Mathematical Methods for Quantitative Finance Course

Lesson 1 • Linear Algebra Essentials

Introduces vectors, matrices, and linear transformations used throughout portfolio and factor models. Provides tools for solving systems of financial equations.

Lesson 2 • Calculus Review for Finance

Reviews differentiation and integration with emphasis on financial applications such as sensitivity analysis. Bridges undergraduate calculus to advanced financial mathematics.

Lesson 3 • Real Analysis and Number Systems

Covers real numbers, limits, and continuity as the backbone of financial modelling. Connects rigorous analysis to pricing and risk measurement.

Lesson 1 • Multivariate Statistical Techniques

Extends analysis to covariance matrices, principal components, and factor structures. Directly supports portfolio construction and risk decomposition.

Lesson 2 • Hypothesis Testing in Finance

Applies classical and robust tests to financial data including normality and autocorrelation checks. Supports model validation and regulatory reporting.

Lesson 3 • Regression Analysis Fundamentals

Builds ordinary least squares regression for factor exposure and return attribution. Introduces diagnostics that ensure model validity.

Lesson 4 • Descriptive Statistics for Returns

Summarises return distributions using moments, quantiles, and shape measures. Establishes baseline data characterisation before inferential methods.

Lesson 1 • Martingale Representation and Girsanov

Covers the martingale representation theorem and Girsanov's change-of-measure technique. Enables risk-neutral pricing in continuous-time models.

Lesson 2 • Ito Calculus and Stochastic Integrals

Develops the Ito integral and Ito's lemma for transforming stochastic processes. Enables derivation of stochastic differential equations for asset prices.

Lesson 3 • Discrete-Time Stochastic Processes

Introduces random walks, Markov chains, and martingales in discrete time. Provides intuition before transitioning to continuous-time models.

Lesson 4 • Brownian Motion Construction

Defines standard Brownian motion and its key properties including continuity and quadratic variation. Forms the foundation for Ito calculus.

Lesson 1 • Black-Scholes-Merton Framework

Derives the Black-Scholes-Merton PDE and closed-form option pricing formula. Connects Ito calculus and risk-neutral measure to practical pricing.

Lesson 2 • Exotic and Path-Dependent Options

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

Lesson 3 • Volatility Modelling and Implied Volatility

Analyses the volatility smile, skew, and term structure from market option prices. Introduces local and stochastic volatility model concepts.

Lesson 4 • Binomial and Lattice Models

Builds discrete-time binomial trees for option pricing and American exercise. Bridges intuition between discrete and continuous models.

Lesson 1 • Term Structure of Interest Rates

Constructs zero-coupon yield curves and forward rate curves from market data. Connects curve shapes to economic theories of interest rates.

Lesson 2 • Heath-Jarrow-Morton Framework

Models the entire forward rate curve evolution using the HJM no-drift condition. Generalises short-rate models to infinite-dimensional rate dynamics.

Lesson 3 • Bond Pricing and Yield Measures

Derives present value formulas for coupon bonds and defines yield, duration, and convexity. Establishes the quantitative toolkit for fixed income analysis.

Lesson 4 • Short-Rate and Affine Models

Develops Vasicek, CIR, and affine term structure models for interest rate dynamics. Enables analytical bond pricing under stochastic rates.

Lesson 1 • Multifactor Models

Extends CAPM to Fama-French and arbitrage pricing theory factor models. Enables return attribution and risk decomposition across multiple factors.

Lesson 2 • Performance Measurement and Attribution

Quantifies portfolio performance using Sharpe, Sortino, and information ratios. Decomposes returns into allocation and selection effects.

Lesson 3 • Capital Asset Pricing Model

Derives the CAPM equilibrium pricing equation and beta as a systematic risk measure. Connects portfolio theory to asset pricing and cost of capital.

Lesson 4 • Constrained Portfolio Optimisation

Solves portfolio problems with long-only, budget, and factor constraints using Lagrangian and quadratic programming. Bridges theory to practical implementation.

Lesson 1 • Copulas and Dependence Modelling

Models joint tail dependence using Gaussian, t, and Archimedean copulas. Addresses the limitations of linear correlation in risk aggregation.

Lesson 2 • Credit Risk Modelling

Quantifies default probability, loss given default, and credit exposure using structural and reduced-form models. Supports credit portfolio management.

Lesson 3 • Hedging Strategies and Greeks Management

Designs delta, gamma, and vega hedges for derivatives portfolios. Connects risk sensitivities to dynamic rebalancing and P&L explanation.

Lesson 4 • Value at Risk and Expected Shortfall

Defines VaR and Expected Shortfall as coherent risk measures and derives them analytically and via simulation. Connects measures to capital adequacy requirements.

Lesson 1 • Model Calibration and Validation

Calibrates pricing models to market quotes and validates fit using statistical diagnostics. Ensures models are both mathematically consistent and market-consistent.

Lesson 2 • Optimisation Algorithms for Finance

Implements gradient descent, Newton-Raphson, and evolutionary algorithms for model calibration and portfolio optimisation. Addresses convergence and local minima issues.

Lesson 3 • Numerical Integration and Fourier Methods

Applies Gaussian quadrature and fast Fourier transform to option pricing under characteristic functions. Enables efficient pricing for models without closed forms.

Lesson 4 • Monte Carlo Simulation Methods

Generates asset price paths and computes option prices and risk measures via simulation. Introduces variance reduction techniques to improve computational efficiency.