Applications

Applications for 2022 entry are now open. 


The course

A one-year (full-time), two-year (part-time) MSc programme, designed to prepare students for a wide range of careers in quantitative finance and risk management. 

Mathematical finance is a subject that is both mathematically challenging and deployed every day by sophisticated practitioners in the financial markets. Our objective is to provide you with everything you need to get into this area at a level where you can understand–and contribute to–industry practice and the latest research.

Our intake consists mainly of recent graduates in mathematical sciences and engineering seeking positions in the financial services sector. However, we welcome applications from candidates already in employment in that area who want to upgrade various aspects of their mathematical and financial expertise and expand their portfolio of skills. We usually recruit a small number of students each year from this category who fulfil our academic requirements.

Further information

Programme overview

Theoretical modules

Candidates reading for the MSc in Mathematics and Finance follow seven compulsory core modules, and choose another five from a menu of elective modules divided into three indicative streams: Derivatives Pricing, Market Microstructure and Machine Learning in Finance that are offered in autumn and spring terms. The modules cover fundamental mathematics, finance and economic background, scientific computing and statistical methodology. The summer period is devoted to the project.


Project and placements

  • A supervised thesis project takes place during the summer months after the theoretical modules have been completed.
  • Students who have achieved an acceptable level of academic competence will be offered as candidates to external sponsors.
  • These industry-based placements take place in banks, consultancies, hedge funds, insurance companies, rating agencies, or financial software companies.
  • Each project is normally based on new areas of possible interest to the sponsor, or extensions to existing lines of work.
  • An academic supervisor and the sponsor work with the student to scope out the project at the start; supervision is a joint activity between the former two.
  • Undertaking the project on site gives students a genuine insight into the reality of the financial marketplace.

Programming in C++

We regard it as essential that students become proficient in object-oriented programming. The computing environment at the college is based on wireless networking. Students must equip themselves, at their own expense, with a laptop running Windows. We will supply the software you need: Microsoft Visual Studio, Microsoft Office and Matlab, a software environment for scientific computing. The teaching in programming stretches over the autumn and spring terms and consists of lectures, laboratory sessions and a series of graded exercises which must be submitted.


Recommended reading and further information

To find out more about the course, including pre-course reading course handbooks, timetables, information on careers support offered and social events, please see the current student pages.

The part-time option

The MSc in Mathematics and Finance can be taken on a part-time basis. Students attend the same lectures as those taking the degree full-time. The courses are spread more or less evenly over two years, instead of one, and the project is taken in the second year. There is a need to attend lectures on about three days a week. Considerable flexibility is needed by those in full-time employment who pursue this option. A typical sample schedule would look like this:

Year 1

Autumn Term
  • Stochastic Processes
  • Mathematical Finance: Introduction to Option Pricing Theory
  • Python
  • Electives, from the list of Elective Modules
Spring Term
  • Simulation Methods for Finance
  • Programming in C++ II: Object oriented programming
  • Electives, from the list of Elective Modules

Year 2 

Autumn Term
  • Statistical Methods in Finance
  • Quantitative Risk Management
  • Electives, from the list of Elective Modules
Spring Term
  • Interest Rate Modelling
  • Electives, from the list of Elective Modules

Summer Period

Derivatives Pricing Stream

Stochastic Calculus for Finance

Malliavin Calculus is an extremely powerful tool in stochastic analysis, extending the classical notion of derivative to the space of stochastic processes. A certain number of results arising from this theory turn out to provide the right framework to analysis several problems in mathematical finance. The module will be divided into two parts: the first one will concentrate on developing the theoretical tools of Malliavin Calculus, including analysis on Wiener space, the Wiener chaos decomposition, the Ornstein-Uhlenbeck semigroup and hypercontractivity, the Malliavin derivative operator and the divergence operator, and Sobolev spaces and equivalence of norms. The second part of the module will focus on understanding how these tools come in handy in order to price and hedge financial derivatives, and to compute their sensitivities. (This module will not be given in 2021-22)

Topics in Derivatives Pricing

Derivatives pricing is at the core of trading and model validation, in so far as traders and quantitative analysts rely on stochastic models to build their trades and monitor their risks. The goal of this module is to introduce the technical tools needed to understand the specificities of these models and their inherent risks.

Selected Topics in Quantitative Finance

Option markets are extremely diverse, spanning several different asset classes and many pricing and hedging strategies. The goal of this module is to complement the other option-flavoured modules, focusing on the specificities of Foreign Exchange and Fixed Income markets. For each of these markets, the module will study their specific characteristics and evolutions, develop the technical tools needed to understand the pricing of derivatives, and explain how to set up trading and hedging strategies therein. A strong emphasis will be given on the actual implementation of the models and their calibration to real data.

