Financial and Computational Mathematics MSc

In this module you will cover three major topics: continuous time modelling for equity derivatives pricing, pricing interest rate derivatives and credit risk modelling. These will be underpinned by the theory of stochastic processes and stochastic differential equations.

You will gain experience of a topic of considerable contemporary importance, both in research and in terms of how it is applied. You will undertake a group project which will involve independent reading, computer simulations and a written report.

During this module four major topics for the computational solution of problems in applied mathematics are considered.

• approximation theory
• numerical solution of nonlinear problems
• numerical solution of ODEs
• numerical solution of PDEs.

The focus is on formulating and understanding computational techniques with illustrations on elementary models from a variety of scientific applications. Specific contents include: approximation theory, multivariate polynomial approximation, Gauss quadrature, splines, trigonometric polynomials, DFTs, FFTs; numerical solution of (systems of) nonlinear equations; numerical differentiation and numerical solution of ODEs; introduction to PDEs and finite difference methods including error analysis. AI based coding assistants and state-of-the-art ML libraries for numerical analysis.

This module introduces the main concepts of financial mathematics, such as pricing and hedging of financial instruments (forwards, futures, options, swaps) with a focus on discrete models. You will develop your knowledge of probability and stochastic calculus. 

The first part of the course introduces the no-arbitrage pricing principle and financial instruments such as forward and futures contracts, bonds and swaps, and options.

The second part of the course considers the pricing and hedging of options and discrete-time discrete-space stochastic processes.

The final part of the module focuses on the Black-Scholes formula for pricing European options and also introduces the Wiener process, Ito integrals and stochastic differential equations.

This module provides an introduction to the programming language C++ with a strong emphasis on scientific computing applications. You will study C++ Language: basic types and control structures, program design and implementation, program comprehension and modification, program testing and documentation; pointers, functions, and reference variables. You will also cover classes, inheritance and derived classes; templates, applications, computer roundoff and its effect on the design of algorithms; polynomial interpolations and numerical integration. You will also gain an understanding of computational linear algebra, including direct and iterative methods.

You will carry out a substantial investigation on a topic in financial mathematics and related subjects. The study will be largely self-directed, with oversight and input provided where necessary by a supervisor from the School of Mathematical Sciences. The topic will be chosen from a list of potential projects provided by the school.

The projects will usually contain three components: a finance-related part; mathematical and/or statistical methods and analysis; element of computing either using existing packages or developing new code to simulate and analyse appropriate mathematical models. The balance of the three components will depend on the nature of a particular project.

A training session covering the oral presentation assessment criteria and some elements of good practice will also be included.

In this module a variety of techniques and areas of mathematical optimization will be covered. You will study topics such as lagrangian methods for optimization, linear programming including the simplex algorithm, dynamic programming both deterministic control problems and stochastic problems. You will also cover network and graph algorithms.

During the module you will gain a rigorous mathematical background and develop the techniques for application through computational examples.

In this module, the fundamental principles and techniques underlying modern statistical and data analysis will be introduced. You will gain experience in using a statistical package and interpreting its output. The course will cover a 'common core' consisting of:

• statistical concepts and methods
• linear models
• probability techniques
• Markov chains

This module is a topic at the interface between statistics and computer science, concerning models that can adapt to, and make predictions based on, data. This module builds on the principles of statistical inference and linear regression to introduce a variety of methods of clustering, dimension reduction, regression and classification.

Much of the focus is on the bias-variance trade-off and on methods to measure and compensate for overfitting. The learning approach is hands-on; you will be using R extensively in studying contemporary statistical machine learning methods, and in applying them to tackle challenging real-world applications.

This module will provide a general introduction to the analysis of data that arise sequentially in time. Several commonly occurring models will be discussed and their properties derived. Methods for model identification for real-time series data will be considered. Techniques for estimating the parameters of a model, assessing its fit and forecasting future values will be developed. You will gain experience of using a statistical package and interpreting its output.

Core techniques of econometric theory, including detailed analysis of the multiple linear regression model; large sample theory; asymptotic testing procedures; non-linear techniques; and mis-specification testing will be covered.

The module extends the coverage of advanced econometric modelling techniques and considers their application through the study of selected topics in finance and macroeconomics, developing familiarity and critical awareness of empirical research in these areas. You will study techniques for the analysis of stationary ARMA processes, Vector Autoregressions (VARs), linear regression models, linear systems of simultaneous equations, cointegration, long-run structural VARs, forecasting, and models of changing volatility. The selected topics include the econometric analysis of business cycle fluctuations, wage, price and (un)employment determination, portfolio choice and stock market returns.

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The course starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner's Dilemma. We look at tree-searching, including alpha-beta pruning, the killer heuristic and its relatives. It then turns to the mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

The module covers fundamental properties of time series and various classes of stochastic processes. Issues in estimation and forecasting of time series models; concepts of contemporary interest to time series econometricians are also covered.

This module offers you an introduction to the economics of corporate finance. It is designed to provide you with the basic theoretical background in this area that is necessary for any applied work. Emphasis is placed on the analysis of simple models and their applications.

The module covers a variety of topics with substantial time devoted for covering issues directly related to the financial needs of firms, such as capital structure, credit rationing and corporate governance.

The module also examines the role of financial intermediaries analyzing bank failures and, consequently, the scope for banking regulations. The last part of the module looks closely at the relationship between the financial sector and the real economy thus offering the background for any applied work related to the link between financial development and economic fluctuations.

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The course starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner's Dilemma. We look at tree-searching, including alpha-beta pruning, the killer heuristic and its relatives. It then turns to the mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

This module covers foundations of consumer theory, decision-making under risk, elements of the theory of imperfect competition, and incomplete information in markets.

The module lays the foundations of macroeconomic models with the use of very stylized one and two-period models in which consumers, firms and governments interact in the goods and factor markets. You will also study the determinants of long-run economic growth. The overarching question that we will consider is: why are some countries so much richer than others?

We will study several mechanisms that can potentially account for these differences, namely, capital accumulation, knowledge production and institutions. Finally, we will embed the consumption and labour supply decision studied at the beginning of the semester into a dynamic stochastic general equilibrium model of the macroeconomy. The standard real business cycle (RBC) model developed by Kydland and Prescott will be considered alongside the investment decision of firms, and you will spend some time studying the search and matching model of unemployment of Mortensen and Pissarides.

This module covers monetary aspects of advanced macroeconomics. It focuses on the theory and practice of central banking, monetary policy and control. It covers concepts such as time inconsistency, the problem of inflation bias with solutions, credibility, transparency and accountability of monetary institutions. You will also cover inflation targeting and price stability, the choice of instruments for monetary policy and their control, and finally monetary transmission. You will combine some theory with evidence and practice. Students that take this module typically aim to work in central banks, financial institutions or government.

This module will focus on advanced Big Data methods and their applications in various economics problems. Topics of the module include:

• nonlinear models
• tree-based models
• support vector machines
• unsupervised learning and applications in international trade
• household finance
• macro forecasting
• labor economics
• text analysis

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The course starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner's Dilemma. We look at tree-searching, including alpha-beta pruning, the killer heuristic and its relatives. It then turns to the mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

This module is intended as an introduction of the methodology and implementation of machine learning methods used widely in economic analysis; the module provides an introduction to the analysis of large datasets, with more up to date Big Data methods introduced later as natural developments. Topics will include:

• an introduction to statistical computing
• generating random variables and random processes
• Monte Carlo integration and variance reduction; high performance computing
• multivariate linear regression
• classification
• resampling methods
• linear model selection and regularization