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 Financial Mathematics MSci Hons (GN1H)
Financial Mathematics MSci Hons  2019 Entry
UCAS Code
GN1H
Entry Year
2019
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 4 Year(s)
Course Overview
Financial mathematicians are interested in making good decisions in the face of uncertainty. Through our rigorous Masters programme, you will develop advanced skills and knowledge to become a major decision maker and influencer in your chosen career.
Financial maths is about learning to interpret and manipulate data to draw valuable conclusions and make predictions. This is particularly important in business and finance, but the skills gained from this programme are widely applicable and highly sought after. Over the first three years, you will be able to draw on expertise from three specialist departments: Mathematics and Statistics; Accounting and Finance; and Economics. Studying at Masterslevel, you will gain an advanced understanding of the discipline and will develop a greater breadth and depth of financial and mathematical expertise.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problemsolving and quantitative reasoning skills. During this year, you will also receive an introduction to accounting and finance, which will provide you with a comprehensive understanding of the concepts and techniques of the discipline, including financial accounting, managerial finance, and financial statement analysis.
In the second year, you will further develop your knowledge in analysis, probability and statistics. During this year, you will also begin to explore advanced finance and will benefit from modules in economics at managerial level. These will provide you with insight into micro and macroeconomics, and their related issues. You will develop realworld knowledge and experience by applying your analytical skills, interpreting events and evaluating policies. This will be instrumental in developing a career related to banking, finance, international payments and exchange rates, and monetary policy.
During your third year, you will delve deeper into the theory of probability and will explore the concept of statistical inference. Alongside this, you will gain valuable industry and employability skills through a dedicated careersfocused module. This final year will also be highly customisable, allowing you to select from a range of modules to suit your own interests and career aspirations.
In the fourth year, you will benefit from expertise from a fourth specialist department, Management Science. This year will offer you advanced modules in financial maths, which will enhance your existing skills, allow you to gain new ones, and develop a greater level of knowledge in the discipline. You will also benefit from a wide range of optional modules, which can allow you to drive your study towards a specific industry, or enhance your skill set more generally. In addition, you will complete a substantial research project, which will give you the opportunity to put everything you have learnt into practice, and will provide you with valuable insight into realworld projects.
Entry Requirements
Grade Requirements
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
IELTS 6.5 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics and/or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. They will see how partial derivatives can help to understand surfaces, while repeated integrals enable them to calculate volumes. The module will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, rates of change are introduced with respect to several quantities. How to find maxima and minima will be explained. Applications include the method of least squares. Finally, various methods for solving differential equations of one variable will be investigated.

Geometry and Calculus
The main focus of this module is vectors in two and threedimensional space. Starting with the definition of vectors, students will discover some applications to finding equations of lines and planes, then they will consider some different ways of describing curves and surfaces via equations or parameters. Partial differentiation will be used to determine tangent lines and planes, and integration will be used to calculate the length of a curve.
In the second half of the course, the functions of several variables will be studied. When attempting to calculate an integral over one variable, one variable is often substituted for another more convenient one; here students will see the equivalent technique for a double integral, where they will have to substitute two variables simultaneously. They will also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, the module will explain how to calculate the areas of various surfaces and the volumes of various solids.

Integration and Differentiation
Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Introduction to Accounting and Finance
This course provides a comprehensive introduction to the basic concepts and techniques of Accounting and Finance, which include financial accounting, managerial finance, and financial statement analysis.
An important element of this course is that it provides exposure to the business and financial environment within which the discipline of Accounting and Finance operates, using realworld financial data for actual companies.
The course covers concepts, techniques and interpretive skills that relate to the external financial reporting of companies and their relationship to the stock market, and to the use of accounting information for internal management purposes.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.
The student will learn how to express a linear transformation of the real euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.

Numbers and Relations
The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Statistics
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.
Core
Year 2

Advanced Principles of Finance
This module provides a detailed analysis of three key finance paradigms: decisionmaking under uncertainty, including utility theory; capital asset pricing and market equilibrium; and option pricing and hedging strategies. Emphasis is placed on financial concepts, theories and models such as portfolio theory, the efficient market hypothesis, and theories of capital structure.

