Mathematics for Quantitative Finance

Mathematics for quantitative finance is a compact set of tools, used again and again: derivatives to measure sensitivity, matrices to handle many assets at once, optimisation to choose portfolios and differential equations to price options. This course revises that set and applies each part to the financial problem it was brought in to solve.

Why a separate mathematics course?

Because the researcher-level courses move fast. Stochastic calculus, volatility models and portfolio theory all assume that calculus, linear algebra and optimisation are automatic. If they are not, the new ideas get lost in the old algebra.

Who is this course for?

Graduates from economics, business or the sciences whose mathematics needs strengthening before the researcher path, and practitioners returning to mathematics after time away. It is the first course of the Fellowship in Computational Finance.

What you will be able to do

• Differentiate and integrate the functions that appear in pricing and risk formulas, including Taylor expansions.
• Use matrices, eigenvalues and decompositions to work with covariance matrices.
• Solve constrained optimisation problems with Lagrange multipliers.
• Solve simple ordinary and partial differential equations and recognise the heat equation.
• Implement each method numerically in Python.

Syllabus, module by module

1. Functions, limits and series

   Exponentials and logarithms in compounding, sequences and series, and Taylor expansions.

2. Differential calculus

   Derivatives, partial derivatives and the chain rule, with sensitivities of prices to inputs.

3. Integral calculus

   Integration techniques and the integrals behind expected values and option prices.

4. Vectors and matrices

   Matrix algebra for portfolios, returns and factor exposures.

5. Eigenvalues and decompositions

   Eigen, Cholesky and singular value decompositions of covariance matrices; principal components.

6. Optimisation

   Unconstrained and constrained optimisation, Lagrange multipliers and the mean-variance problem.

7. Ordinary differential equations

   First and second order equations, with interest rate and growth models.

8. Partial differential equations

   The heat equation, its solution and why it appears in option pricing.

9. Numerical methods

   Root finding, numerical integration and finite differences in Python.

10. Review and application

    A worked derivation from a portfolio problem and a pricing problem, using every tool in the course.

How the certificate is earned

Each module ends with a graded quiz, and the course ends with a graded problem set of derivations and numerical exercises. On a pass you receive the Course certificate in Mathematics for Quantitative Finance, issued by the Institute with a verification code. Read how certification works.

Before you start

School-level calculus and algebra. The course revises everything beyond that.