Financial Mathematics BSc

Core modules:

Mathematics Core

Mathematics Core covers topics which continue school mathematics and which are used throughout the degree programmes: calculus and linear algebra, developing the framework for higher-dimensional generalisation.  This material is central to many topics in subsequent courses.  At the same time, small-group tutorials with the Personal Tutor aim to develop core skills, such as mathematical literacy and communication, some employability skills and problem-solving skills.

40 credits

Computing and Analysis

This module collects together a number of the most useful topics for future modules in financial mathematics. The first semester will cover some computing, using Python to write simple programmes and LaTeX to write mathematical documents.  The second semester is given to analysis, a branch of pure mathematics with applications in financial mathematics.  The two semesters are independent of each other.

20 credits

Probability and Data Science

Probability theory is branch of mathematics concerned with the study of chance phenomena. Data science involves the handling and analysis of data using a variety of tools: statistical inference, machine learning, and graphical methods. The first part of the module introduces probability theory, providing a foundation for further probability and statistics modules, and for the statistical inference methods taught here. Examples are presented from diverse areas, and case studies involving a variety of real data sets are discussed. Data science tools are implemented using the statistical computing language R.

20 credits

Economics pathway:

Economic Analysis and Policy

This is a compulsory module for all single and dual honours students in Economics. The module provides students with an introduction to microeconomic and macroeconomic analysis together with examples of their application in order to develop students' understanding of the roles of both in economic policy making.

40 credits

Management pathway:

Introduction to Financial Accounting

Financial Accounting is concerned with the ways in which the financial transactions of a business are recorded and summarised in financial statements. This module provides an introduction to the construction of financial statements and an understanding and evaluation of the principles and concepts on which they are underpinned. Once the principles have been established, the module further develops the practical understanding of the preparation of financial statements and focuses on the preparation, interpretation and limitations of company financial statements and the regulatory framework in which they are prepared.

20 credits

Foundations in Financial Management

This module aims to create a foundation of knowledge in the subject area of financial management, creating the required framework of skill and knowledge for financial decision making and to provide a base of knowledge for the related modules in levels 2 and 3. This module will achieve this by introducing the essential principles, theories and calculations within financial management. It will also introduce the contemporary issues and developments in financial markets. The module design and content expects to at least cover the contents of foundation level financial management related module syllabus of professional accounting bodies.

20 credits

Core modules:

Mathematics Core II

Building on Level 1 Mathematics Core, Mathematics Core II will focus on foundational skills and knowledge for both higher mathematics and your future life as a highly skilled, analytically-astute worker.  Mathematical content will focus on topics that are vital for all areas of the mathematical sciences (pure, applied, statistics), such as vector calculus and linear algebra.  This will help develop your analytic and problem solving skills.  Alongside this, you will continue to develop employability skills, building on Level 1 Core.  Finally, there will be opportunities to learn and reflect on social, ethical, and historical aspects of mathematics, which will enrich your understanding of the importance of mathematics in the modern world.

30 credits

Stochastic Modelling

Many things about life are unpredictable.  Consequently, it often makes sense to incorporate some randomness in mathematical models of natural and physical processes. Such models are called 'stochastic models' and are the study object of this module.  We will learn about a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might include the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. We will cover various techniques for analysis of such models, setting the student up for further study of stochastic processes and probability at levels 3 and 4.

10 credits

Statistical Inference and Modelling

Statistical inference and modelling are at the heart of data science, a field of rapidly-growing importance in the modern word.  This module develops methods for analysing data, and provides a foundation for further study of probability and statistics at higher Levels.  You will learn about a range of standard probability distributions beyond those met at Level 1, including multivariate distributions. You will learn about sampling theory and summary statistics, and their relation to data analysis. You will discover how to parametrise various types of statistical model, learn techniques for determining whether one model is 'better' than another for understanding a dataset, and learn how to ascertain how good a statistical model is at explaining trends in data. The software package R will be used throughout.

20 credits

A student will take 20 credits from either this group:

Analysis and Algebra

This module will build on the theory built in Level 1 'Foundations of Pure Mathematics', focusing on the twin pillars of analysis and algebra.  These are not only fundamental for pure mathematics at higher levels, but provide rigorous theory behind core concepts that are used throughout the mathematical sciences.  Whilst to some extent you have been doing analysis and algebra since you were at school, here you will be going much deeper.  You will examine why familiar tools, like differentiation and integration, actually work.  Familiar objects, such as vectors, differential operators, and matrices, will be unpacked; powerful, formal properties of these objects proved.  Ultimately, this rigorous foundation will enable you to extend these tools and concepts to tackle a far greater set of problems than before.