In order to do so, the module will constantly strike a fair balance between the mathematical framework and specific tools (stochastic analysis, Fourier methods, fractional calculus), the numerical aspects (actual implementation of the models, optimization routines) and the data (calibration on real data, backward testing of hedging strategies)

Numerical Methods in Finance

Numerical Methods are at the very core of quantitative modelling, no matter which area one considers. The goal of this module is to complement the Core module on Simulation Methods to investigate other techniques that are widely spread among the financial industry. We shall investigate two popular techniques, namely PDE methods and Fourier methods.

For each case, we will start with a theoretical framework, explaining how the option pricing problem can be turn, depending on the model used, either into a PDE problem, or into a Fourier integration issue. We shall then focus our attention to the actual numerical methods needed to implement these two approaches, and test them on real models and real data.

Market Microstructure Stream

Convex Optimisation

Many optimisation problems can be reformulated as convex optimisation problems (COP). These have many highly desirable features, in particular it is possible to numerically solve them efficiently and reliably. Morever, COP have a rich, and beautiful theory behind them: convex analysis, i.e. the theory of convex functions and sets, which leads to duality techniques and optimality and complimentary conditions crucial to solving a COP. Moreover, COP also admit several important subclasses of problems with additional nice properties: Linear, Quadratic and Conical optimisation problems, to just mention a few.

This module is devoted to explaining how to identify and solve several classes of COP. To better do this, we cover some of the underlying convex analysis, e.g. we talk about subgradients and Fenchel-Moreau conjugates. In particular, we cover in some detail the geometry of linear programming, discussing for example Farkas' lemma, the Bipolar theorem and the Minkowski-Weyl characterisation of polyhedra.

Moreover, as COP occur naturally throughout mathematical finance, participants of this course will learn a spectrum of COP applications through a series of examples of practical relevance; in particular, we will cover cash flow matching, mean-variance portfolio optimization (in the way of Markowitz), robust portfolio optimisation etc.

In this module we will at times use Python and CVXOPT to numerically solve some COP. We will discuss the simplex algorithm, and its theoretical relevance to the study of Linear optimisation problems. We will also briefly talk of numerical methods for COP in more generality.

Stochastic Control in Finance

Many problems in mathematical finance (and in other areas) are essentially optimisation problems subject to random perturbations, where some controls play the role of a performance criterion. The goal of this module is to bring the main concepts and techniques from dynamic stochastic optimisation and stochastic control theory to the realm of quantitative finance. It will therefore naturally start with a theoretical part focussing on required elements of stochastic analysis, and with a motivation through several examples of control problems in Finance. We will then turn to the classical PDE approach of dynamic programming, including controlled diffusion processes, dynamic programming principle, the Hamilton-Jacobi-Bellman equation and its verification theorem. We will finally see how to derive an solve dynamic programming equations for various financial problems such as the Merton portfolio problem, pricing under transaction costs, super-replication with portfolio constraints, and target reachability problems.

Algorithmic and High-Frequency Trading

The increase in computer power over the last decades has given rise to prices being quoted and stocks being traded at an ever-increasing pace. Since humans are not able to place orders at this speed, algorithms have replaced classical traders to optimise portfolios and investments. In this module, we will study specificities of this market, and in particular, we shall develop the mathematical tools required to develop such algorithms in this high-frequency framework. The module will start with a short review of stochastic optimal control, which forms the mathematical background. We shall then move on to study optimal execution, namely how and when to place buy/sell orders in this market, both assuming continuous trading and in the context of limit and market orders. The last part of the module will be dedicated to the concept of market making and statistical arbitrage in high-frequency settings.  Pre-requisite MATH97232 Stochastic Control in Finance

Market Microstructure

The goal of the module is to develop thorough understanding of how trades occur in financial markets. The main market types will be described as well as traders’ main motives for why they trade. Market manipulation and high-frequency trading strategies have received a lot of attention in the press recently, so the module will illustrate them and examine recent developments in regulations that aim to limit them. Liquidity is a key theme in market microstructure, and the students will learn how to measure it and to recognise the recent increase in liquidity fragmentation and hidden, “dark” liquidity. The Flash Crash of 6 May 2010 will be analysed as a case study of sudden loss of liquidity. The remaining part of the module focuses on statistical analysis of market microstructure, concentrating on statistical modelling of tick-by-tick data, measurement of price impact and volatility estimation using high-frequency data.

