Advanced Mathematics Learning Path

Created By


54 Reviews

20304 Hours
Media Formats:

Text Book, Video

Enrolled Students Count:
Dropped Out Students Count:
Completed Students Count:
Job Placement Rate:
Accredited By:

Not Accredited

Pacing Type:
Learning Methodology:

Top down, Task based education, Self-paced education

Visit this Pathway

Advanced Mathematics Learning Path

I am often asked in emails how to go about learning the necessary mathematics for getting a job in quantitative finance or data science if it isn't possible to head to university. This article is a response to such emails. I want to discuss how you can become a mathematical autodidact using nothing but a range of relatively reasonably priced textbooks and resources on the internet. While it is far from easy to sustain the necessary effort to achieve such a task outside of a formal setting, it is possible with the resources (both paid and free) that are now available.


No Featuers

The pathway does not offer any specific features.

Main Modules

Most top-tier UK undergraduate courses have a Foundations module of some description. The goal of the course is to provide you with a detailed overview of the nature of university mathematics, including the notions of proof (such as proof by induction and proof by contradiction), the concept of a map or function, as well as the differing types such as the injection, surjection and bijection.
Year1:Real Analysis - Sequences and Series
Real Analysis is a staple course in first year undergraduate mathematics. It is an extremely important topic, especially for quants, as it forms the basis for later courses in stochastic calculus and partial differential equations. The subject is primarily about real numbers and functions between sets of real numbers. The main topics discussed include sequences, series, convergence, limits, calculus and continuity.
Year1:Linear Algebra
Linear Algebra is one of the most important, if not the most important, subjects to learn for a prospective quant or data scientist. In an abstract sense Linear Algebra is about the study of linear maps between vector spaces. It teaches us that in certain cases linear maps and matrices are actually equivalent. This latter result makes it extremely useful when dealing with matrix equations, of which there are many within quantitative finance and data science.
Year1:Ordinary Differential Equations - Introduction
The subject of differential equations permeates wide areas of quantitative finance. They are an extremely important subject for a prospective quant to learn, as stochastic differential equations play a large part in options pricing theory.
Year1:Geometry - Euclidean
Geometry is one of the most fundamental areas of mathematics. It is absolutely essential for many areas of deeper mathematics, including those related to quantitative finance. Many undergraduate courses introduce Euclidean geometry to students in their first year, and it is also an appropriate place to start for the autodidact.
Year1:Algebra - Group Theory
Groups are one of the most important algebraic structures found in mathematics. They provide the basis for studying more complex structures such as rings, fields, vector spaces (which we mentioned above in Linear Algebra). They are also strongly related to the idea of mathematical symmetry.
Year1: Probability
Along with Linear Algebra and Real Analysis (Calculus), introductory Probability is the most important first year course for a quant to know. This applies for quantitative traders, quantitative analysts (derivatives pricers), risk managers (VaR, CVA etc) and data scientists. I cannot stress enough how important it is for a practising quant to have an intuitive grasp of probabilistic concepts. Time spent studying here will pay dividends over a quant career.
Year1: Mathematical Computing
What is Mathematical Computing? Broadly, it is carrying out mathematical analysis using computer programs. This is essentially the definition of a quant! Hence, it is absolutely essential that you gain a grounding in programming algorithms at the earliest possible stage.
Year2:Real Analysis - Riemann Integral
The first year courses on real analysis tend to concentrate on sequences, series, functions of a single real variable (i.e. f:R->R ), continuity of those functions as well as properties and results related to their derivatives.
Year2:Metric Spaces
A course in Metric Spaces is often the first introduction to the more abstract ideas from the branch of mathematics known as topology. A metric space is a mathematical set along an associated function of two points within the set that defines a sense of distance or metric between them.
Year2:Vector Calculus
Vector calculus is one of the most practically relevant courses for a prospective quant to have studied. It deals with the concept of change in scalar and vector fields. Many concepts in mathematics, physics and quant finance can be modelled as fields and as such the machinery of vector calculus is highly applicable.
Year2:Ordinary Differential Equations - Non-linearity and Chaos
Differential equations are a large research area in their own right. Once beyond the first year material, which generally finishes up discussing second order linear ODE with constant coefficients, more interesting ODE found in real-life applications appear. These include the areas of mechanics, electronics and mathematical biology. Such ODE possess non-linear and chaotic behaviour.
Year2:Geometry - Non-Euclidean
The familiar geometry of everyday life is three-dimensional Euclidean geometry. In the second year students are often introduced to projective geometry, elliptic geometry and hyperbolic geometry. These geometries arise when Euclids fifth postulate is relaxed, which allows parallel lines to cross or diverge, unlike in Euclidean geometry.
Year2:Abstract Algebra
In Part 1 we saw that students will often be exposed to abstract groups through a Foundations module. In Year 2 a more thorough treatment of abstract algebra is provided, which covers groups in depth and often leads onto the study of rings.
Year2:Stochastic Processes
Stochastic Processes is generally offered as an option module at university and as such is not core. However it is clearly extremely relevant to quantitative finance particularly in the area of derivatives pricing.
Year2:Numerical Analysis
As mentioned briefly above the second year of an undergraduate mathematics degree extends the discussion of analysis to the Riemann integral, which is the usual integral that is familiar from highschool, engineering and physics. This is in contrast to the Lebesgue integral, which is discussed in Measure Theory during the third year.
