Financial Mathematics - BSc (Hons)

Introduction to R and investigating data sets. Basic use of R (Input and manipulation of data). Graphical representations of data. Numerical summaries of data.

Sampling and sampling distributions. ?² distribution. t-distribution. F-distribution. Definition of sampling distribution. Standard error. Sampling distribution of sample mean (for arbitrary distributions) and sample variance (for normal distribution) .

Point estimation. Principles. Unbiased estimators. Bias, Likelihood estimation for samples of discrete r.v.s

Interval estimation. Concept. One-sided/two-sided confidence intervals. Examples for population mean, population variance (with normal data) and proportion.

Hypothesis testing. Concept. Type I and II errors, size, p-values and power function. One-sample test, two sample test and paired sample test. Examples for population mean and population variance for normal data. Testing hypotheses for a proportion with large n. Link between hypothesis test and confidence interval. Goodness-of-fit testing.

Association between variables. Product moment and rank correlation coefficients. Two-way contingency tables. ?² test of independence.

Time value of money: Basic concepts, Compounding to determine future values, Inflation,

Financial valuation and cash flow analysis: Discounting, Interest rates and time requirements, Future and Present value. Project Evaluation

Characteristics of different financial securities: Debt capital, bonds and stocks, valuation of bonds and stocks

Terminology in finance: Securities markets, primary market, secondary securities markets, the role of the various financial markets.

This module serves as an introduction to algebraic methods and linear algebra methods. These are central in modern mathematics, having found applications in many other sciences and also in our everyday life.

Indicative module content:

Basic set theory, Functions and Relations, Systems of linear equations and Gaussian elimination, Matrices and Determinants, Vector spaces and Linear Transformations, Diagonalisation, Orthogonality.

This module introduces widely-used mathematical methods for functions of a single variable. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Complex numbers: Complex arithmetic, the complex conjugate, the Argand diagram, de Moivre's Theorem, modulus-argument form; elementary functions

Polynomials: Fundamental Theorem of Algebra (statement only), roots, factorization, rational functions, partial fractions

Single variable calculus: Differentiation, including product and chain rules; Fundamental Theorem of Calculus (statement only), elementary integrals, change of variables, integration by parts, differentiation of integrals with variable limits

Scalar ordinary differential equations (ODEs): definition; methods for first-order ODEs; principle of superposition for linear ODEs; particular integrals; second-order linear ODEs with constant coefficients; initial-value problems

Curve sketching: graphs of elementary functions, maxima, minima and points of inflection, asymptotes

This module introduces widely-used mathematical methods for vectors and functions of two or more variables. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Vectors: Cartesian coordinates; vector algebra; scalar, vector and triple products (and geometric interpretation); straight lines and planes expressed as vector equations; parametrized curves; differentiation of vector-valued functions of a scalar variable; tangent vectors; vector fields (with everyday examples)

Partial differentiation: Functions of two variables; partial differentiation (including the chain rule and change of variables); maxima, minima and saddle points; Lagrange multipliers

Integration in two dimensions: Double integrals in Cartesian coordinates; plane polar coordinates; change of variables for double integrals; line integrals; Green's theorem (statement – justification on rectangular domains only)

The aim of this module is to introduce students to the basic tools used by economists to study resource allocation, price formation and decision making. The module focuses on classical theories of consumer and producer behaviour and on the theory of competitive equilibrium.

The module starts with an analysis of the optimisation problems of price-taking consumers and firms. We then analyse market interaction and the formation of prices in the framework of perfect competition. We will introduce decision making under uncertainty and apply it to problems in finance. We will conclude with a brief introduction to game theory.

Introduction to Probability. Concepts of events and sample space. Set theoretic description of probability, axioms of probability, interpretations of probability (objective and subjective probability).

Theory for unstructured sample spaces. Addition law for mutually exclusive events. Conditional probability. Independence. Law of total probability. Bayes' theorem. Permutations and combinations. Inclusion-Exclusion formula.

Discrete random variables. Concept of random variable (r.v.) and their distribution. Discrete r.v.: Probability function (p.f.). (Cumulative) distribution function (c.d.f.). Mean and variance of a discrete r.v. Examples: Binomial, Poisson, Geometric.

Continuous random variables. Probability density function; mean and variance; exponential, uniform and normal distributions; normal approximations: standardisation of the normal and use of tables. Transformation of a single r.v.

Joint distributions. Discrete r.v.'s; independent random variables; expectation and its application.

