Financial Mathematics MMath

Compulsory

• Fundamentals of Mathematics

  Aims to manipulate mathematical expressions accurately, as well as recall and use mathematical formulae in areas of interest for Year 1. To develop skills in handling summation notation. To introduce students to fundamental results in mathematics. To develop an understanding of the need for rigour in definitions and proofs. To introduce the language of formal mathematics, in particular sets and functions.

• Calculus 1
  This module aims to familiarise students with the basic results, techniques and elementary functions of differential and integral calculus, to introduce students to rigorous definitions, arguments and proofs through many simple examples, to develop students’ manipulative skills in performing operations in differential calculus through work on many simple examples, and to illustrate the solution techniques of first order differential equations.
• Calculus 2

  Aims to further develop skills in differential and integral calculus and associated applications. To further develop students’ manipulative skills in performing operations in differential calculus through work on examples, including solution techniques of ordinary differential equations.

• Linear Algebra
  This module aims to enable students to understand and become proficient in basic linear algebra and the algebra of complex numbers, to determine the eigenvalues and eigenvectors of matrices and understand their role in the theory of similar matrices and diagonalization, and to see and practice applications of linear algebra.
• Elements of Applied Mathematics 1
  The first of a sequence of blocks aimed at developing modelling skills. The main aim of this module is for students to develop a facility for mathematical modelling by examining a problem in its original form, extracting the principal features, formulating and solving appropriate mathematical models, and interpreting the results in terms of the original problems.
• Elements of Applied Mathematics 2

  Second of a pair of modules developing modelling skills. Furthering a facility for mathematical modelling by examining a problem in its original form, extracting the principal features, formulating and solving appropriate mathematical models and interpreting the results in terms of the original problems.

Compulsory

• Elements of Investment Mathematics

  Aims to introduce the mathematics of deterministic cash flow streams. To introduce the concepts of optimum allocation of financial resources under uncertainty and issues of financial planning and with the models that provide mathematical descriptions of these investment problems. To introduce financial risk measures and illustrate how they can be incorporated in financial planning models. To implement simulations in software.

• Linear and Abstract Algebra

  Aims to enlarge the set of technical tools of linear algebra and develop its applications to different problems, including construction and analysis of linear models. To introduce basic algebraic structures, concentrating on group theory. To exemplify their power, relevance and importance in real life applications.

• Applied Statistics

  Aims to introduce and consolidate the notions of single and multivariable probability. Introduce statistical tools and explain their use in extracting and/or inferring meaning from data. To introduce sampling and inference, along with confidence intervals and hypothesis tests.

• Calculus 3

  Aims to develop ideas and methods of multivariable calculus, including Taylor series, extrema, the use of Lagrange multipliers, and the integration of functions of several variables. To understand the extension from single variable to several variables of basic concepts such as continuity and differentiability.

Compulsory

• Mathematical Finance
  This module aims to introduce the students to the main aspects of modern mathematical finance, in particular to the Black-Scholes-Merton theory of option pricing, the ideas of arbitrage pricing, replication and dynamic hedging. It aims to illustrate the necessary ideas from continuous time stochastic process theory by using discrete time models to, in particular, outline the concept of a lognormal random walk.
• Financial Mathematics Final Year Project
  The aim of this module is to stimulate independent learning and critical thinking by the student, both as a means for studying their chosen topic and for approaching other real-life problems of use in finance. This enables the student to plan, execute and present a substantial piece of work, gain experience in the written communication of complex ideas and concepts, and foster an interest in the study of more advanced topics in financial mathematics.

Optional

• Ordinary and Partial Differential Equations
  This module aims to introduce students to the mathematics of differential equations; techniques of analysing such equations, and methods of solving them, exactly or approximately.
• Numerical Methods for Differential Equations
  This module aims to introduce numerical methods used to solve problems in financial mathematics, in particular option pricing, and implement them. Most examples treated will be taken from finance, but the module will be suitable for students mostly interested in advanced numerical methods, in particular finite differences algorithms used to approximate solutions of PDEs. The Matlab implementations of the algorithms will be an important part of the module.
• Stochastic Models
  This module aims to introduce students to the concept of a stochastic process, so that they may develop an understanding of the theory underlying some of the standard models and acquire knowledge of methods of applying these models to solve problems. Students will further develop their general ability to think abstractly, to generalise, to formulate and structure stochastic problems, and to apply their knowledge of analytical and numerical mathematical techniques to solving a variety of problems in stochastic modelling.
• Decision Making in the Face of Risk
• Practical Machine Learning
• Experimental Design and Regression
• Deep Learning

  Within this module, an in-depth introduction will be provided to the area of learning using deep neural networks. A wide variety of the architectures of deep neural networks and their learning methods will be covered, including convolutional networks, recurrent networks, generative models and deep reinforcement learning etc. The main focus of the module is to develop students’ skill in analysing of problem requirements, applying appropriate deep learning methods to real-world problems, and evaluating the effectiveness of the adopted approach.

Compulsory

• Probability and Stochastics
  This module aims to equip students with the basic measure-theoretic and probabilistic concepts and techniques needed for them to be able to apply the modern mathematical theory of finance, and to enable students to use methods of stochastic calculus based on Brownian motion in such a way that they are able to carry out the necessary mathematical manipulations and calculations required for use and critical assessment of the various financial models introduced in other modules of the programme.
• Computer Intensive Statistical Methods
  This module aims to introduce the students to a range of computational intensive statistical methods, to further develop their skills in correct interpretation and clear reporting of results, and to enable the students to create algorithms for regression models (parametric regression and nonparametric regression) to cope with massive data.
• Quantitative Data Analysis
  The aim of this module is to develop knowledge and skills of the quantitative data analysis methods that underpin data science. Content covers a practical understanding of core methods in data science application and research, such as bivariate and multivariate methods, regression and graphical models. A focus is also placed on learning to evaluate the strengths and weaknesses of methods alongside an understanding of how and when to use or combine methods.

Optional

• Interest Rate Theory
  The purpose of this module is to equip students with a basic familiarity of fixed-income securities markets and, in particular, with the structures of key financial products traded in such markets. It aims to enable students to value derivatives in a number of basic models for interest rates and discount bonds.
• Fundamentals of Machine Learning
  This module aims to equip students with the knowledge and ability to use modern regression and classification methods with different types of data, to enable students to apply a range of models and tools to variable selection and model selection.
• Time Series Modelling
  This module aims to equip students with the ability to employ different methods for modelling and forecasting time series data, in particular in the context of financial data and forecasting financial risk, and to enable students to apply a range of models and tools to make financial decisions such as risk assessment.
• Option Pricing Theory
  This module aims to equip students with the notion of risk-neutral valuation and the relation between physical and risk-neutral probability measures in the Brownian context, and to enable students to price vanilla options and basic barrier options in the geometric Brownian motion model.
• Big Data Analytics
  The aim of this module is to develop the reflective and practical understanding necessary to extract value and insight from large heterogeneous data sets. Focus is placed on the analytic methods/techniques/algorithms for generating value and insight from the (real-time) processing of heterogeneous data. Content will cover approaches to data mining alongside machine learning techniques, such as clustering, regression, support vector machines, boosting, decision trees and neural networks.