Portfolio Management

This module gives students a foundation for quantitative portfolio management and for understanding market price determination. Key concepts include risk measurement, risk-reward trade-offs, portfolio optimization, benchmarking, equilibrium asset pricing, market efficiency, and pricing anomalies. Specific portfolio management tools include mean-variance optimization, CAPM and APT asset pricing, factor models (e.g., Fama-French), momentum strategies, and performance evaluation. The course will present essential theories and formulas and will also review important institutional and empirical facts about equity, bond, and commodity markets.

Machine Learning in Finance Stream

Algorithmic Trading and Machine Learning

The aim of the course is to present in some detail a series of models/techniques used in the algorithmic trading space. For each topic, we shall emphasize both theoretical aspects as well as practical applications. The course consists of two main blocks: 1) optimal execution theory and 2) machine learning for finance.

Optimal execution techniques are typically used by quantitative brokers to buy/sell large numbers of securities. Machine learning algorithms are often used by hedge fund and trading desks to generate trading signals, quote on exchange and hedge complex portfolios.

The basic optimal execution problem consists of an agent (e.g. a bank or a broker) who needs to buy or sell a pre-specified number of units of a given asset within a fixed time frame (e.g. an hour, a day, etc.). Assuming that the purchase or sale of the asset will have an adverse impact on its price, what is the execution policy which minimizes market impact? This problem can be formulated as a trade-off between the expected execution cost and the price risk due to exogenous factors. We shall solve the optimization problem using different types of impact models (temporary, transient, permanent) and risk functions (variance, VaR).

Machine learning techniques are becoming increasingly popular in the financial industry. For example, they are used to help predict asset prices, improve the hedging and pricing of complex portfolios. In the lectures we shall analyse in detail some of the most popular supervised learning algorithms such as LASSO/Ridge regression, logistic regression and support vector machines. We shall also introduce unsupervised learning techniques such as clustering and PCA. We will talk about issues related to model selection, overfitting and explore ways to deal with other problems such as selection bias. Trading applications will be presented during the course. Students will be requested to implement some of the models presented in the lectures in Python. (this module will not be given in 2021-22)

Advances in Machine Learning

This course is an introduction to data analysis and ‘machine learning’ techniques, a vast and hugely active area of research and applications.

The emphasis will be on methodology and each technique will be illustrated by applications in finance. Implementation of key algorithms in R, Python or MATLAB will be discussed. The course will first provide an overview clarifying the distinction between estimation and statistical learning. A major component of the course consists of learning methods for linear relations, regression and its refinements, and principal components analysis. The course will then proceed to non-linear learning, where statistical classification methods, kernel estimation will be introduced. The remainder of the course deals with statistical learning for real-time algorithms, focusing on online learning and bandits.

Advanced Topics in Data Science: Rough Path in Machine Learning

Rough path theory was developed in the 1990s in order to understand the structure and information content of a given path (be it a financial time series, a hand-drawn character or the route taken by a vehicle). It turned out to be one of the key developments in stochastic analysis over the past 20 years, and has allowed for a better understanding (and new proofs) to many problems in this field. The goal of this module is to provide students with a flavour of this powerful theory and to understand how it can efficiently be applied in machine learning, one of the fast-developing techniques in the financial industry nowadays. One of the key elements in this exploration is the so-called signature of a path, of which we shall study the algebraic properties, the faithfulness, as well as the inversion and asymptotic properties. We shall further see how this signature is in fact a feature set in machine learning, and illustrate these results in mathematical finance (in particular to predict financial time series), as well as in other areas (handwriting recognition, computer vision, classification problems in medical data).

Deep Learning

Deep learning is subfield of Machine Learning that applies deep neural nets to represent and predict complex data. It has recently revolutionised several areas such as image recognition and artificial intelligence and it is currently gaining traction also in the financial industry. The module will first introduce the multi-layer neural nets and explain their universal approximation property. Subsequently, the module proceeds to the training of neural nets, starting from the derivation of the gradient of a neural net and its evaluation through backpropagation, culminating in the stochastic gradient descent and related modern optimisation methods. Techniques to avoid overfitting in training are also elucidated. The remainder of the module focuses on the practical implementation and training of deep neural nets using Keras and TensorFlow, with examples in computational and statistical finance. Time permitting, elements of recurrent neural nets are also sketched.

Quantum Computing in Finance

Quantitative Finance is a rapidly changing environment, and the financial industry is always on the lookout for new techniques and new technologies able to harness the rise of big data and the availability of computing power. Quantum computing, though not a recent field, has gained huge popularity in the past few years with the development of small-scale quantum computers and quantum annealers. These have in turn pushed for new algorithms, hybrid between classical and quantum, and tailored for such computers. The financial industry is now looking at such developments and there is a common agreement that this will be one of the leading advances in the coming decade.