Further Topics in Economics for Managers
The module further develops microeconomic issues relating to labour, organisations and markets, together with macroeconomic issues relating to employment and aggregate demand management.
It examines the essential features of a money economy:
 banking
 the national debt, interest rates, inflation, seigniorage
 the balance of international payments
 fixed and flexible currency exchange rates
 monetary policy
 money and business cycles

Introduction to Economics for Managers
Looking at microeconomic issues relating to markets and firms, and macroeconomic issues relating to money, banking and monetary policy, this module helps you to analyse economic issues from a business perspective. It demonstrates why economic concepts and principles are relevant to business issues by applying introductory economic theory to a range of issues that affect economic aspects of the business environment. Particular emphasis is given to interpreting the economic behaviour of individuals and firms, using theory to interpret events and evaluate policies.

Probability II
Probability provides the theoretical basis for statistics and is of interest in its own right.
Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.
Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Real Analysis
A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics II
Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
The focus will be on likelihoodbased parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.
Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.
Core

Principles of Finance
This module covers project evaluation methods as well as risk, return and the cost of capital, including the capital asset pricing model. Corporate financing, including dividend policy and capital structure, options, and working capital management will also be investigated.
Optional
Year 3

Likelihood Inference
Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.
This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.
Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.
Core

Bayesian Inference
Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.
Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.

Corporate Finance
This module examines corporate financing and investment decisions, focusing in particular on settings where companies’ assets and liabilities contain embedded options. Topics covered include valuation of options, investment appraisal, valuation of warrants and convertibles, capital structure, and mergers and restructuring.

Financial Markets
Fixed income securities are one of the major asset classes, and recent developments in debt markets (bankruptcies and reorganisation of key global players) call for deeper understanding of this key area of the financial spectrum.
This module develops your intellectual and practical understanding of the organisation and structure of bond markets, introducing you to the main problems and issues relevant in the management of interest rate risk and the principles governing the valuation of fixed income securities and their derivatives.

International Financial and Risk Management
This module provides knowledge that is important to those concerned with financial management in a multinational setting. Areas covered include the relationships between exchange rates, interest rates and inflation rates, forward, futures and options markets, and corporate exchange rate risk management.

Investments
This module covers the fundamental concepts and techniques of modern investment theory and practice. Topics include security analysis, equity and bond portfolio management, asset allocation, performance evaluation, estimation of risk measures and hedging. There is also an emphasis on some of the practical issues in portfolio management.

Medical Statistics: study design and data analysis
The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.
Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.
They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

Quantitative Finance
This module helps you to understand how econometric models can be used to learn about the future behaviour of the prices of financial assets by using information on the history of asset prices and the prices of derivative securities.
It describes time series models for financial market prices and shows how these models can be applied by banks and investors. It covers random walk tests and forecasting price volatility for financial asset prices.

Statistical Models
The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).
Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

Stochastic Processes
Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.
As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).
The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Time series analysis
Modern statistics is characterised by computerintensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.
Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.
Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.
Lastly,students will spend time looking at ChangePoint Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
Optional
Year 4

Derivatives Pricing
This module provides extensive coverage of methods used for valuing derivative securities in the investment banking industry, and includes an introduction to stochastic calculus.
Topics covered include:

Discretetime vs. continuoustime finance

Arbitrage pricing

Continuous processes

Stochastic calculus and Itô’s lemma

Hedging issues

Investment in derivatives

Continuous dividends

Black and Scholes model

Interest rate derivatives

Exotic derivatives
This module contributes to the following CFA syllabus areas:

Derivative Investments (CFA levels I and II)


Optimisation
Information for this module is currently unavailable.

Spreadsheet Modelling for Quantitative Finance
This module will cover the following topics:
 Introduction to Excel and principles of model design (week 1)
 Introduction to VBA and simple macros and tools (weeks 2 and 3)
 Case Study 1: Monte Carlo simulation model (week 4)
 Case Study 2: Optimization model using solver (week 5)
Core

Assessing Financial Risk: Extreme Value Methods
Assessment of financial risk requires accurate estimates of the probability of rare events. Estimating the probability of such "extreme" events is challenging, as by nature they are sufficiently rare that there is little direct empirical evidence on which to base inference. Instead we have to extrapolate based on the past frequency of the occurrence of less extreme events. This module covers ideas from Extreme Value Theory which give a sound mathematical basis to such extrapolation, and shows practically how it can be used to give accurate assessments of financial risk in a wide range of scenarios.
In this module, students will study topics related to the understanding of special models to describe the extreme values of a financial times series, and they will learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. Extreme value models will be used to evaluate Value at Risk and gain an understanding of the impact of heavy tailed data on standard statistical diagnostic tools.

