Mathematical Finance Course

Lesson 1 • Probability and Statistics Review

Introduces probability spaces, random variables, and key distributions relevant to asset returns. Provides the statistical vocabulary needed for stochastic modelling later.

Lesson 2 • Linear Algebra for Finance

Covers vectors, matrices, and systems of equations as tools for portfolio construction and factor models. Connects matrix operations to multi-asset pricing problems.

Lesson 3 • Calculus Essentials for Pricing

Reviews differentiation, integration, and optimisation techniques applied to financial functions. Prepares students for deriving pricing formulas and sensitivity measures.

Lesson 1 • Bond Pricing Fundamentals

Derives bond prices from discounted cash flows and links coupon structure to market yield. Anchors fixed income analysis in the time value framework from Chapter 1.

Lesson 2 • Duration, Convexity, and Sensitivity

Quantifies price sensitivity to yield changes using duration and convexity measures. Enables hedging of interest rate exposure in fixed income portfolios.

Lesson 3 • Interest Rate Risk Management

Applies duration and convexity to hedge bond portfolios against parallel and non-parallel yield shifts. Introduces key rate durations for granular risk control.

Lesson 1 • Capital Asset Pricing Model

Introduces systematic vs. idiosyncratic risk and derives the CAPM security market line. Provides a benchmark for expected returns used in equity valuation.

Lesson 2 • Multifactor Models

Extends CAPM to Fama-French and arbitrage pricing theory frameworks for richer return attribution. Enables factor-based portfolio construction and risk decomposition.

Lesson 3 • Return and Risk Measurement

Defines arithmetic and geometric returns, volatility, and downside risk metrics for equities. Establishes the statistical foundation for portfolio optimisation.

Lesson 4 • Mean-Variance Optimisation

Derives the efficient frontier using Markowitz optimisation and identifies the minimum-variance portfolio. Connects linear algebra from Chapter 1 to multi-asset allocation.

Lesson 1 • Brownian Motion and Random Walks

Defines Wiener processes and their properties as the continuous-time limit of random walks. Provides the probabilistic engine for all continuous-time finance models.

Lesson 2 • Ito's Lemma and Change of Variables

Derives Ito's lemma and applies it to transform functions of stochastic processes. Enables derivation of the Black-Scholes PDE and other pricing equations.

Lesson 3 • Stochastic Differential Equations

Formulates SDEs for asset prices and interest rates and classifies their drift and diffusion terms. Builds the modelling language used in options pricing and rate models.

Lesson 1 • Implied Volatility and the Volatility Surface

Extracts implied volatility from market prices and analyses the volatility smile and skew. Reveals Black-Scholes model limitations and motivates extensions.

Lesson 2 • Greeks and Sensitivity Analysis

Computes delta, gamma, vega, theta, and rho and interprets their role in hedging and risk management. Enables dynamic hedging strategies for options portfolios.

Lesson 3 • Option Contracts and Payoff Structures

Defines call and put options, moneyness, and payoff diagrams for standard contracts. Grounds derivative pricing in contractual mechanics before introducing models.

Lesson 4 • Black-Scholes Model Derivation

Derives the Black-Scholes PDE using delta hedging and Ito's lemma, then solves for the closed-form price. Connects stochastic calculus from Chapter 4 to a practical pricing formula.

Lesson 1 • Monte Carlo Simulation Fundamentals

Generates asset price paths under risk-neutral measure and estimates option prices by averaging payoffs. Provides a flexible pricing engine for path-dependent products from Chapter 6.

Lesson 2 • Variance Reduction Techniques

Applies importance sampling, stratified sampling, and quasi-random sequences to improve simulation efficiency. Reduces computational cost while maintaining pricing accuracy.

Lesson 3 • Fourier and Transform Methods

Uses characteristic functions and fast Fourier transform to price options under non-normal return models. Extends pricing capability beyond Black-Scholes to jump and stochastic volatility models.

Lesson 4 • Calibration and Model Fitting

Fits model parameters to market prices using least-squares and gradient-based optimisation. Connects numerical methods to practical model deployment in trading and risk systems.

Lesson 1 • Interest Rate Derivatives

Prices caps, floors, swaptions, and interest rate swaps using market models and Black's formula. Extends fixed income knowledge from Chapter 2 into the derivatives space.

Lesson 2 • Structured Products and Embedded Options

Deconstructs convertible bonds, callable bonds, and equity-linked notes into component derivatives. Enables valuation and risk assessment of retail and institutional structured products.

Lesson 3 • Path-Dependent Options

Analyses Asian, barrier, lookback, and digital options whose payoffs depend on the price path. Requires Monte Carlo and PDE methods introduced in Chapter 7.

Lesson 1 • Stress Testing and Scenario Analysis

Designs historical and hypothetical stress scenarios to assess portfolio resilience under extreme conditions. Completes the risk framework with forward-looking tools beyond statistical models.

Lesson 2 • Credit Risk Modelling

Models default probability, loss given default, and credit exposure using structural and reduced-form approaches. Enables pricing of credit derivatives and calculation of credit capital.

Lesson 3 • Counterparty Credit Risk and XVA

Quantifies counterparty exposure through CVA, DVA, and FVA adjustments on derivative portfolios. Reflects post-crisis industry practice for pricing bilateral credit risk.

Lesson 4 • Value at Risk Methodologies

Computes VaR using historical simulation, variance-covariance, and Monte Carlo approaches. Establishes the primary risk metric used in trading book risk management.