Mathematics for Quantitative Finance: Calculus and Linear Algebra

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, optimization 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 optimization are automatic. If they are not, the new ideas get lost in the old algebra.

Why do quants need linear algebra?

Linear algebra for finance is the language of portfolios, risk and regression. A portfolio is a vector of weights, its risk is a quadratic form in a covariance matrix, and nearly every factor model and estimator in this catalog is a matrix calculation. This course teaches that language on the data it is used for.

Because the work is done in matrices. Mean-variance optimization inverts a covariance matrix. Principal component analysis finds the few movements that explain most of a yield curve. Monte Carlo simulation of correlated assets uses a Cholesky factor. Without the algebra these methods are black boxes, and their failures, such as an unstable inverse, go unnoticed.

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.

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 optimization problems with Lagrange multipliers.
• Solve simple ordinary and partial differential equations and recognize the heat equation.
• Implement each method numerically in Python.
• Work with vectors and matrices by hand and in NumPy, including products, inverses and solving linear systems.

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. Vectors and portfolios

   Weights and returns as vectors, dot products, norms and the return of a portfolio.

6. Matrices and linear systems

   Products, transposes, inverses, rank and solving for hedge ratios.

What every course contains

Every paid course comes with the first seven and the certificate, whichever level it sits at. The proctored exam and the credential come when the course is taken as part of a Certificate, Diploma or Fellowship. Everything is self-paced: the material is yours to study in any order, at any hour, for the whole of the license.

1. Workbooks. Written to train the thinking patterns and the important concepts of the course, with worked problems and the reasoning behind each answer, so that you learn to think the way the subject is thought.
2. Audio edition. Focused, well-presented recordings with explainers and examples of the material, made to be interesting, relevant and entertaining, so that the general subject can be learned away from the desk.
3. Notes. Pre-produced notes with every fact the course relies on, written out in full, so that nothing has to be transcribed from the films or the audio.
4. Cheat sheet. One sheet with every important learning concept of the course on it, to keep beside you during the work and to revise from before the exam.
5. Named sources. Every fact in the notes names its source, the paper, the book or the dataset, for anyone who wants to dig in further.
6. Explainer films. Focused films, in the style of the sample films on the catalog page, that walk through the most difficult concepts of the course with visual explanations of a high standard.
7. Transcript. A full transcript of every film and every audio recording, searchable and quotable.
8. Proctored exam. Taken as part of a Certificate, Diploma or Fellowship, the course ends in a proctored multiple-choice questionnaire, with a pass mark of 60% and three retakes included. A course taken on its own is assessed by its coursework and has no exam.
9. Verifiable certificate. A certificate of completion that a third party can verify: an employer sends the code on it to the credentials office and receives written confirmation of the course, the date and the result.
10. Credential of the program. The Certificate, Diploma or Fellowship of the program, track or path the course belongs to, awarded when every course in it is passed. Which one depends on the program you choose.

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. A course taken as part of a Certificate, Diploma or Fellowship ends in a proctored online exam with a pass mark of 60%. The fee includes three retakes; each attempt after that carries a reduced proctored exam fee. A course taken on its own is assessed by its coursework and has no exam. Read how certification works.

Before you start

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