Quantitative Finance and Risk Modeling

Lesson 1 • Linear Algebra in Finance

Matrix operations, eigenvalues, and decompositions underpin covariance modeling and factor analysis. Connects directly to portfolio construction.

Lesson 2 • Time Series Fundamentals

Stationarity, autocorrelation, and basic ARMA modeling of financial return sequences. Prepares students for advanced econometric modeling.

Lesson 3 • Probability Theory and Distributions

Establishes probability spaces, random variables, and key distributions used in asset return modeling. Foundation for all stochastic methods ahead.

Lesson 4 • Statistical Inference and Estimation

Maximum likelihood and Bayesian estimation methods applied to financial data. Enables calibration of models to observed market prices.

Lesson 1 • Interest Rate Conventions and Curves

Explains day-count conventions, compounding, and bootstrapping of discount curves. Essential for fixed-income pricing and risk calculations.

Lesson 2 • Equity and Fixed-Income Basics

Explains stock pricing, bond valuation, and yield curve construction. Provides the instrument-level knowledge required for pricing model inputs.

Lesson 3 • Derivative Instruments Overview

Introduces forwards, futures, swaps, and options with payoff diagrams. Sets the stage for rigorous derivative pricing in later chapters.

Lesson 1 • Ito's Lemma and Stochastic Differential Equations

Derives Ito's lemma and applies it to transform SDEs governing asset prices. Directly enables derivation of the Black-Scholes equation.

Lesson 2 • Risk-Neutral Pricing and Measure Changes

Introduces Girsanov's theorem and the risk-neutral measure for arbitrage-free pricing. Unifies derivative valuation under a single framework.

Lesson 3 • Brownian Motion and Stochastic Processes

Defines Wiener processes, quadratic variation, and martingales. These are the building blocks for all continuous-time pricing models.

Lesson 4 • Black-Scholes Model and Extensions

Derives the Black-Scholes PDE and closed-form solution, then extends to dividends and currency options. Core benchmark for all option pricing.

Lesson 1 • Portfolio Optimization Techniques

Applies quadratic programming, robust optimization, and Black-Litterman to real allocation problems. Addresses estimation error and constraints.

Lesson 2 • Mean-Variance Framework

Derives the efficient frontier and minimum-variance portfolio using matrix algebra. Establishes the core trade-off between return and risk.

Lesson 3 • Capital Asset Pricing Model

Develops CAPM from equilibrium assumptions and tests its empirical validity. Provides a benchmark for expected return estimation.

Lesson 4 • Factor Models and Risk Decomposition

Builds multi-factor models to decompose portfolio risk into systematic and idiosyncratic components. Enables targeted risk management.

Lesson 1 • Stress Testing and Scenario Analysis

Designs historical and hypothetical stress scenarios to reveal tail exposures. Complements statistical risk measures with narrative-driven analysis.

Lesson 2 • Expected Shortfall and Coherent Risk Measures

Defines Expected Shortfall and proves its coherence properties over VaR. Aligns with current regulatory capital frameworks.

Lesson 3 • Sensitivity-Based Risk Measures

Computes DV01, PV01, and option Greeks for granular risk decomposition. Links position-level sensitivities to portfolio-level risk.

Lesson 4 • Value at Risk Methodologies

Derives parametric, historical, and Monte Carlo VaR and compares their assumptions. Establishes the primary risk metric used across the industry.

Lesson 1 • Structural and Reduced-Form Default Models

Contrasts Merton's structural model with intensity-based reduced-form approaches. Provides two complementary frameworks for default probability estimation.

Lesson 2 • Counterparty Credit Risk and CVA

Quantifies counterparty exposure through EPE profiles and computes CVA adjustments. Integrates credit risk into derivative valuation.

Lesson 3 • Loss Given Default and Recovery Modeling

Estimates recovery rates by seniority and models LGD distributions. Feeds directly into expected and unexpected loss calculations.

Lesson 4 • Credit Portfolio Models and Correlation

Applies Gaussian copula and factor models to capture default correlation in portfolios. Enables CDO tranche pricing and economic capital estimation.

Lesson 1 • Implied Volatility and the Volatility Surface

Extracts implied volatility from option prices and analyzes smile and skew patterns. Motivates the need for models beyond constant volatility.

Lesson 2 • Stochastic Volatility Models

Develops Heston and SABR models with mean-reverting variance processes. Captures volatility clustering and smile dynamics simultaneously.

Lesson 3 • Local Volatility Models

Derives Dupire's local volatility equation and calibrates it to the observed surface. Provides a complete market model consistent with all vanilla prices.

Lesson 4 • Exotic Derivatives Pricing

Prices barrier, Asian, and lookback options using analytical and numerical methods. Applies advanced models to structured product valuation.

Lesson 1 • Binomial and Trinomial Trees

Constructs recombining trees for option pricing and interest rate models. Offers intuitive discretization for American and Bermudan options.

Lesson 2 • High-Performance Computing in Finance

Leverages vectorization, parallel processing, and GPU acceleration for large-scale risk calculations. Bridges model correctness and computational scalability.

Lesson 3 • Monte Carlo Simulation Methods

Designs Monte Carlo engines for pricing and risk, including variance reduction techniques. Enables valuation of high-dimensional and path-dependent instruments.

Lesson 4 • Calibration and Optimization Algorithms

Applies gradient-based and global optimization to fit model parameters to market prices. Addresses ill-posedness and regularization in calibration.