Bayesian Inference
Bayesian statistics is a framework for rational decision making using imperfect knowledge, expressed through probability distributions. Bayesian principles are applied in the fields of navigation, control, automation and artificial intelligence. The aim of decision makers is to make rational decisions that maximise some personal utility function which may represent quantities such as money which are related to the wealth of an individual.
Within the Bayesian framework, knowledge of the world, (the prior) is updated as fresh observations arrive to yield a posterior distribution which shows the revised knowledge. The evidence for the model is expressed by calculating a marginal likelihood. Future behaviour and the fit of the model are assessed using a predictive distribution. This includes sampling uncertainty and uncertainty of our knowledge.
In this module students will look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multiparameter fully conjugate models. The range of belief types that can be modelled by using mixtures of conjugate priors will be extended, and will also explore the use of nonconjugate formulations of models and use MonteCarlo integration, importance sampling and rejection sampling for calculating and simulating from these distributions.

Behavioural Finance
This module extends the analytical tools used for evaluating strategic and investment decisions learnt in other modules by deviating from the paradigm of rational decision making. It focuses on the implications of investor behaviour and capital market imperfections (such as limits to arbitrage) for investment management. The concepts you will cover on this module provide a foundation for value investing, arbitrage, asset management and opportunistic corporate finance. Insights from psychology and behavioural finance are used to complement traditional market frictions and explain the behaviour of capital markets.

Clinical Trials
Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancyinduced hypertension; or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients. Treatments may be procedural, for example, surgery or methods of care.
This module combines the study of technical methodology with discussion of more general research issues. First , the relative advantages and disadvantages of different types of medical studies will be discussed. Then students will explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. They will also cover crossover trials, concepts of sample size determination, and equivalence trials. The module also includes a brief introduction to sequential trial designs and metaanalysis.

Computer Intensive Methods
Students will be introduced to Markov chain Monte Carlo methods and how to use them as a powerful technique for performing Bayesian inference on complex stochastic models.
The first part of the module looks in detail at the necessary concepts and theory for finite statespace Markov chains, before introducing analogous concepts and theory for continuous statespace Markov chains. In the second part of the course module the MetropolisHastings algorithm for sampling from a distribution known up to a constant of proportionality will be investigated.
In the third (and largest) part, students will take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. They will also examine the two most common MetropolisHastings algorithms (the random walk and the independence sampler). Examples will include hierarchical models, random effects models, and mixture models.

Dissertation
Information for this module is currently unavailable.

Financial Econometrics
This module explains how econometric methods can be used to learn about the future behaviour of the prices of financial assets by using the information in the history of asset prices and in the prices of derivative securities. It also gives you practical experience of analysing market prices.
It will help you to understand the important features of time series of market prices, appreciate the relevance of efficient market theory to predicting prices, and make you familiar with appropriate methods for forecasting price volatility. You will also learn how to use option prices to make statements about the distributions of future asset prices, gain experience of applying computational methods in Excel to stock market and currency prices, and develop your knowledge of a broad range of econometric methods that are applied in finance research.

Forecasting
Every managerial decision concerned with future actions is based upon a prediction of some aspects of the future. Therefore Forecasting plays an important role in enhancing managerial decision making.
After introducing the topic of forecasting in organisations, time series patterns and simple forecasting methods (naïve and moving averages) are explored. Then, the extrapolative forecasting methods of exponential smoothing and ARIMA models are considered. A detailed treatment of causal modelling follows, with a full evaluation of the estimated models. Forecasting applications in operations and marketing are then discussed. The module ends with an examination of judgmental forecasting and how forecasting can best be improved in an organisational context. Assessment is through a report aimed at extending and evaluating student learning in causal modelling and time series analysis.

Generalised Linear Models
In this module students will learn techniques for formulating sensible models for data, enabling you to tackle problems such as the probability of success for a particular treatment, and how this depends on the patient's age, weight, blood pressure, and so on.
Students will be introduced to a large family of models, called the generalised linear models (GLMs), including the standard linear regression model as a special case, and will have the opportunity to discuss and investigate the theoretical properties of these models.
A common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters will be studies. Using the statistical package ‘R’, you’ll fit and check these models, and will produce confidence intervals and tests corresponding to questions of interest.

