to test a famous financial model.
Decomposing the components of a time series (Trend, Seasonality, Cycles, and Noise).
The most important property for modeling time series data.
The key tools for identifying the structure of a time series.
Modeling how past values influence the present.
Modeling how past forecast errors influence the present.
Combining AR and MA models to capture complex dynamics.
Incorporating differencing to model real-world data like stock prices.
The systematic process for identifying, estimating, and validating ARIMA models.
Introducing models where variance depends on past errors.
The industry-standard model for volatility forecasting.
A real-world project to model and forecast the volatility of a major stock index.
Exploring the theory that market prices are unpredictable.
The formal mathematical definition of a "fair game" and its implications for financial markets.
The standard model for stock price paths used in the Black-Scholes formula.
Using simulation to solve problems that are too hard for pure math.
A powerful computational method for assessing the uncertainty of an estimate when theory fails.
A related resampling technique for bias and variance estimation.
Modeling the dynamics of multiple time series at once.
A statistical test for finding stable, long-term relationships between non-stationary time series (the basis of pairs trading).
A model that combines long-run equilibrium (cointegration) with short-run dynamics (VAR).
A complete, real-world project to find a cointegrated pair of stocks and build a basic trading strategy.
A framework for updating beliefs with new evidence.
Modeling a single trial with two outcomes.
Modeling a series of success/fail trials.
Modeling the frequency of rare events.
Modeling trials until the first success.
Modeling sampling without replacement.
Modeling trials until a set number of successes.
Modeling where all outcomes are equally likely.
Generalizing the Binomial for multiple outcomes.
Modeling waiting times and skewed data.
Modeling probabilities, percentages, and proportions.
Modeling the time between events in a Poisson process.
Modeling extreme events and 'fat-tailed' phenomena.
Modeling with a sharp peak and 'fat tails'.
Modeling time-to-failure and event durations.
A key distribution in machine learning and growth modeling.
Understanding the building blocks of probability.
Techniques for counting outcomes and possibilities.
How the occurrence of one event affects another.
Updating beliefs in the face of new evidence.
Mapping outcomes of a random process to numbers.
Calculating the center, spread, and shape of a distribution.
Exploring Bernoulli, Binomial, and Poisson distributions.
Modeling the behavior of multiple random variables at once.
Measuring how two random variables move together.
Why casino averages are so stable.
Why the normal distribution is everywhere.
Finding the distribution of a function of a random variable.
Quantifying information with Entropy and KL Divergence.
Understanding random phenomena that evolve over time.
Modeling memoryless state transitions.
Modeling the timing of random events.
The mathematical foundation of stock price movements.
The rigorous foundation of modern probability.
A more powerful theory of integration.
The formal model of a fair game.
The calculus of random walks, essential for derivatives pricing.
Sharpen your calculation speed and accuracy for interviews.
Mean (μ) as the "expected value" or "center" of a distribution. Variance (σ²) as the measure of "spread" or "messiness" of all possible outcomes.
Standard Deviation (σ = √σ²) as the "fix" for variance. Why we use σ instead of σ².
The "bell curve" as the mathematical "recipe" for pure randomness. How μ and σ² affect its shape.
A review of df/dt as the "instantaneous rate of change" and an introduction to partial derivatives.
Reviewing ∫f(x)dx as "summing up an infinite number of tiny pieces" to find a total area or total change.
Why we can ignore second-order terms like (Δt)² in normal calculus. This is the key rule that is about to be broken.
Discover the "infinitely wiggly" path of a stock and why df/dt (a "slope") doesn't exist for a random path.
Defining our 'perfectly random' path and its 4 key properties: W₀=0, continuous path, independent increments, and the key rule: Wt - Ws ~ N(0, t-s).
Deriving the "typical size" of a single random step. Why does Wt ~ N(0, t) mean the Standard Deviation is √t?
Building the SDE: dSₜ = μSₜdt + σSₜdWₜ. The "Drift" (μ) is the river's current, and "Diffusion" (σ) is the "jiggliness" of the water.
Proving why a Brownian Motion has no derivative by showing that the "path length," Σ|ΔW|, is proportional to Σ√Δt, which goes to infinity.
If Σ|ΔW| fails, what about Σ(ΔW)²? Showing that Σ(ΔW)² is proportional to Σ(√Δt)² = ΣΔt = T. The "squared" path is finite.
Turning our discovery into rules for our infinitesimal steps: (dt)²=0, dt·dWt=0, and the "Weird Rule" (dWt)²=dt.
How do these new rules change integration? We'll solve ∫WₜdWₜ to show that ∫WₜdWₜ = ½Wₜ² - ½T. That extra term is the "cost of randomness."
A review of our key tool. How to get Δf ≈ f'(x)Δx + ½f''(x)(Δx)². This is the foundation for Itô's Lemma.
What's the chain rule for a function of just Wt? We'll plug ΔW into our Taylor formula and apply our "weird algebra" rules. Result: df = f'(Wₜ)dWₜ + ½f''(Wₜ)dt.
What if our function depends on two variables, like f(t, S_t)? We will show all 5 terms of the expansion.
The "final boss" of our theory. We'll combine all our tools: the 2-variable Taylor expansion, the SDE (dSt = a dt + b dWt), and our "weird algebra" rules. We will go step-by-step, showing which of the 5 Taylor terms "survive" and which "die" (go to 0). Result: df = (∂f/∂t + a∂f/∂S + ½b²∂²f/∂S²)dt + (b∂f/∂S)dWt
A physical meaning for all 4 terms in the full Itô's Lemma. (Time Decay, Drift Effect, Itô Correction, and the new Random Part).
Constructing the delta-hedged portfolio (Π = -V + ΔS) and finding its SDE, dΠ.
How to choose a "magic" value for Δ that makes the entire dWt ("Random") bin equal zero. We'll solve for Δ and find that Δ = ∂V/∂S.
Plugging our "magic" Δ back into the drift part of our portfolio to watch the subjective μ term vanish completely.
Our portfolio is now risk-free, so it must earn the risk-free rate, r. We'll set our two equations for dΠ equal to each other and rearrange to get the Black-Scholes-Merton PDE.
Explaining the physical meaning of the famous formula: (Expected Benefit) - (Expected Cost).
Δ = ∂V/∂S. How much V moves when S moves $1. How to use Delta to hedge.
Γ = ∂²V/∂S². How much Δ moves when S moves $1. Why Gamma is the "risk of your hedge."
ν = ∂V/∂σ. How much V moves when volatility σ changes by 1%. Why "panic is good" for an option holder.
Θ = ∂V/∂t. The rate of time decay. The cost of waiting.
ρ = ∂V/∂r. The sensitivity to changes in the risk-free rate.
Introducing the risk-neutral measure Q and the fundamental theorem of asset pricing.
Using simulation to price complex derivatives that have no closed-form solution.
Modeling volatility itself as a random process to capture market dynamics like the "volatility smile."
Adding "jumps" to our random walk to account for sudden market crashes and shocks.
Modeling the risk-free rate itself as a random process for pricing long-term bonds and derivatives.