 The goal of this new Elective (so far not given in any similar MSc programmes around the world) is to introduce students to this new technology and these new algorithms and show them how they can be used to solve financial problems, in particular

  • For portfolio optimisation,
  • For data generation,
  • For Machine learning and neural network.

The module will strike a fair balance between theoretical concepts of Quantum Computing, their implementation (in Python using IBM’s Qiskit framework) and their application to real financial problems. 

Data Science for Fintech Regtech and Suptech: Methodological Foundations and Key Applications 

Advances and innovations in computational technology have allowed data scientists to explore and understand increasingly complex financial problems. However, emerging opportunities require financial professionals to update their analytical skills and embrace new technologies, methods, and data sources. The goal of this module is to provide students with an interest in quantitative finance an overview of the evolution of data science in the context of Fintech, RegTech and Suptech, as well as to equip them with the skills to apply new analytical techniques to real world challenges. The emphasis will be on practical applications; and to this end, the module will be led by industry experts and include regular hands-on exercises involving the use of advanced data analytics

 

 

Alumni Testimonials

 JordanJordan Anaya (MSc Mathematics and Finance 2018-2019)

Quant Analyst at Velador Associates, a data science firm specialized in financial litigation services

Why did you choose the MSc in Mathematics and Finance at Imperial College?

After 4 years working as Derivatives Pricing and Accounting consultant I decided I wanted to deepen my knowledge in some areas of Mathematics and Finance that I did not cover in my undergrad and that would help me go further in my career. I chose the MSc at Imperial as it is one of the best in the world with the latest research topics in the industry and with the Elective modules that allow you to tailor the course on the topics you are most interested.

What did you enjoy most during the MSc?

I enjoyed a lot the Practitioners' Lectures as they share with you their latest research on topics applied in practice, as well as the opportunity to do the thesis project with an external supervisor from the industry. 

What do you think are the strongest points of the MSc?

The lecturers are amazing, they engage with the students throughout the lectures, and you can notice their passion and in-depth knowledge. On top of that they are part of one of the world's leading research groups in Mathematics and Finance and they are always helpful if you approach them. Also, the focus of the Masters in coding helps you improve your skills which is very useful in practice.

What piece of advice would you give to anyone thinking of applying to the programme?

Do some study before the Masters as it is a course from which you learn a lot but you have to review a large quantity of information for the exams. Practice to improve your coding skills as you will use it for the projects and will make them easier.


 

ceciliaCécilia Auburn (MSc Mathematics and Finance 2019-2020)

PhD student with the Econophysics Chaire in Polytechnique Paris, working on endogeneous liquidity crisis

Why did you choose the MSc in Mathematics and Finance at Imperial College?

When looking at quant job offers from hedge funds or banks, they ask for many different skills and the MSc Mathematics and Finance of Imperial tackles all of those, with a bonus: it is well connected with the financial industry.

What did you enjoy most during the MSc?

I really enjoyed the Careers meetings. First, there are food and drinks! But mostly, we get to meet HR, Quants, and have a privileged contact with them to discuss about their jobs and the recruitment processes. It is also a great moment to share with the other MSc students!

What do you think are the strongest points of the MSc?

Probably, the availability of its people. It is really easy to get a meeting to discuss or ask questions with anyone of the MSc programme. It is really helpful.

What piece of advice would you give to anyone thinking of applying to the programme?

Take part in as many MSc events you can, there are plenty of opportunities, you do not want to miss them!


ElisaElisa Barbaro (MSc Mathematics and Finance 2013-2014)

Exotics trader at Citigroup

 

 Why did you choose the MSc in Mathematics and Finance at Imperial College?

The teaching quality had good reputation and at the same time I thought it would help me bridge the gap between academia and work (because of the work placement project and being in London).

What did you enjoy most during the MSc?

The work placement project.

What do you think are the strongest points of the MSc?

I think it can vary. In my case, during my BSc, I studied mainly the more theoretical aspects of Maths/Engineering, because in Italy (except for a couple of Business Schools) students don’t usually get taught how to apply for a job or what opportunities one can look for. The strongest point in my view was to bridge this gap between academia and work (especially in finance). I remember other students didn’t know how to code and the course helped them for that, which is also important. It depends on each individual situation. 

What piece of advice would you give to anyone thinking of applying to the programme?

The course provides you with a lot of information and events, across a wide range of topics. You will not achieve in-depth knowledge on any specific topic, but likely, you don’t need it, as it is more important to know what various subjects involve and what ultimately you like to pursue. You might not learn enough concepts to interview for jobs as a hardcore quant, but I think many students start their career thinking that they want to be a quant to then discover that they prefer other roles, and this course gives you the chance to apply to most jobs out there. You can also continue to study more and still be a hardcore quant some years later!