Industrial Dissertation
At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.
In the first term there will be weekly supervisor meetings and students will be guided into their indepth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

International Money and Banking
The first four sessions aim to establish an understanding of banks’ behaviour and balance sheets, including capital structure, lending decisions and attitudes to risk. These sessions also study the banks’ role in transmitting the monetary policy decisions of the central bank, i.e. the choice of official interest rates and ‘quantitative easing’. This enables a discussion of the effects of monetary and fiscal policy on the main macroeconomic variables. Session four looks at the origins of the financial crisis and the policy responses.
Sessions five to seven cover developments in international banking regulation before and since the crisis. This includes the regulation of capital and liquidity under the Basel accords, the attempts to address the moral hazard and the ‘toolargeto fail’ problems, and the influence of regulation on the shadow banking industry.
The final three sessions study banking and monetary policy in the international context, with a particular focus on problems in the Eurozone and the operation of the eurosystem of central banks.
In this course, the treatment will generally be nontechnical and will be based on developing an understanding of institutional practices and their implications.

International Money and Finance
Beginning with the basic international parity relationships, this module examines the nature of business exposure to foreign exchange risk and the techniques available for hedging these risks. In addition to reviewing forward and futures contracts, several sessions are devoted to the theory and application of options contracts in the context of forex risk hedging.

Introduction to Intelligent Data Analysis (Data Mining)
This module develops modelling skills on synthetic and empirical data by showing simple statistical methods and introducing novel methods from artificial intelligence and machine learning.
The module will cover a wide range of data mining methods, including simple algorithms such as decision trees all the way to state of the art algorithms of artificial neural networks, support vector regression, knearest neighbour methods etc. We will consider both Data Mining methods for descriptive modelling, exploration & data reduction that aim to simplify and add insights to large, complex data sets, and Data Mining methods for predictive modelling that aim to classify and cluster individuals into distinct, disjoint segments with different patterns of behaviour.
The module will also include a series of workshops in which you will learn how to use the SAS Enterprise Miner software for data mining (a software skill much sought after in the job market) and how to use it on real datasets in a real world scenario.

Lebesgue Integration
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Longitudinal Data Analysis
Longitudinal data arise when a timesequence of measurements is made on a response variable for each of a number of subjects in an experiment or observational study. For example, a patient’s blood pressure may be measured daily following administration of one of several medical treatments for hypertension.
Typically, the practical objective of most longitudinal studies is to find out how the average value of the response varies over time, and how this average response profile is affected by different experimental treatments. This module presents an approach to the analysis of longitudinal data, based on statistical modelling and likelihood methods of parameter estimation and hypothesis testing.

Mathematics Dissertation
At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.
In the first term they will meet their supervisor weekly and will be guided into their indepth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

Microeconomics for Money, Banking and Finance
This module introduces students to those aspects of microeconomics upon which the modern understanding of financial markets, assetprice determination, and financial intermediation is built.

Principles of Epidemiology
This module focuses on the basic principles of epidemiology, including its methodology and application to prevention and control of disease.The concepts and strategies used in epidemiologic studies will be examined, and students will come to appreciate a historical and general overview of epidemiology and related strategies for study design. They will also develop their knowledge of how to apply basic epidemiologic methods, and will gain confidence in assessing the validity of epidemiologic studies with respect to their design and inferences. Most inference will be likelihood based, although the emphasis is on conceptual considerations and interpretation.

Statistics Dissertation
At the end of Year 3 students will fill in a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student's degree scheme and choice.
During the first term students will meet their supervisor on a weekly basis and will be guided into an indepth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Stochastic Processes
This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.
In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.
Careers
Careers
Financial maths graduates are very versatile, having indepth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including data analysis and manipulation, logical thinking, problemsolving and quantitative reasoning, as well as adept knowledge of the discipline. As a result, financial mathematicians are sought after in a range of industries. However, graduates of this programme are ideally placed to forge rewarding careers in the business and finance, and government sectors.
The starting salary for many financial maths graduate roles is highly competitive, and career options include:
 Actuary
 Data Analyst
 Finance Manager
 Financial Analyst
 Insurance Underwriter
 Investment Analyst
 Market Researcher
Alternatively, you may wish to undertake postgraduate study at Lancaster and pursue a career in research and teaching.
Fees and Funding
Fees
We set our fees on an annual basis and the 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU  Overseas 

£9,250  £18,195 
Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.