20 credits

Or this group:

Scientific Computing

The ability to programme is a central skill for any highly-numerate person in the 21st century.  This module builds on skills learned at Level 1 in 'Mathematical Investigation Skills' by developing skills in computer programming and independent investigation. You will learn how to solve various mathematical problems in programming languages commonly-used by mathematicians, for example Python. You will learn basic computational and visualisation methods for exploring numerical solutions to equations (including differential equations), and then apply this knowledge to explore the behaviour of example physical systems that these equations might model.

10 credits

Analysis

This module will build on the theory built in Level 1 'Foundations of Pure Mathematics', focusing specifically on analysis.  This provides rigorous theory behind core concepts that are used throughout the mathematical sciences.  Whilst to some extent you have been doing analysis since you were at school, in the form of calculus, here you will be going much deeper.  You will examine why familiar tools, like differentiation and integration, actually work.  This allows powerful, formal properties of these objects to be proved.  Ultimately, this rigorous foundation will enable you to extend these tools and concepts to tackle a far greater set of problems than before.

10 credits

Or this group:

Mathematics and Statistics in Action

In this project module, you will investigate one or more case studies of using mathematics and statistics for solving empirical (i.e. 'real world') problems. These case studies will illustrate the process of mathematical and statistical modelling, whereby real-world questions are translated to mathematical and/or statistical questions. Students will see how techniques learned earlier in their degree can be used to explore these problems. There will be a mix of individual and group projects to choose from, and some projects may  involve the use of R or Python, but 'MAS116 Mathematical Investigations Skills' is not a prerequisite. Students will be expected to work independently (either individually or in a small group). However, the topic and scope of each piece of project work will be clearly defined by the lecturer in charge of the topic.

10 credits

Analysis

This module will build on the theory built in Level 1 'Foundations of Pure Mathematics', focusing specifically on analysis.  This provides rigorous theory behind core concepts that are used throughout the mathematical sciences.  Whilst to some extent you have been doing analysis since you were at school, in the form of calculus, here you will be going much deeper.  You will examine why familiar tools, like differentiation and integration, actually work.  This allows powerful, formal properties of these objects to be proved.  Ultimately, this rigorous foundation will enable you to extend these tools and concepts to tackle a far greater set of problems than before.

10 credits

A student will take 40 credits from either the economics pathway:

Intermediate Microeconomics

This module builds on Level 1 modules in microeconomics and mathematical economics, using the mathematical training to allow a more rigorous investigation of the principles of microeconomics. It aims to develop an understanding and ability to undertake economic analysis of models of the behaviour and interaction of economic agents (consumers, firms and government) in a market economy, the functioning of different types of industries, decision making under uncertainty and economic welfare.

20 credits

Intermediate Macroeconomics

The aims of this course are to provide firm grounding in the analytical tools of modern macroeconomics; to use these tools to understand critically the conduct of economic policy nationally and internationally. The course builds on level 1 modules in macroeconomics. The main subject areas covered are: Basic macroeconomic models: consumption/leisure choice, closed economy one period-macro models, models of search and unemployment; Savings, investment and government deficits: consumption/savings choice (two-period model), credit market imperfections, real intertemporal model with investment; Money and business cycles: flexible price models, New Keynesian economics (sticky prices), inflation; International macroeconomics: international trade, money in open economy; Economic growth: Malthus and Solow growth models, convergence, endogenous growth model.

20 credits

Or the management pathway:

Introduction to Corporate Finance and Asset Pricing

This course builds on the concepts developed in MGT212. The course focuses on the more quantitative and advanced aspects of finance and is aimed at those students who intend to specialize in finance. The purpose of the course is to give a solid foundation in principles of corporate finance and asset pricing to understand and analyze the major issues affecting the financial policies of corporations. More specifically, the following topics will be dealt with: the time value of money, capital budgeting techniques, cash flows, risk/return trade-offs, portfolio theory, market efficiency, capital structure, payout policy, and option pricing. The course is a prerequisite for the final-year finance modules MGT321 (Corporate Finance) and MGT375 (Introduction to Financial Derivatives).

20 credits

Financial Management

This unit takes the key themes and techniques that were introduced in MGT141, Introduction to Financial Management, and locates them in their institutional and intellectual setting to enable students to reflect critically on understandings of financial institutions and phenomena. The resulting understanding and skills of critique will enhance students' capabilities to reflect on the more specialist bodies of knowledge encountered in financial management and finance units in subsequent semesters. The unit uses a combination of conventional lectures to familiarise students with ideas and tutorials in which students are encouraged to show their understanding of and critically evaluate content of lectures.

20 credits

Core modules:

Stochastic Processes and Finance

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.

20 credits

Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the Black-Scholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the Black-Scholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance.

