Financial Engineering: Optimization Methods Course

Lesson 1 • Core Concepts in Quantitative Finance

Introduces financial instruments, market mechanics, and risk-return tradeoffs. Establishes vocabulary and context for all subsequent optimisation work.

Lesson 2 • Introduction to Optimisation Theory

Defines optimisation problems, objective functions, and constraint types. Connects abstract theory to concrete financial decision-making scenarios.

Lesson 3 • Mathematical Prerequisites for Optimisation

Covers linear algebra, calculus, and probability essential for financial models. Provides the computational backbone for formulating optimisation problems.

Lesson 1 • The Simplex Method

Explains the simplex algorithm step by step, including pivoting and basis changes. Students apply the method to financial LP problems by hand and with software.

Lesson 2 • Duality Theory and Sensitivity Analysis

Introduces dual LP problems and shadow prices with financial interpretations. Sensitivity analysis reveals how optimal solutions respond to parameter changes.

Lesson 3 • Financial Applications of Linear Programming

Applies LP to asset-liability matching, cash flow planning, and index replication. Students build and solve complete financial LP models from problem statements.

Lesson 1 • Performance Evaluation of Optimal Portfolios

Assesses optimised portfolios using risk-adjusted metrics and out-of-sample testing. Connects optimisation outputs to practical investment performance measurement.

Lesson 2 • Extensions and Practical Constraints

Incorporates transaction costs, factor constraints, and ESG limits into QP models. Students build industry-realistic portfolio optimisation models.

Lesson 3 • Solving Quadratic Programs

Covers active-set, interior-point, and critical-line algorithms for QP. Students implement solvers and interpret convergence and solution quality.

Lesson 4 • Mean-Variance Framework

Formalises the mean-variance model, including expected return and portfolio variance. Establishes the theoretical basis for all quadratic portfolio optimisation.

Lesson 1 • Convex Sets and Functions

Defines convexity rigorously for sets and functions with financial examples. Provides the theoretical foundation for guaranteeing global optimality.

Lesson 2 • Optimality Conditions and Duality

Covers KKT conditions, Lagrangian duality, and strong duality for convex problems. Students verify optimality and interpret dual variables in financial contexts.

Lesson 3 • Gradient and Subgradient Methods

Introduces first-order algorithms including gradient descent and subgradient methods. Students implement these algorithms on financial objective functions.

Lesson 4 • Second-Order and Interior-Point Methods

Covers Newton's method, barrier methods, and primal-dual interior-point algorithms. Students apply these to large-scale convex financial optimisation problems.

Lesson 1 • Integer Programming Fundamentals

Defines integer and binary variables and their role in financial decision models. Contrasts IP complexity with LP and motivates specialised solution methods.

Lesson 2 • Branch-and-Bound Algorithm

Explains the branch-and-bound framework for solving integer programs to optimality. Students trace algorithm execution on financial MIP examples.

Lesson 3 • Financial MIP Applications

Models cardinality-constrained portfolios, capital budgeting, and facility location as MIPs. Students build and solve complete financial integer programs using solvers.

Lesson 4 • Cutting Plane Methods

Introduces Gomory cuts and valid inequalities to tighten LP relaxations. Students apply cutting planes to accelerate MIP solution in financial models.

Lesson 1 • Uncertainty Modelling in Finance

Introduces sources of financial uncertainty and their probabilistic representations. Motivates the need for optimisation methods that explicitly handle randomness.

Lesson 2 • Simulation-Based Optimisation

Combines Monte Carlo simulation with optimisation for complex financial problems. Students implement sample average approximation and stochastic gradient methods.

Lesson 3 • Risk Measures and CVaR Optimisation

Defines coherent risk measures and formulates CVaR minimisation as a linear program. Students optimise portfolios under tail-risk constraints using scenario data.

Lesson 4 • Stochastic Programming Fundamentals

Covers two-stage and multi-stage stochastic programs with recourse decisions. Students formulate financial planning problems as stochastic programs.

Lesson 1 • Continuous-Time Optimal Control

Covers the Hamilton-Jacobi-Bellman equation and Pontryagin's maximum principle. Students derive analytical solutions for classic continuous-time finance problems.

Lesson 2 • Principles of Dynamic Programming

Introduces Bellman's principle of optimality and the value function concept. Establishes the recursive structure underlying all dynamic financial optimisation.

Lesson 3 • Discrete-Time Financial Models

Applies DP to binomial trees, multi-period portfolio choice, and option pricing. Students implement backward induction on discrete financial state spaces.

Lesson 4 • Applications in Hedging and Execution

Uses DP and optimal control for dynamic hedging strategies and optimal trade execution. Students build models for real-world sequential financial decision problems.

Lesson 1 • Algorithmic Trading Strategy Design

Applies optimisation to signal combination, execution scheduling, and strategy allocation. Students build quantitative trading strategies grounded in optimisation principles.

Lesson 2 • Multi-Objective Optimisation in Finance

Addresses problems with competing objectives such as return, risk, and liquidity. Students compute Pareto frontiers and apply scalarisation and goal programming.

Lesson 3 • Optimisation Under Market Frictions

Incorporates bid-ask spreads, market impact, and liquidity constraints into models. Students build realistic optimisation frameworks that account for trading costs.

Lesson 4 • Capstone: End-to-End Financial Optimisation

Integrates all course methods in a comprehensive financial engineering project. Students deliver a complete optimisation solution with documentation and performance analysis.