Generating functions. Idea of generating functions. Probability generating functions (pgfs) and moment generating functions (mgfs). Finding moments from pgfs and mgfs. Sums of independent random variables.

Laws of Large Numbers. Weak law of large numbers. Central Limit Theorem.

Real Numbers: Rational and real numbers, absolute value and metric structure on the real numbers, induction, infimum and supremum.

Limits of Sequences: Sequences, definition of convergence, epsilon terminology, uniqueness, algebra of limits, comparison principles, standard limits, subsequences and non-existence of limits, convergence to infinity.

Completeness Properties: Cantor's Intersection Theorem, limit points, Bolzano-Weierstrass theorem, Cauchy sequences.

Continuity of Functions: Functions and basic definitions, limits of functions, continuity and epsilon terminology, sequential continuity, Intermediate Value Theorem.

Differentiation: Definition of the derivative, product rule, quotient rule and chain rule, derivatives and local properties, Mean Value Theorem, L'Hospital's Rule.

Taylor Approximation: Taylor's Theorem, remainder term, Taylor series, standard examples, limits using Taylor series.

Constructing suitable models for data is a key part of statistics. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. Even if the temperature and pressure are fixed, there will be variation in the yield which motivates the use of a statistical model which includes a random component. In this module, we study how suitable models can be constructed, how to fit them to data and how suitable conclusions can be drawn. Both theoretical and practical aspects are covered, including the use of R.

In this module we will study linear partial differential equations, we will explore their properties and discuss the physical interpretation of certain equations and their solutions. We will learn how to solve first order equations using the method of characteristics and second order equations using the method of separation of variables.

Introduction to linear PDEs: Review of partial differentiation; first-order linear PDEs, the heat equation, Laplace's equation and the wave equation, with simple models that lead to these equations; the superposition principle; initial and boundary conditions

Separation of variables and series solutions: The method of separation of variables; simple separable solutions of the heat equation and Laplace’s equation; Fourier series; orthogonality of the Fourier basis; examples and interpretation of solutions

Solution by characteristics: the method of characteristics for first-order linear PDEs; examples and interpretation of solutions; characteristics of the wave equation; d’Alembert’s solution, with examples; domains of influence and dependence; causality.

The aim of this module is to introduce students to a set of tools that they can use to analyse macroeconomic issues in the short run and long run, and to help them understand which modelling techniques are appropriate for particular macroeconomic problems.

This module enhances the students' ability to understand the determinants of key macroeconomic variables including output, inflation, interest rates and exchange rates. Students’ technical skills are developed and they are expected to apply their knowledge to real-world policymaking and link it to financial issues.

Probability: Joint distributions of two or more discrete or continuous random variables. Marginal and conditional distributions. Independence. Properties of expectation, variance, covariance and correlation. Poisson process and its application. Sums of random variables with a random number of terms.

Transformations of random variables: Various methods for obtaining the distribution of a function of a random variable —method of distribution functions, method of transformations, method of generating functions. Method of transformations for several variables. Convolutions. Approximate method for transformations.

Sampling distributions: Sampling distributions related to the Normal distribution — distribution of sample mean and sample variance; independence of sample mean and variance; the t distribution in one- and two-sample problems.

Statistical inference: Basic ideas of inference — point and interval estimation, hypothesis testing.

Point estimation: Methods of comparing estimators — bias, variance, mean square error, consistency, efficiency. Method of moments estimation. The likelihood and log-likelihood functions. Maximum likelihood estimation.

Hypothesis testing: Basic ideas of hypothesis testing — null and alternative hypotheses; simple and composite hypotheses; one and two-sided alternatives; critical regions; types of error; size and power. Neyman-Pearson lemma. Simple null hypothesis versus composite alternative. Power functions. Locally and uniformly most powerful tests. Composite null hypotheses. The maximum likelihood ratio test.

Interval estimation: Confidence limits and intervals. Intervals related to sampling from the Normal distribution. The method of pivotal functions. Confidence intervals based on the large sample distribution of the maximum likelihood estimator – Fisher information, Cramer-Rao lower bound. Relationship with hypothesis tests. Likelihood-based intervals.

This module is an introduction to the methods, tools and ideas of numerical computation. In mathematics, one often encounters standard problems for which there are no easily obtainable explicit solutions, given by a closed formula. Examples might be the task of determining the value of a particular integral, finding the roots of a certain non-linear equation or approximating the solution of a given differential equation. Different methods are presented for solving such problems on a modern computer, together with their applicability and error analysis. A significant part of the module is devoted to programming these methods and running them in MATLAB.