Statistics is probably the most sought after quantitative skill in the commercial sector that can be studied on a mathematics degree. It provides the basis for understanding uncertainty and measuring risk, both of which are absolutely crucial to practising quants.
Year3:Complex Analysis
Complex numbers are a generalisation of real numbers motivated by the need to define the concept of 𝕚=√-1. This comes about because of the solution to particular equations, such as the familiar quadratic equation from highschool algebra, which possesses complex roots when b^2-4ac<0 in the solution to the equation ax^2+bx+c=0.
Topology is a subject of study in mathematics that attempts to determine particular properties of abstract spaces that are preserved under continuous transformations. The continuous aspect of these transformations means that only concepts such as stretching or bending are considered. If holes are teared or parts are glued together, these are not considered topological transformations.
Year3:Ring Theory
In the first and second year of a traditional undergraduate degree it is common place to study the abstract algebraic concept of a group in some depth. Groups have a huge range of mathematical and physical applications and are one of the most fundamental mathematical structures.A ring, however, is similar to a group except that rather than simply being a set with a single associated binary set operation it is a set with two binary operations, which attempt to generalise the concepts of addition and multiplication.
Year3:Fluid Dynamics
Fluid dynamics is formulated via the principle of conservation laws taken from theoretical physics. By assuming continuity of mass, momentum and energy, it is possible construct the compressible Navier-Stokes equations, which are the main equations used within fluid dynamics research.
Year3:Measure Theory
Measure Theory is usually considered a difficult course by many undergraduates. However it is an absolutely essential prerequisite for a quant who wishes to be an expert at derivatives pricing.
Year3:Linear Functional Analysis
Linear Functional Analysis is primarily concerned with extending the ideas from finite-dimensional vector spaces, learned about in Year 1, to infinite-dimensional spaces, often with some form of structural addition, such as an inner product, a norm or a topology. Such spaces are motivated by the need to identify a setting for solutions of differential and integral equations. An infinite-dimensional vector space over the real or complex numbers often provides such a setting.
Year3:Elementary Differential Geometry
Elementary differential geometry is predominantly concerned with curves and surfaces lying in three-dimensional space, that is R^3 . For curves the notions of curvature and torsion allow us to determine how a curve can twist in R^3. However, surfaces in R^3 require additional notions in order to specify their behaviour, namely the Guassian and Mean curvatures.
Year3:Partial Differential Equations
At university Partial Differential Equations (PDE) were my favourite area of study and were one of the original reasons that I eventually became a quant, namely through numerical solution of Black-Scholes type models. They are one of the most diverse and fascinating areas of applied mathematics and touch on a huge range of other mathematical topics. I cannot emphasise enough how important they are in mathematics and quantitative finance in general.
Year3:Numerical Linear Algebra
Numerical Linear Algebra is a more specialised subject for a mathematics degree, but I have included it since it was a module offered on my own undergraduate course, as well as being extremely relevant for computational finance.
Year4:Brownian Motion
Brownian Motion (or, the Wiener Process) is an indispensible concept within quantitative finance. It is widely used as a model for stock price path evolution. Hence gaining familiarity with its mathematical properties is an essential prerequisite for more advanced study in derivatives pricing.
Year4:Stochastic Analysis
Stochastic Analysis (or Stochastic Calculus) is the theory that underpins modern mathematical finance. It provides a natural framework for carrying out derivatives pricing. While quantitative finance is one of the main application areas of stochastic analysis, it also has a rich research history in the fields of pure mathematics, theoretical physics and engineering.
Year4:Stochastic Calculus for Finance
Stochatic Calulus for Finance is another course widely found at mathematics and statistics departments within the fourth year syllabus. It is usually shared by those taking a Masters in Financial Engineering.
Year4:Stochastic Optimal Control
Stochastic Optimal Control is a very useful and interesting interdisciplinary field across mathematics, physics, engineering and finance. It provides the underlying theory to certain control problems, which deal with the design of models to control continously operating dynamical systems that are subject to 'noise' in the current estimates of system state.
Year4:Statistical Modeling
Many mathematics and statistics departments provide some form of advanced statistical modeling course, often applied in nature, that goes beyond linear statistical models. Such courses often focus on the theory and application of Generalised Linear Models (GLM), which are an extension of standard linear regression to problems where the expectation of the response variable is not simply given by a linear combination of predictors.
Year4:Statistical Machine Learning
Machine learning has provided significant leaps in performance on many challenging tasks including image recognition, natural language processing and even video games competitions, that until relatively recently were deemed unsolveable via a computational approach.
Year4:Markov Chains
Markov Chains (discrete time) and Markov Processes (continous time) are stochastic processes, which describe a sequence of events, where the probability of a subsequent event occuring is only dependent upon the current state. This aspect of Markov Processes is known as the Markov Property.
Year4:High Performance Computing
High Performance Computing (HPC) is the study of efficient parallelisation of computational tasks requiring significant CPU or memory beyond that provided by a typical laptop or desktop workstation. HPC is prevalent in quantitative finance within the realms of derivatives pricing and trading simulation.