10 credits

Time Series

Time series are observations made in time, for which the time aspect is potentially important for understanding and use. The course aims to give an introduction to modern methods of time series analysis and forecasting as applied in economics, engineering and the natural, medical and social sciences. The emphasis will be on practical techniques for data analysis, though appropriate stochastic models for time series will be introduced as necessary to give a firm basis for practical modelling. Appropriate computer packages will be used to implement the methods.

10 credits

Optional maths modules include:

Bayesian Statistics

This module develops the Bayesian approach to statistical inference. The Bayesian method is fundamentally different in philosophy from conventional frequentist/classical inference and is becoming the approach of choice in many fields of applied statistics. This course will cover both the foundations of Bayesian statistics, including subjective probability, inference, and modern computational tools for practical inference problems, specifically Markov Chain Monte Carlo methods and Gibbs sampling. Applied Bayesian methods will be demonstrated in a series of case studies using the software package R.

10 credits

Codes and Cryptography

The word 'code' is used in two different ways. The ISBN code of a book is designed in such a way that simple errors in recording it will not produce the ISBN of a different book. This is an example of an 'error-correcting code' (more accurately, an error-detecting code). On the other hand, we speak of codes which encrypt information - a topic of vital importance to the transmission of sensitive financial information across the internet. These two ideas, here labelled 'Codes' and 'Cryptography', each depend on elegant pure mathematical ideas: codes on linear algebra and cryptography on number theory. This course explores these topics, including the real-life applications and the mathematics behind them.

10 credits

Combinatorics

Combinatorics is the mathematics of selections and combinations. For example, given a collection of sets, when is it possible to choose a different element from each of them? That simple question leads to Hall's Theorem, a far-reaching result with applications to counting and pairing problems throughout mathematics.

10 credits

Complex Analysis

It is natural to use complex numbers in algebra, since these are the numbers we need to provide the roots of all polynomials. In fact, it is equally natural to use complex numbers in analysis, and this course introduces the study of complex-valued functions of a complex variable. Complex analysis is a central area of mathematics. It is both widely applicable and very beautiful, with a strong geometrical flavour. This course will consider some of the key theorems in the subject, weaving together complex derivatives and complex line integrals. There will be a strong emphasis on applications. Anyone taking the course will be expected to know the statements of the theorems and be able to use them correctly to solve problems.

10 credits

Game Theory

The module will give students the opportunity to apply previously acquired mathematical skills to the study of Game Theory and to some of the applications in Economics.

10 credits

Generalised Linear models

This module introduces the theory and application of generalised linear models. These models can be used to investigate the relationship between some quantity of interest, the 'dependent variable', and one more 'explanatory' variables; how the dependent variable changes as the explanatory variables change. The term 'generalised' refers to the fact that these models can be used for a wide range of different types of dependent variable: continuous, discrete, categorical, ordinal etc. The application of these models is demonstrated using the programming language R.

10 credits

Graph Theory

A graph is a simple mathematical structure consisting of a collection of points, some pairs of which are joined by lines. Their basic nature means that they can be used to illustrate a wide range of situations. The aim of this course is to investigate the mathematics of these structures and to use them in a wide range of applications.

10 credits

Knots and Surfaces

The course studies knots, links and surfaces in an elementary way. The key mathematical idea is that of an algebraic invariant: the Jones polynomial for knots, and the Euler characteristic for surfaces. These invariants will be used to classify surfaces, and to give a practical way to place a surface in the classification. Various connections with other sciences will be described.

10 credits

Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of 'Big Data', statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.

10 credits

Probability with Measure

Probability is a relatively new part of mathematics, first studied rigorously in the early part of 20th century. This module introduces the modern basis for probability theory, coming from the idea of 'measuring' an object by attaching a non-negative number to it. This might refer to its length or volume, but also to the probability of an event happening. We therefore find a close connection between integration and probability theory, drawing upon real analysis. This rigorous theory allows us to study random objects with complex or surprising properties, which can expand our innate intuition for how probability behaves. The precise material covered in this module may vary according to the lecturer's interests.

10 credits

Medical Statistics

This course comprises sections on Clinical Trials and Survival Data Analysis. The special ethical and regulatory constraints involved in experimentation on human subjects mean that Clinical Trials have developed their own distinct methodology. Students will, however, recognise many fundamentals from mainstream statistical theory. The course aims to discuss the ethical issues involved and to introduce the specialist methods required. Prediction of survival times or comparisons of survival patterns between different treatments are examples of paramount importance in medical statistics. The aim of this course is to provide a flavour of the statistical methodology developed specifically for such problems, especially with regard to the handling of censored data (eg patients still alive at the close of the study). Most of the statistical analyses can be implemented in standard statistical packages.