Introduction: Importance of numerical methods; short description of flops, round-off error, conditioning

Solution of linear and non-linear equations: bisection, Newton-Raphson, fixed point iteration

Interpolation and polynomial approximation: Taylor polynomials, Lagrange interpolation, divided differences, splines

Numerical integration: Newton-Cotes rules, Gaussian rules

Numerical differentiation: finite differences

Introduction to initial value problems for ODEs: Euler methods, trapezoidal method, Runge-Kutta methods.

Formulation/Mathematical modelling of optimisation problems

Linear Optimisation: Graphical method, Simplex method, Phase I method, Dual problems,

Transportation problem.

Non-linear Optimisation: Unconstrained one dimensional problems, Unconstrained high dimensional problems, Constrained optimisation.

This module introduces the basic ideas to solve certain ordinary differential equations, like first order scalar equations, second order linear equations and systems of linear equations. It mainly considers their qualitative and analytical aspects. Outline syllabus includes: First-order scalar ODEs; Second-order scalar linear ODEs; Existence and Uniqueness of Solutions; Autonomous systems of two linear first-order ODEs.

This module covers aspects of Statistics which are particularly relevant to insurance. Some topics (such as risk theory and credibility theory) have been developed specifically for actuarial use. Other areas (such as Bayesian Statistics) have been developed in other contexts but now find applications in actuarial fields. Stochastic processes of events such as accidents, together with the financial flow of their payouts underpin much of the work. Since the earliest games of chance, the probability of ruin has been a topic of interest. Outline Syllabus includes: Decision Theory; Bayesian Statistics; Loss Distributions; Reinsurance; Credibility Theory; Empirical Bayes Credibility theory; Risk Models; Ruin Theory; Generalised Linear Models; Run-off Triangles.

This module introduces and explores a range of topics relating to corporate finance which are fundamental to understanding why and how companies raise money to start a business or expand an existing one. The module covers the different ways that the money can be raised, for example from a bank or through a stocks and shares market, and the interest rate or investment return that an investor will expect to receive from a company in order to provide the money required. This is a very practical module to the extent that it will help students develop business awareness in the field of company finance. Reference will often be made to actual happenings in the financial markets in support of the material covered.

The aim of this module is to provide a grounding in the principles of modelling as applied to actuarial work – focusing particularly on stochastic asset liability models. These skills are also required to communicate with other financial professionals and to critically evaluate modern financial theories.

Indicative topics covered by the module include theories of financial market behaviour, measures of investment risk, stochastic investment return models, asset valuations, and liability valuations.

This module will cover a number of syllabus items set out in Subject CM2 – Actuarial Mathematics published by the Institute and Faculty of Actuaries.

Introduction: Principles and examples of stochastic modelling, types of stochastic process, Markov property and Markov processes, short-term and long-run properties. Applications in various research areas.

Random walks: The simple random walk. Walk with two absorbing barriers. First–step decomposition technique. Probabilities of absorption. Duration of walk. Application of results to other simple random walks. General random walks. Applications.

Discrete time Markov chains: n–step transition probabilities. Chapman-Kolmogorov equations. Classification of states. Equilibrium and stationary distribution. Mean recurrence times. Simple estimation of transition probabilities. Time inhomogeneous chains. Elementary renewal theory. Simulations. Applications.

Continuous time Markov chains: Transition probability functions. Generator matrix. Kolmogorov forward and backward equations. Poisson process. Birth and death processes. Time inhomogeneous chains. Renewal processes. Applications.

Queues and branching processes: Properties of queues - arrivals, service time, length of the queue, waiting times, busy periods. The single-server queue and its stationary behaviour. Queues with several servers. Branching processes. Applications.

Marks on this module can count towards exemption from the professional examination CT4 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

Multivariate normal distribution, Inference from multivariate normal samples, principal component analysis, mixture models, factor analysis, clustering methods, discrimination and classification, graphical models, the use of appropriate software.

