Subtopic
Mathematics
34 lessons to explore at your own pace.
Lessons in Mathematics
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Self-paced lesson · ~16 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 · ~15 min read
Present Value Relations: Cash Flows, Compounding, Inflation, and Bonds
A dollar today and a dollar next year have the same currency label but different economic values. Present value gives us a common measuring date so that we can compare investments,…
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Self-paced lesson · ~17 min read
Fixed-Income Securities: Bond Pricing, Yield Curves, Duration, and Credit Risk
A bond is a contract for dated payments. To understand its value, ask three questions: What payments are promised? What discount factors price those dates? What could change the payments…
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Self-paced lesson · ~16 min read
Equity: Dividend Valuation, Growth Opportunities, P/E Ratios, and Index Futures
Equity valuation connects the money a business can distribute to its owners with the price investors pay today. Growth matters because retained earnings can finance future distributions.…
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Self-paced lesson · ~16 min read
Forward and Futures Contracts: Pricing, Margin, Hedging, and an Introduction to Options
A forward or futures contract lets you agree on a future transaction price today. That can help a business stabilize costs or revenues, or let an investor change market exposure without…
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Self-paced lesson · ~17 min read
Options: Payoffs, Strategies, Replication, and Risk-Neutral Pricing
An option lets its buyer choose whether to transact at an agreed price. That choice changes the shape of the payoff: the buyer can benefit from favorable outcomes while declining an…
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Self-paced lesson · ~15 min read
Risk and Return: Diversification, Portfolio Risk, and the Efficient Frontier
An investment's average return is only part of its story. You also need to know how uncertain that return is and how it moves with the rest of your portfolio. An asset that looks risky…
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Self-paced lesson · ~15 min read
Portfolio Theory: Markowitz Optimization, CAPM, Beta, Alpha, and Capital Budgeting
Portfolio theory begins with a choice: how should an investor combine uncertain investments? CAPM adds an equilibrium question: if investors make those choices together, what expected…
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Self-paced lesson · ~17 min read
The CAPM and APT: Beta, Alpha, Factor Risk, and No-Arbitrage Pricing
Two investments can have the same volatility and still require different expected returns. The important distinction is which risks remain when investors combine assets into portfolios.…
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Self-paced lesson · ~18 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 · ~15 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 · ~15 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 · ~10 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 · ~14 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 · ~17 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 · ~17 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 · ~16 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 · ~17 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 · ~17 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 · ~16 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 · ~18 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 · ~17 min read
Neural Networks and Deep Learning: Training Deep Neural Networks
A neural network learns a prediction rule by adjusting weights and biases. A deep network composes several transformations, allowing it to learn useful intermediate representations as…
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Self-paced lesson · ~16 min read
Training Deep Neural Networks: Keras, TensorFlow, and Tabular Data
Training a neural network means turning a mathematical prediction rule into an experiment: prepare the data, choose an objective, update parameters, and measure performance on unseen…
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Self-paced lesson · ~17 min read
Deep Learning for Computer Vision: Building Convolutional Neural Networks from Scratch
Images contain structure: neighboring pixels form edges, edges combine into shapes, and shapes help identify objects. A convolutional neural network, or CNN, builds that structure into…
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Self-paced lesson · ~16 min read
Deep Learning for Computer Vision: Transfer Learning, Fine-Tuning, and Hugging Face
Training every visual feature from random initialization can be wasteful when you have only a few hundred labeled images. Transfer learning starts from representations learned on another…
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Self-paced lesson · ~16 min read
Deep Learning for Natural Language: The Basics
Text carries explanations, opinions, stories, instructions, and conversations. Natural language processing, or NLP, turns this unstructured material into representations that…
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Self-paced lesson · ~17 min read
Deep Learning for Natural Language: Word Embeddings
A word ID tells a model which token it has encountered, but says nothing about what that token means. Word embeddings replace isolated token indicators with learned dense vectors. This…
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Self-paced lesson · ~15 min read
Deep Learning for Natural Language: Transformers
A static embedding gives a token one vector. A Transformer combines token representations with information about their positions and surrounding tokens, producing a contextual state at…
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Self-paced lesson · ~14 min read
Deep Learning for Natural Language: Transformers and Self-Supervised Learning
A Transformer specifies how token representations interact. A learning objective specifies what those interactions should help predict. Self-supervised learning connects the two by…
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Self-paced lesson · ~15 min read
Generative AI: Large Language Models and Retrieval-Augmented Generation
A language model can produce fluent answers from patterns learned during training, but an application often needs information that is private, recent, or too extensive to fit in one…
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Self-paced lesson · ~15 min read
Generative AI: Adapting LLMs with Parameter-Efficient Fine-Tuning
Adapting a large language model can mean changing its instructions, supplying external evidence, or training parameters for a target task. Parameter-efficient fine-tuning, or PEFT, makes…
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Self-paced lesson · ~17 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 · ~19 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…