Subtopic
Mathematics
18 lessons to explore at your own pace.
Lessons in Mathematics
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Self-paced lesson · ~15 min read
Financial Markets, Financial Terms and Concepts, and Bond Mathematics
The central question is simple: how do we compare money, assets, and risks across time? First understand where trading happens and who trades; then learn to translate future cash flows…
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Self-paced lesson · ~17 min read
Linear Algebra for Finance: Vectors, Replication, Markov Chains, and PCA
Linear algebra studies vectors, linear transformations, and systems of equations. In finance, these become portfolios, scenario payoffs, transition models, and risk factors. The same…
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Self-paced lesson · ~14 min read
Probability Theory for Finance: Distributions, Limit Theorems, CAPM, and Martingales
Probability theory turns uncertainty into a model we can reason about. In finance, it helps describe defaults, measure portfolio risk, estimate average returns, and price contingent…
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Self-paced lesson · ~17 min read
Linear Rates, Products, and Models: SOFR, Swaps, Yield Curves, and Hedging
Interest-rate markets connect today's money to future cash flows. Their basic products—deposits, bonds, futures, and swaps—provide the prices used to construct yield curves and hedge…
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Self-paced lesson · ~14 min read
Stochastic Processes for Finance: Martingales, Stopping Times, and Markov Chains
A stochastic process is a collection of random variables indexed by time. A price at one date is a random variable; its evolution across dates is a process. The central questions are:…
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Self-paced lesson · ~14 min read
Stochastic Processes: Brownian Motion, Diffusion, and Hitting Times
Brownian motion is the continuous-time reference model for accumulated small random shocks. It connects random walks, diffusion equations, barrier events, statistical tests, and…
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Self-paced lesson · ~13 min read
Stochastic Calculus: Itô Integrals, Itô's Formula, and Stochastic Differential Equations
Ordinary calculus integrates against time or sufficiently regular paths. Brownian paths are continuous but too irregular for the usual pathwise rules to apply unchanged. Stochastic…
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Self-paced lesson · ~9 min read
Stochastic Differential Equations: Black–Scholes, Heat Kernels, and Mean Reversion
A stochastic differential equation (SDE) specifies how a state changes through a predictable drift and unpredictable shocks. Solving it gives an entire random trajectory, rather than one…
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Self-paced lesson · ~13 min read
Black–Scholes Formula and Risk-Neutral Valuation: Replication, PDEs, and Pricing
An option's payoff is uncertain, but a portfolio can sometimes reproduce that payoff state by state. If so, the option and its replicating portfolio must have the same price in an…
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Self-paced lesson · ~16 min read
Volatility Modeling: OHLC Estimators, EWMA, GARCH, and Implied Volatility
Volatility describes the scale of uncertainty in a specified financial variable over a specified horizon. Estimating past volatility, forecasting tomorrow's conditional variance, and…
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Self-paced lesson · ~16 min read
Regression Analysis for Finance: OLS, Inference, Robust Methods, and Regularization
Regression connects an outcome to explanatory variables. In finance, it can estimate market exposure, explain credit spreads, or predict returns. The central question is which…
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Self-paced lesson · ~15 min read
Regression Analysis II: Hypothesis Tests, Structural Breaks, and CAPM Regimes
A regression coefficient describes a relationship in a model. Hypothesis testing asks whether the data are compatible with a specific restriction on that relationship. Structural-break…
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Self-paced lesson · ~16 min read
Regression Analysis III: Stationarity, Autocorrelation, and ARMA–ARIMA Models
Ordinary regression relates an outcome to predictors. Time-series modeling also asks how the past predicts the future and whether that relationship remains stable through time. A trend,…
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Self-paced lesson · ~16 min read
Time Series Analysis: Likelihood, Unit Roots, Seasonality, and Volatility Forecasts
A time-series model can fit a financial series well while answering the wrong forecasting question. A rolling volatility measure, for example, is partly predictable because most of its…
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Self-paced lesson · ~15 min read
Principal Component Analysis in Finance: Yield-Curve Factors, Hedging, and Relative Value
Hundreds of financial series can share a few major sources of variation. Principal component analysis (PCA) finds directions that summarize that variation. In rates, those directions…
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Self-paced lesson · ~17 min read
Machine Learning for Finance: Validation, Neural Pricing, and Reinforcement-Learning Hedges
Machine learning fits patterns from data to support predictions or decisions. In finance, a model can estimate a return, approximate an expensive pricing engine, organize risk exposures,…
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Self-paced lesson · ~16 min read
Portfolio Management: Allocation, Rebalancing, Liquidity, and Position Sizing
Portfolio management turns investment ideas into positions that can survive uncertainty. Choosing a promising asset is only part of the task: how much to own, how to finance it, when to…
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Self-paced lesson · ~18 min read
Counterparty Risk Optimization: Collateral, Risk Measures, and Margin Networks
Two portfolios can have the same aggregate market exposure but very different collateral requirements. The difference comes from where trades sit: across counterparties, enforceable…