Financial Engineering Course

Lesson 1 • Introduction to No-Arbitrage Pricing

Introduces the law of one price and replication arguments as the core logic of derivative pricing. Students apply no-arbitrage to simple forward contracts.

Lesson 2 • Essential Mathematical Tools

Covers calculus, linear algebra, and probability as used in quantitative finance. These tools underpin every pricing and risk model in the course.

Lesson 3 • Financial Markets and Instruments Overview

Surveys equity, fixed-income, currency, and commodity markets. Provides the market context needed to apply engineering techniques meaningfully.

Lesson 4 • Role and Scope of Financial Engineering

Defines financial engineering and its position within finance and applied mathematics. Establishes the professional context for all subsequent technical content.

Lesson 1 • Probability Theory and Measure Basics

Introduces sigma-algebras, filtrations, and measure-theoretic probability. Provides the formal language used throughout stochastic modeling.

Lesson 2 • Stochastic Differential Equations

Formulates and solves SDEs relevant to finance, including geometric Brownian motion. Connects SDE solutions to asset price dynamics.

Lesson 3 • Brownian Motion and Its Properties

Defines standard Brownian motion and derives its key statistical properties. Brownian motion is the building block for all continuous-time price models.

Lesson 4 • Change of Measure and Girsanov's Theorem

Introduces equivalent martingale measures and Girsanov's theorem for drift removal. This is the theoretical basis for risk-neutral pricing introduced in the next chapter.

Lesson 1 • Greeks and Sensitivity Analysis

Defines and computes delta, gamma, vega, theta, and rho for standard options. Greeks are essential for hedging strategies covered in later chapters.

Lesson 2 • Risk-Neutral Valuation Framework

Formalizes the risk-neutral measure and the fundamental theorem of asset pricing. Establishes the pricing framework used for all derivative instruments.

Lesson 3 • Binomial Tree Models

Constructs single- and multi-period binomial trees for option pricing. Demonstrates convergence to BSM and enables American option valuation.

Lesson 4 • Exotic and Path-Dependent Options

Prices barrier, Asian, lookback, and digital options using analytical and simulation methods. Extends the BSM framework to non-standard payoff structures.

Lesson 1 • Heath-Jarrow-Morton Framework

Introduces the HJM framework for modeling the entire forward rate curve. Demonstrates how HJM encompasses short-rate models as special cases.

Lesson 2 • Term Structure of Interest Rates

Analyzes spot rates, forward rates, and yield curve shapes using bootstrapping. Provides the fixed-income foundation for all rate derivative pricing.

Lesson 3 • Short-Rate Models

Develops Vasicek, Cox-Ingersoll-Ross, and Hull-White models for interest rate dynamics. Students derive bond prices and calibrate models to market data.

Lesson 4 • LIBOR Market Model

Develops the LIBOR market model for pricing caps, floors, and swaptions. Students calibrate the model to cap volatility surfaces.

Lesson 1 • Implied Volatility and the Volatility Surface

Extracts implied volatility from market option prices and analyzes smile and skew patterns. Motivates the need for models beyond Black-Scholes.

Lesson 2 • Realized Volatility and Variance Swaps

Defines realized variance, variance swaps, and the VIX methodology. Connects the volatility surface to tradeable variance products.

Lesson 3 • Stochastic Volatility Models

Develops Heston and SABR models with mean-reverting variance processes. Students calibrate these models and compare their smile-generating properties.

Lesson 4 • Local Volatility Models

Derives Dupire's local volatility equation and calibrates it to the implied volatility surface. Students implement local vol grids for option pricing.

Lesson 1 • Fundamentals of Credit Risk

Defines default probability, loss given default, and exposure at default. Establishes the credit risk components used in all subsequent pricing models.

Lesson 2 • Credit Default Swaps and Indices

Values single-name CDS and credit index products using standard market conventions. Connects reduced-form models to liquid credit derivative markets.

Lesson 3 • Reduced-Form Credit Models

Models default as a Poisson arrival with stochastic intensity, enabling flexible calibration. Students price defaultable bonds and CDS using intensity-based methods.

Lesson 4 • Structured Credit and Correlation Products

Prices CDO tranches using Gaussian copula and factor models for portfolio credit risk. Students analyze correlation sensitivity and tranche risk profiles.

Lesson 1 • Hedging Strategies and Immunization

Designs delta, gamma, and duration hedges for equity and fixed-income portfolios. Connects Greeks and duration measures to practical hedging execution.

Lesson 2 • Counterparty Credit Risk and XVA

Quantifies CVA, DVA, and FVA adjustments for OTC derivative portfolios. Students compute XVA metrics and understand their impact on pricing and capital.

Lesson 3 • Portfolio Theory and Factor Models

Applies mean-variance optimization and multi-factor models to portfolio construction. Students build efficient frontiers and decompose portfolio risk.

Lesson 4 • Value at Risk and Expected Shortfall

Derives VaR and ES using parametric, historical, and Monte Carlo methods. Compares their statistical properties and regulatory roles.

Lesson 1 • Finite Difference Methods for PDEs

Solves the Black-Scholes PDE using explicit, implicit, and Crank-Nicolson schemes. Students implement finite difference grids for American and barrier options.

Lesson 2 • Monte Carlo Simulation Methods

Develops Monte Carlo pricing for derivatives with complex payoffs and path dependence. Students apply variance reduction techniques to improve computational efficiency.

Lesson 3 • Calibration and Optimization Techniques

Formulates model calibration as a nonlinear optimization problem and applies gradient and evolutionary algorithms. Students calibrate BSM, Heston, and SABR models.

Lesson 4 • Fourier and Transform Methods

Applies characteristic functions and FFT to price options under affine models. Enables fast calibration of Heston and other stochastic volatility models.