 

GeorgeGeorge Lambert (MSc Mathematics and Finance 2012-2013)

Founder of Lambert Labs, a Python-focused software development agency based just around the corner from Imperial in Earls Court. We build software solutions for companies ranging in size from start-ups all the way through to global corporations. Our solutions are very much cross-industry too; we have clients in the education, finance, legal, hospitality and logistics sectors.

Why did you choose the MSc in Mathematics and Finance at Imperial College?

I worked as a secondary school Maths teacher for three years after doing a Maths undergraduate degree but was looking for a way to combine my academic/career experience in Maths with a passion for technology and an interest in Finance. Training for a career as a quant seemed like the natural choice and after researching similar courses at various UK and international universities, I decided that the MSc in Mathematics and Finance at Imperial looked like the perfect course for me. 

What did you enjoy most during the MSc?

As part of the MSc I did a project internship at Citigroup over the summer. This gave me exposure to how quantitative methods are used in industry on a day-to-day basis. This exposure was absolutely invaluable at the time, and led to getting a part-time job Junior Quantitative Analyst role at Citigroup while studying part-time for a PhD at Imperial (unfortunately a serious sports injury put a stop to this role and the PhD at a later date!).

What do you think are the strongest points of the MSc?

There isn’t one single thing that I want to put my finger on and say, ‘this area was the strongest’. However, the course as a whole was excellent. There is no doubt in my mind that I learnt more in my year on the MSc than i any other year in my life. The combination of lectures, office hours, exams, library study, coursework and research projects is intense, but extremely valuable.

What piece of advice would you give to anyone thinking of applying to the programme? 

Do as much research as possible before applying. This could involve reading about Mathematics and Finance in academia or Mathematics and Finance in industry, or both. If you are accepted onto the programme and haven’t coded before, it is a good idea to learn some programming basics before you start the course, just so you can hit the ground running!


AitorAitor Muguruza (MSc Mathematics and Finance 2015-2016)

 Head of Quant Modelling at a Hedge Fund

 

Why did you choose the MSc in Mathematics and Finance at Imperial College?

Among the different choices in the EU and UK, London is probably the strongest Quant Hub in the continent. Imperial offered the best curriculum and the best job market.

What did you enjoy most during the MSc?

I enjoyed the balance and depth on both numerical and theoretical courses.

What do you think are the strongest points of the MSc?

A constantly updated curriculum that is always on top of the market requirements. When I did my MSc few years ago, Python was picking up in the industry which I had already been exposed to through different modules in the course. The same story applies to Machine Learning courses and other topics, it's not just about basics also what is currently relevant.

What piece of advice would you give to anyone thinking of applying to the programme?

I would say there is a lot to learn to be a Quant, a lot of asset classes, diverse roles in different technology areas. I believe it is helpful to start narrowing down what your ideal role is, to be able to make the most of the MSc in terms of elective modules. This does not mean that by day 1 you should know what you are aiming at, but it is certainly important to do some market research before and during the first term.


Niklas

Niklas Walter (MSc Mathematics and Finance 2019-2020)

PhD student in Financial Mathematics at the University of Munich in Germany focusing on rough volatility

 

Why did you choose the MSc in Mathematics and Finance at Imperial College?

I think that the programme at Imperial perfectly combines academic excellence and the opportunity to get insights about how the taught knowledge is in fact applied in the industry. Moreover, the university’s location in the heart of Europe’s centre of the financial industry is a daily motivation to learn new things.

What did you enjoy most during the MSc?

I think one of the most important aspects of the programme is the social one. Studying together with a very international cohort is quite enrichened and interesting. Moreover, the entire staff of Imperial and in particular of the MSc is very motivating and supportive. Lastly, the weekly Career events followed by some more casual pub visits are helpful to start building a professional network and to gain interesting insides about current topics driving the quant world. 

What do you think are the strongest points of the MSc?

At first, I want to mention the weekly Career events again. They are indeed very well organised and a good opportunity to find an industry partner for one’s final project. In addition, the range of Elective courses is quite broad and oriented towards the current needs and topics in the industry. Lastly, the programme offers a good blend between pure mathematical techniques and their real-world applications.

What piece of advice would you give to anyone thinking of applying to the programme?

One advice is to reach out to former students to get a better impression of the programme. In general, it is important to carefully look at the programme structure and course contents to get a clear idea about why you want to apply and if it is even the right MSc for you.

Frequently asked questions (FAQ)

Read answers to our frequently asked questions from prospective students.