Course Overview
Course Overview
Financial mathematicians are interested in making good decisions in the face of uncertainty. Through our rigorous Masters programme, you will develop advanced skills and knowledge to become a major decision maker and influencer in your chosen career.
Financial maths is about learning to interpret and manipulate data to draw valuable conclusions and make predictions. This is particularly important in business and finance, but the skills gained from this programme are widely applicable and highly sought after. Over the first three years, you will be able to draw on expertise from three specialist departments: Mathematics and Statistics; Accounting and Finance; and Economics. Studying at Masterslevel, you will gain an advanced understanding of the discipline and will develop a greater breadth and depth of financial and mathematical expertise.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problemsolving and quantitative reasoning skills. During this year, you will also receive an introduction to accounting and finance, which will provide you with a comprehensive understanding of the concepts and techniques of the discipline, including financial accounting, managerial finance, and financial statement analysis.
In the second year, you will further develop your knowledge in analysis, probability and statistics. During this year, you will also begin to explore advanced finance and will benefit from modules in economics at managerial level. These will provide you with insight into micro and macroeconomics, and their related issues. You will develop realworld knowledge and experience by applying your analytical skills, interpreting events and evaluating policies. This will be instrumental in developing a career related to banking, finance, international payments and exchange rates, and monetary policy.
During your third year, you will delve deeper into the theory of probability and will explore the concept of statistical inference. Alongside this, you will gain valuable industry and employability skills through a dedicated careersfocused module. This final year will also be highly customisable, allowing you to select from a range of modules to suit your own interests and career aspirations.
In the fourth year, you will benefit from expertise from a fourth specialist department, Management Science. This year will offer you advanced modules in financial maths, which will enhance your existing skills, allow you to gain new ones, and develop a greater level of knowledge in the discipline. You will also benefit from a wide range of optional modules, which can allow you to drive your study towards a specific industry, or enhance your skill set more generally. In addition, you will complete a substantial research project, which will give you the opportunity to put everything you have learnt into practice, and will provide you with valuable insight into realworld projects.

Entry Requirements
Entry Requirements
Grade Requirements
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
IELTS 6.5 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics and/or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk

Course Structure
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. They will see how partial derivatives can help to understand surfaces, while repeated integrals enable them to calculate volumes. The module will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, rates of change are introduced with respect to several quantities. How to find maxima and minima will be explained. Applications include the method of least squares. Finally, various methods for solving differential equations of one variable will be investigated.

Geometry and Calculus
The main focus of this module is vectors in two and threedimensional space. Starting with the definition of vectors, students will discover some applications to finding equations of lines and planes, then they will consider some different ways of describing curves and surfaces via equations or parameters. Partial differentiation will be used to determine tangent lines and planes, and integration will be used to calculate the length of a curve.
In the second half of the course, the functions of several variables will be studied. When attempting to calculate an integral over one variable, one variable is often substituted for another more convenient one; here students will see the equivalent technique for a double integral, where they will have to substitute two variables simultaneously. They will also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, the module will explain how to calculate the areas of various surfaces and the volumes of various solids.

Integration and Differentiation
Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Introduction to Accounting and Finance
This course provides a comprehensive introduction to the basic concepts and techniques of Accounting and Finance, which include financial accounting, managerial finance, and financial statement analysis.
An important element of this course is that it provides exposure to the business and financial environment within which the discipline of Accounting and Finance operates, using realworld financial data for actual companies.
The course covers concepts, techniques and interpretive skills that relate to the external financial reporting of companies and their relationship to the stock market, and to the use of accounting information for internal management purposes.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.
The student will learn how to express a linear transformation of the real euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.

Numbers and Relations
The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Statistics
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.
Core
Year 2

Advanced Principles of Finance
This module provides a detailed analysis of three key finance paradigms: decisionmaking under uncertainty, including utility theory; capital asset pricing and market equilibrium; and option pricing and hedging strategies. Emphasis is placed on financial concepts, theories and models such as portfolio theory, the efficient market hypothesis, and theories of capital structure.

Further Topics in Economics for Managers
The module further develops microeconomic issues relating to labour, organisations and markets, together with macroeconomic issues relating to employment and aggregate demand management.
It examines the essential features of a money economy:
 banking
 the national debt, interest rates, inflation, seigniorage
 the balance of international payments
 fixed and flexible currency exchange rates
 monetary policy
 money and business cycles

Introduction to Economics for Managers
Looking at microeconomic issues relating to markets and firms, and macroeconomic issues relating to money, banking and monetary policy, this module helps you to analyse economic issues from a business perspective. It demonstrates why economic concepts and principles are relevant to business issues by applying introductory economic theory to a range of issues that affect economic aspects of the business environment. Particular emphasis is given to interpreting the economic behaviour of individuals and firms, using theory to interpret events and evaluate policies.

