Financial Mathematics BSc

Core modules:

Introduction to Probability and Statistics

The module provides an introduction to the fields of probability and statistics, which form the basis of much of applicable mathematics and operations research. The theory behind probability and statistics will be introduced, along with examples occurring in diverse areas. Some of the computational statistical work may make use of the statistics package R.

20 credits

Mathematics Core 1

The module explores topics in mathematics which will be used throughout many degree programmes. The module will consider techniques for solving equations, special functions, calculus (differentiation and integration), differential equations, Taylor series, complex numbers and finite and infinite series.

20 credits

Mathematics Core II

The module continues the study of core mathematical topics begun in MS4F1015, which will be used throughout many degree programmes. The module will discuss 2-dimensional co-ordinate geometry, discussing the theory of matrices geometrically and algebraically, and will define and evaluate derivatives and integrals for functions which depend on more than one variable, with an emphasis on functions of two variables.

20 credits

Numbers and Groups

The module provides an introduction to more specialised Pure Mathematics. The first half of the module will consider techniques of proof, and these will be demonstrated within the study of properties of integers and real numbers. The second semester will study symmetries of objects, and develop a theory of symmetries which leads to the more abstract study of groups.

20 credits

Optional modules:

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

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

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

Core modules:

Advanced Calculus and Linear Algebra

Advanced Calculus and Linear Algebra are basic to most further work in pure and applied mathematics and to much of statistics. This course provides the basic tools and techniques and includes sufficient theory to enable the methods to be used in situations not covered in the course.
The material in this course is essential for further study in mathematics and statistics.

20 credits

Analysis

This course is a foundation for the rigorous study of continuity, differentiation and integration of functions of one real variable. As well as providing the theoretical underpinnings of calculus, we develop applications of the theory and examples that show why the rigour is needed, even if we are focused on applications.

The material in this course is vital to further studies in metric spaces, measure theory, parts of probability theory, and functional analysis.

20 credits

Statistical Inference and Modelling

This unit develops methods for analysing data, and provides a foundation for further study of probability and statistics at Level 3. It introduces some standard distributions beyond those met in MAS113, and proceeds with study of continuous multivariate distributions, with particular emphasis on the multivariate normal distribution. Transformations of univariate and multivariate continuous distributions are studied, with the derivation of sampling distributions of important summary statistics as applications. The concepts of likelihood and maximum likelihood estimation are developed. Data analysis is studied within the framework of linear models. There will be substantial use of the software package R.

20 credits

Probability Modelling

The course introduces a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might be the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. The aim is to familiarize students with an important area of probability modelling.

10 credits

Optional modules:

Econometrics

The aim of this module is to consolidate the statistical theory covered in earlier modules for economists, introduce students to the role of econometrics in economic analysis and to give an introduction to the conduct of empirical work in econometrics.

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

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

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

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

Career Development Skills

This unit will equip students with the necessary skills to support them in gaining employment upon graduation. Students will learn how to construct covering letters, CV writing and complete applications to enhance their success when applying for jobs. Skills such as how to communicate mathematics to non-mathematicians and the need for attention to detail will also be introduced.

10 credits

Mathematics and Statistics in Action

This module will demonstrate, in a series of case studies, the use of applied mathematics, probability and statistics in solving a variety of real-world problems. The module 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, as well as simple computer programming, can be used to explore these problems. There will be a mix of individual and group projects, and some projects will involve the use of R or Python, but MAS115 is not a prerequisite.

10 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

Metric Spaces

This unit explores ideas of convergence of iterative processes in the more general framework of metric spaces. A metric space is a set with a distance function which is governed by just three simple rules, from which the entire analysis follows. The course follows on from MAS207 'Continuity and Integration', and adapts some of the ideas from that course to the more general setting. The course ends with the Contraction Mapping Theorem, which guarantees the convergence of quite general processes; there are applications to many other areas of mathematics, such as to the solubility of differential equations.

10 credits

Optional modules:

Advanced Macroeconomics

This unit is designed to cover topics which illustrate and amplify the core teaching in macroeconomics at Level 2. 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

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

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

Further Econometrics

This module is designed to introduce students to a number of important topics in econometrics. The aims of the module are to provide an introduction to further econometric techniques, an overview of modern econometric methodology 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 behavioural financial decision making, in particular the effects of framing and loss aversion on the decision to invest in stocks, and the decision as to which stocks to buy and sell, and when;
Provide students with an understanding of the practical operations and economic rationale underlying financial markets;
Introduce students to current issues in futures market analysis.

20 credits

Practical and Applied Statistics

The overall aim of the course is to give students practice in the various stages of dealing with a real problem: objective definition, preliminary examination of data, modelling, analysis, computation, interpretation and communication of results. It could be said that while other courses teach how to do statistics, this teaches how to be a statistician. There will be a series of projects and other exercises directed towards this aim. Projects will be assessed, but other exercises will not.

20 credits

Applied Probability

The unit will link probability modelling to Statistics. It will explore a range of models that can be constructed for random phenomena that vary in time or space - the evolution of an animal population, for example, or the number of cancer cases in different regions of the country. It will illustrate how models are built and how they might be applied: how likelihood functions for a model may be derived and used to fit the model to data, and how the result may be used to assess model adequacy. Models examined will build on those studied in MAS275

10 credits

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

Differential Geometry

What is differential geometry? In short, it is the study of geometric objects using calculus. In this introductory course, the geometric objects of our concern are curves and surfaces. Besides calculating such familiar quantities as lengths, angles and areas, much of our focus is on how to measure the 'curvature' of a geometric object. The story is relatively simple for curves, but naturally becomes more involved for surfaces - and more interesting too.

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

Group Project

This unit will provide students with opportunities to improve their transferable skills by working in groups of (normally) four students to investigate a mathematical project topic. Students will be expected to register for SOM369 in pre-formed groups of four. With the aid of the Library and the internet each group will produce a (single) written account of the group's investigations into the topic, and contribute to an oral presentation of their work. Topics will be proposed by members of staff, but groups may propose their own. The module Coordinator will provide guidance about working in groups, and on appropriate techniques for the written and oral presentation of mathematical topics.

10 credits

History of Mathematics

The course aims to introduce the student to the history of mathematics. The topics discussed are Egyptian and Babylonian mathematics, early Greek mathematics, Renaissance mathematics, and the early history of the calculus.

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

Measure and Probability

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a useful precursor or companion course to the Level 4 courses MAS436 (Functional Analysis) and MAS452 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas

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

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

Topics in Number Theory

In this module we study intergers, primes and equations. Topics covered include linear and quadratic congruences, Fermat Little Theorem and Euler's Theorem, the RSA cryptosystem, Quadratic Reciprocity, perfect numbers, continued fractions and others.

10 credits