10 credits

Operations Research

Mathematical Programming is the title given to a collection of optimisation algorithms that deal with constrained optimisation problems. Here the problems considered will all involve constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable; special algorithms have to be developed. The module considers not only the solution of such problems but also the important area of post-optimality analysis; i.e. given the solution can one answer questions about the effect of small changes in the parameters of the problem (such as values of the cost coefficients)?

10 credits

Sampling Theory and Design of Experiments

The results of sample surveys through opinion polls are commonplace in newspapers and on television. The objective of the Sampling Theory section of the module is to introduce several different methods for obtaining samples from finite populations. Experiments which aim to discover improved conditions are commonplace in industry, agriculture, etc. The purpose of experimental design is to maximise the information on what is of interest with the minimum use of resources. The aim of the Design section is to introduce some of the more important design concepts.

10 credits

Economics pathway:

Advanced Microeconomics

This module is designed to further develop students' understanding of core microeconomic principles by exploring a number of advanced topics in microeconomics. The course material will be predominately theoretical with a substantial mathematical component and some evaluation of empirical evidence. Indicative topics include: decision-making under uncertainty; insurance markets, principal-agent theory, risk aversion and risky asset holdings; cooperative and non-cooperative bargaining; economics of sporting contests.

20 credits

Advanced Macroeconomics

This module is designed to cover topics which illustrate and amplify the core teaching in macroeconomics at level two. Topics include Dynamic general equilibrium theory of consumption and saving, Consumption theory, Macroeconomic Risk, Real business cycle and fiscal policy, Financial frictions and credit constraints, Nominal economy and monetary policy, Economic growth and finance.

20 credits

Further Econometrics

This module is designed to introduce students to a number of important advanced topics in econometrics. The aims of the module are to provide: an overview of modern econometric methodology; an introduction to further econometric techniques; and an introduction to applied econometric research methods. The module will cover topics in both microeconometrics and times series econometrics.

20 credits

Modern Finance

The aim of this module is to introduce some of the main principles of modern finance. This is an analytical module which reflects the quantitative nature of the subject and each topic is developed from first principles. The topics covered include: the time value of money and its applications; risk return and diversification; introduction to portfolio selection; the capital asset pricing model (CAPM) and its use; and an introduction to the role of utility theory in finance and company capital structure. The aims of the module are to: Provide an introduction to portfolio theory, i.e., the concept of financial risk and behaviour of rational, risk-averse investors; Leading to a general equilibrium picture of financial asset returns and prices; Explore corporate financial decision making in the major areas of Capital Structure (the mix of equity and debt financing used to finance the firm’s investments); Introduce students to concepts of Stock Market Efficiency and Option Pricing, considering in particular alternative pricing models.

20 credits

Management pathway:

Modern Finance

The aim of this module is to introduce some of the main principles of modern finance. This is an analytical module which reflects the quantitative nature of the subject and each topic is developed from first principles. The topics covered include: the time value of money and its applications; risk return and diversification; introduction to portfolio selection; the capital asset pricing model (CAPM) and its use; and an introduction to the role of utility theory in finance and company capital structure. The aims of the module are to: Provide an introduction to portfolio theory, i.e., the concept of financial risk and behaviour of rational, risk-averse investors; Leading to a general equilibrium picture of financial asset returns and prices; Explore corporate financial decision making in the major areas of Capital Structure (the mix of equity and debt financing used to finance the firm’s investments); Introduce students to concepts of Stock Market Efficiency and Option Pricing, considering in particular alternative pricing models.

20 credits

Corporate Finance

The course unit covers more advanced topics in corporate finance - such as financing and investment decisions under asymmetric information - and valuation techniques for investment appraisal - such as real option pricing. Some of the fundamental assumptions underlying corporate finance such as the efficient market hypothesis are also challenged and an alternative approach to finance, behavioural finance, is reviewed. Financial operations such as mergers and acquisitions and initial public offerings are also discussed. As this course unit is highly quantitative, it requires a good knowledge of the basic mathematical concepts (e.g. probability calculus and derivatives), statistics (e.g. regression analysis, normal distributions and variance analysis) and the financial concepts reviewed in MGT230.

20 credits

Financial Derivatives

Over the last thirty years, the worldwide derivatives market has grown enormously in size and importance. This growth is due in part to the long-term consequences of the now famous option pricing formula developed by Black, Scholes and Merton and published in 1973 and the increase in the volatility of many financial instruments over the last 30 years. Futures and options, which are both derivative securities, are increasingly used by many participants in financial markets. This includes bankers, fund managers, security and currency traders in the world's major financial centres, but also increasingly extends to the finance departments of public and private sector organizations. This module aims to provide an introduction to the pricing and use of some of the basic types of derivative securities. Reflecting the subject, the module is analytical in nature. All concepts are taught from first principles. The course is self-contained to a large extent and includes lectures on the underlying financial economics as well as necessary mathematics and statistics.

20 credits