Probability II
Probability provides the theoretical basis for statistics and is of interest in its own right.
Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.
Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Real Analysis
A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics II
Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
The focus will be on likelihoodbased parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.
Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.
Core

Principles of Finance
This module covers project evaluation methods as well as risk, return and the cost of capital, including the capital asset pricing model. Corporate financing, including dividend policy and capital structure, options, and working capital management will also be investigated.
Optional
Year 3

Likelihood Inference
Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.
This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.
Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.
Core

Bayesian Inference
Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.
Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.

Corporate Finance
This module examines corporate financing and investment decisions, focusing in particular on settings where companies’ assets and liabilities contain embedded options. Topics covered include valuation of options, investment appraisal, valuation of warrants and convertibles, capital structure, and mergers and restructuring.

Financial Markets
Fixed income securities are one of the major asset classes, and recent developments in debt markets (bankruptcies and reorganisation of key global players) call for deeper understanding of this key area of the financial spectrum.
This module develops your intellectual and practical understanding of the organisation and structure of bond markets, introducing you to the main problems and issues relevant in the management of interest rate risk and the principles governing the valuation of fixed income securities and their derivatives.

International Financial and Risk Management
This module provides knowledge that is important to those concerned with financial management in a multinational setting. Areas covered include the relationships between exchange rates, interest rates and inflation rates, forward, futures and options markets, and corporate exchange rate risk management.

Investments
This module covers the fundamental concepts and techniques of modern investment theory and practice. Topics include security analysis, equity and bond portfolio management, asset allocation, performance evaluation, estimation of risk measures and hedging. There is also an emphasis on some of the practical issues in portfolio management.

Medical Statistics: study design and data analysis
The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.
Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.
They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

Quantitative Finance
This module helps you to understand how econometric models can be used to learn about the future behaviour of the prices of financial assets by using information on the history of asset prices and the prices of derivative securities.
It describes time series models for financial market prices and shows how these models can be applied by banks and investors. It covers random walk tests and forecasting price volatility for financial asset prices.

Statistical Models
The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).
Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

Stochastic Processes
Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.
As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).
The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Time series analysis
Modern statistics is characterised by computerintensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.
Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.
Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.
Lastly,students will spend time looking at ChangePoint Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
Optional
Year 4

Derivatives Pricing
This module provides extensive coverage of methods used for valuing derivative securities in the investment banking industry, and includes an introduction to stochastic calculus.
Topics covered include:

Discretetime vs. continuoustime finance

Arbitrage pricing

Continuous processes

Stochastic calculus and Itô’s lemma

Hedging issues

Investment in derivatives

Continuous dividends

Black and Scholes model

Interest rate derivatives

Exotic derivatives
This module contributes to the following CFA syllabus areas:

Derivative Investments (CFA levels I and II)


Optimisation
Information for this module is currently unavailable.

Spreadsheet Modelling for Quantitative Finance
This module will cover the following topics:
 Introduction to Excel and principles of model design (week 1)
 Introduction to VBA and simple macros and tools (weeks 2 and 3)
 Case Study 1: Monte Carlo simulation model (week 4)
 Case Study 2: Optimization model using solver (week 5)
Core

Assessing Financial Risk: Extreme Value Methods
Assessment of financial risk requires accurate estimates of the probability of rare events. Estimating the probability of such "extreme" events is challenging, as by nature they are sufficiently rare that there is little direct empirical evidence on which to base inference. Instead we have to extrapolate based on the past frequency of the occurrence of less extreme events. This module covers ideas from Extreme Value Theory which give a sound mathematical basis to such extrapolation, and shows practically how it can be used to give accurate assessments of financial risk in a wide range of scenarios.
In this module, students will study topics related to the understanding of special models to describe the extreme values of a financial times series, and they will learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. Extreme value models will be used to evaluate Value at Risk and gain an understanding of the impact of heavy tailed data on standard statistical diagnostic tools.

Bayesian Inference
Bayesian statistics is a framework for rational decision making using imperfect knowledge, expressed through probability distributions. Bayesian principles are applied in the fields of navigation, control, automation and artificial intelligence. The aim of decision makers is to make rational decisions that maximise some personal utility function which may represent quantities such as money which are related to the wealth of an individual.
Within the Bayesian framework, knowledge of the world, (the prior) is updated as fresh observations arrive to yield a posterior distribution which shows the revised knowledge. The evidence for the model is expressed by calculating a marginal likelihood. Future behaviour and the fit of the model are assessed using a predictive distribution. This includes sampling uncertainty and uncertainty of our knowledge.
In this module students will look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multiparameter fully conjugate models. The range of belief types that can be modelled by using mixtures of conjugate priors will be extended, and will also explore the use of nonconjugate formulations of models and use MonteCarlo integration, importance sampling and rejection sampling for calculating and simulating from these distributions.