Futures Markets: Mechanics, Hedging strategies

Interest rates: Type of rates, LIBOR, Repo, Spot, Forward

Determination of Forward and Future Prices: Short and long positions, forward prices

Interest Rate Derivatives and Swaps

Option Markets: Mechanics, Properties, trading strategies

Overview of statistical methods. Stationary time series. Autocovariance and autocorrelation functions. Partial autocorrelation functions. ARMA processes. ARIMA model building, testing and estimation. Criteria for choosing between models. Forecasting. Cointegration. Prediction bounds. Asset return and risk. Term structure of interest rates. Distributional properties of asset returns. Testing for CAPM. Testing random walk hypothesis and predicting asset return. Sharpe ratio and efficient portfolio. Cross-section modelling and GMM. Estimate multifactor models. Financial applications of AR, MA, and ARMA. ARCH and GARCH models. Volatility processes. Simple applications of these techniques using R.

Statistics methods contribute significantly to areas such as biology, ecology, sociology and economics. The real data collected does not always follow standard statistical models. This module looks at modern statistical models and methods that can be utilised for such data, making use of R programs to execute these methods.

Indicative module content: Motivating examples; model fitting through maximum likelihood for specific examples; function optimization methods; profile likelihood; score tests; Wald tests; confidence interval construction; latent variable models; EM algorithm; mixture models; simulation methods; importance sampling; kernel density estimation; Monte Carlo inference; bootstrap; permutation tests; R programs.

There is no specific mathematical syllabus for this module; students will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their coursework assessments (different topics for levels 6 and 7). The coursework is supported by a series of workshops covering various forms of written and oral communication. These may include critically evaluating the following: a research article in mathematics, statistics or finance; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

There is no specific mathematical syllabus for this module. Students will study a topic in mathematics or statistics, either individually or within a small group, and produce an individual or group project on the topic as well as individual coursework assignments. Projects will be chosen from published lists of individual and of group projects. The coursework and project-work are supported by a series of workshops covering various forms of written and oral communication and by supervision from an academic member of staff.

The workshops may include critically evaluating the following: a research article in mathematics or statistics; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

This is a practical module to develop the skills required by a professional statistician (report writing, consultancy, presentation, wider appreciation of assumptions underlying methods, selection and application of analysis method, researching methods).

Software: R, SPSS and Excel (where appropriate/possible). Report writing in Word. PowerPoint for presentations.

• Presentation of data

• Report writing and presentation skills

• Hypothesis testing: formulating questions, converting to hypotheses, parametric and non-parametric methods and their assumptions, selection of appropriate method, application and reporting. Use of resources to explore and apply additional tests. Parametric and non-parametric tests include, but are not limited to, t-tests, likelihood ratio tests, score tests, Wald test, chi-squared tests, Mann Whitney U-test, Wilcoxon signed rank test, McNemar's test.

• Linear and Generalised Linear Models: simple linear and multiple regression, ANOVA and ANCOVA, understanding the limitations of linear regression, generalised linear models, selecting the appropriate distribution for the data set, understanding the difference between fixed and random effects, fitting models with random effects, model selection.

• Consultancy skills: group work exercise(s)

In this module we study the fundamental concepts and results in game theory. We start by analysing combinatorial games, and discuss game trees, winning strategies, and the classification of positions in so called impartial combinatorial games. We then move on to discuss two-player zero-sum games and introduce security levels, pure and mixed strategies, and prove the famous von Neumann Minimax Theorem. We will see how to solve zero-sum two player games using domination and discuss a general method based on linear programming. Subsequently we analyse arbitrary sum two-player games and discuss utility, best responses, Nash equilibria, and the Nash Equilibrium Theorem. The final part of the module is devoted to multi-player games and cooperation; we analyse coalitions, the core of the game, and the Shapley value.

The aim of this module is to provide a grounding in the principles of modelling as applied to actuarial work – focusing particularly on the valuation of financial derivatives. These skills are also required to communicate with other financial professionals and to critically evaluate modern financial theories.

Indicative topics covered by the module include theories of stochastic investment return models and option theory.

This module will cover a number of syllabus items set out in Subject CM2 – Actuarial Mathematics published by the Institute and Faculty of Actuaries.

Most differential equations which arise from physical systems cannot be solved explicitly in closed form, and thus numerical solutions are an invaluable way to obtain information about the underlying physical system. The first half of the module is concerned with ordinary differential equations. Several different numerical methods are introduced and error growth is studied. Both initial value and boundary value problems are investigated. The second half of the module deals with the numerical solution of partial differential equations. The syllabus includes: initial value problems for ordinary differential equations; Taylor methods; Runge-Kutta methods; multistep methods; error bounds and stability; boundary value problems for ordinary differential equations; finite difference schemes; difference schemes for partial differential equations; iterative methods; stability analysis.