Behavioural Finance
This module extends the analytical tools used for evaluating strategic and investment decisions learnt in other modules by deviating from the paradigm of rational decision making. It focuses on the implications of investor behaviour and capital market imperfections (such as limits to arbitrage) for investment management. The concepts you will cover on this module provide a foundation for value investing, arbitrage, asset management and opportunistic corporate finance. Insights from psychology and behavioural finance are used to complement traditional market frictions and explain the behaviour of capital markets.

Clinical Trials
Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancyinduced hypertension; or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients. Treatments may be procedural, for example, surgery or methods of care.
This module combines the study of technical methodology with discussion of more general research issues. First , the relative advantages and disadvantages of different types of medical studies will be discussed. Then students will explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. They will also cover crossover trials, concepts of sample size determination, and equivalence trials. The module also includes a brief introduction to sequential trial designs and metaanalysis.

Computer Intensive Methods
Students will be introduced to Markov chain Monte Carlo methods and how to use them as a powerful technique for performing Bayesian inference on complex stochastic models.
The first part of the module looks in detail at the necessary concepts and theory for finite statespace Markov chains, before introducing analogous concepts and theory for continuous statespace Markov chains. In the second part of the course module the MetropolisHastings algorithm for sampling from a distribution known up to a constant of proportionality will be investigated.
In the third (and largest) part, students will take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. They will also examine the two most common MetropolisHastings algorithms (the random walk and the independence sampler). Examples will include hierarchical models, random effects models, and mixture models.

Dissertation
Information for this module is currently unavailable.

Financial Econometrics
This module explains how econometric methods can be used to learn about the future behaviour of the prices of financial assets by using the information in the history of asset prices and in the prices of derivative securities. It also gives you practical experience of analysing market prices.
It will help you to understand the important features of time series of market prices, appreciate the relevance of efficient market theory to predicting prices, and make you familiar with appropriate methods for forecasting price volatility. You will also learn how to use option prices to make statements about the distributions of future asset prices, gain experience of applying computational methods in Excel to stock market and currency prices, and develop your knowledge of a broad range of econometric methods that are applied in finance research.

Forecasting
Every managerial decision concerned with future actions is based upon a prediction of some aspects of the future. Therefore Forecasting plays an important role in enhancing managerial decision making.
After introducing the topic of forecasting in organisations, time series patterns and simple forecasting methods (naïve and moving averages) are explored. Then, the extrapolative forecasting methods of exponential smoothing and ARIMA models are considered. A detailed treatment of causal modelling follows, with a full evaluation of the estimated models. Forecasting applications in operations and marketing are then discussed. The module ends with an examination of judgmental forecasting and how forecasting can best be improved in an organisational context. Assessment is through a report aimed at extending and evaluating student learning in causal modelling and time series analysis.

Generalised Linear Models
In this module students will learn techniques for formulating sensible models for data, enabling you to tackle problems such as the probability of success for a particular treatment, and how this depends on the patient's age, weight, blood pressure, and so on.
Students will be introduced to a large family of models, called the generalised linear models (GLMs), including the standard linear regression model as a special case, and will have the opportunity to discuss and investigate the theoretical properties of these models.
A common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters will be studies. Using the statistical package ‘R’, you’ll fit and check these models, and will produce confidence intervals and tests corresponding to questions of interest.

Industrial Dissertation
At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.
In the first term there will be weekly supervisor meetings and students will be guided into their indepth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

International Money and Banking
The first four sessions aim to establish an understanding of banks’ behaviour and balance sheets, including capital structure, lending decisions and attitudes to risk. These sessions also study the banks’ role in transmitting the monetary policy decisions of the central bank, i.e. the choice of official interest rates and ‘quantitative easing’. This enables a discussion of the effects of monetary and fiscal policy on the main macroeconomic variables. Session four looks at the origins of the financial crisis and the policy responses.
Sessions five to seven cover developments in international banking regulation before and since the crisis. This includes the regulation of capital and liquidity under the Basel accords, the attempts to address the moral hazard and the ‘toolargeto fail’ problems, and the influence of regulation on the shadow banking industry.
The final three sessions study banking and monetary policy in the international context, with a particular focus on problems in the Eurozone and the operation of the eurosystem of central banks.
In this course, the treatment will generally be nontechnical and will be based on developing an understanding of institutional practices and their implications.

International Money and Finance
Beginning with the basic international parity relationships, this module examines the nature of business exposure to foreign exchange risk and the techniques available for hedging these risks. In addition to reviewing forward and futures contracts, several sessions are devoted to the theory and application of options contracts in the context of forex risk hedging.

Introduction to Intelligent Data Analysis (Data Mining)
This module develops modelling skills on synthetic and empirical data by showing simple statistical methods and introducing novel methods from artificial intelligence and machine learning.
The module will cover a wide range of data mining methods, including simple algorithms such as decision trees all the way to state of the art algorithms of artificial neural networks, support vector regression, knearest neighbour methods etc. We will consider both Data Mining methods for descriptive modelling, exploration & data reduction that aim to simplify and add insights to large, complex data sets, and Data Mining methods for predictive modelling that aim to classify and cluster individuals into distinct, disjoint segments with different patterns of behaviour.
The module will also include a series of workshops in which you will learn how to use the SAS Enterprise Miner software for data mining (a software skill much sought after in the job market) and how to use it on real datasets in a real world scenario.

Lebesgue Integration
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Longitudinal Data Analysis
Longitudinal data arise when a timesequence of measurements is made on a response variable for each of a number of subjects in an experiment or observational study. For example, a patient’s blood pressure may be measured daily following administration of one of several medical treatments for hypertension.
Typically, the practical objective of most longitudinal studies is to find out how the average value of the response varies over time, and how this average response profile is affected by different experimental treatments. This module presents an approach to the analysis of longitudinal data, based on statistical modelling and likelihood methods of parameter estimation and hypothesis testing.

Mathematics Dissertation
At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.
In the first term they will meet their supervisor weekly and will be guided into their indepth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

Microeconomics for Money, Banking and Finance
This module introduces students to those aspects of microeconomics upon which the modern understanding of financial markets, assetprice determination, and financial intermediation is built.

Principles of Epidemiology
This module focuses on the basic principles of epidemiology, including its methodology and application to prevention and control of disease.The concepts and strategies used in epidemiologic studies will be examined, and students will come to appreciate a historical and general overview of epidemiology and related strategies for study design. They will also develop their knowledge of how to apply basic epidemiologic methods, and will gain confidence in assessing the validity of epidemiologic studies with respect to their design and inferences. Most inference will be likelihood based, although the emphasis is on conceptual considerations and interpretation.

Statistics Dissertation
At the end of Year 3 students will fill in a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student's degree scheme and choice.
During the first term students will meet their supervisor on a weekly basis and will be guided into an indepth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Stochastic Processes
This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.
In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.

Calculus

Careers
Careers
Financial maths graduates are very versatile, having indepth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including data analysis and manipulation, logical thinking, problemsolving and quantitative reasoning, as well as adept knowledge of the discipline. As a result, financial mathematicians are sought after in a range of industries. However, graduates of this programme are ideally placed to forge rewarding careers in the business and finance, and government sectors.
The starting salary for many financial maths graduate roles is highly competitive, and career options include:
 Actuary
 Data Analyst
 Finance Manager
 Financial Analyst
 Insurance Underwriter
 Investment Analyst
 Market Researcher
Alternatively, you may wish to undertake postgraduate study at Lancaster and pursue a career in research and teaching.

Fees and Funding
Fees and Funding
Fees
We set our fees on an annual basis and the 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU Overseas £9,250 £18,195 Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.
The Department
Download the latest Mathematics & Statistics brochure, featuring all of the information you could need. The includes detailed course information about each of our courses.
Our Courses
Our courses offer academic excellence. We have 3 singlehonours degrees and 10 combinedhonours degrees. We have focused our courses in the first two years, so you can choose from a wide range of third and fourthyear topics. Our academic research shapes our courses with the latest developments.
Friendly and Collaborative
The department has around 50 academic staff and 500 undergraduate students. Throughout your course, you will have weekly workshops and marked assignments. You will have opportunity to discuss each topic and get feedback from expert tutors. We also offer onetoone sessions during lecturers' office hours, so you soon get to know us and vice versa.
Supportive Environment
You will receive the support you need to achieve your full academic potential. We are a friendly department and foster a supportive learning environment. You will have an academic tutor who will assist you throughout your degree. We also encourage you to attend our weekly Maths Café, lead by fellow students.
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11.8 hours
Typical time in lectures, seminars and similar per week during term time
27%
Average assessment by coursework