School of Mathematical Sciences

Modules

Programme Structure

Below is a full list of all modules which we expect to be available to students on this programme across the semesters.  Please note that this list, and the contents of the individual modules, is for information only and may be subject to change. The programme structure is flexible, so you can choose to focus on either computational or mathematical modules, depending on your background, interests and future plans.

Module outlines

MTH773P Advanced Computing in Finance

This module covers the advanced programming techniques in C++ that are widely used by professional software engineers and quantitative analysts & developers. The most important of these techniques is object-oriented programming, embracing the concepts of encapsulation, inheritance and polymorphism. We then use these techniques to price a wide range of financial derivatives numerically, using several different pricing models and numerical methods. On completion of this module, you will have acquired the key skills needed to apply for your first role as a junior ‘quant’ or software developer in a financial institution.

Topics include:

  • Advanced programming in C++: Classes and objects, dynamic memory allocation, templates, the C++ standard library, strings, container classes, smart pointers, design patterns
  • Stochastic models for asset prices (GBM, local volatility, stochastic volatility, jump diffusion)
  • Financial derivatives, including options on shares (e.g. European, American, digital, barrier, Asian, lookback, compound, chooser)
  • Implied volatility and the construction of the volatility smile
  • Fixed income and rates (bonds and yield-to-maturity, discount factor curve bootstrapping, stochastic interest rate models)
  • Numerical methods (interpolation, numerical quadrature, non-linear solvers, binomial trees (Cox-Ross-Rubinstein), Monte Carlo methods, finite-difference methods for PDEs)

MTH787P Advanced Derivatives Pricing and Risk Management

This module covers a number of advanced topics in the pricing and risk-management of various types of derivative securities that are of key importance in today's financial markets. In particular, the module covers models for interest rate derivatives (short-rate and forward-curve models), and looks at the multi-curve framework. It then considers credit risk management and credit derivatives (both vanilla and exotic). Finally, it also discusses credit valuation adjustment (CVA) and related concepts.

Topics include:

  • Interest rate modelling (instruments, curve bootstrapping, Black's model, short-rate models, the Heath-Jarrow-Morton framework, the LIBOR market model)
  • Credit risk and credit derivatives (instruments, models for default risk, bootstrapping hazard rate curves, copula models for correlation products)
  • Credit risk management and valuation adjustment (general concepts, netting and collateral, CVA, models for wrong-way risk, DVA, FVA and other xVA)

MTH793P Advanced Machine Learning

This module builds on the earlier module "Machine Learning with Python", covering a number of advanced techniques in machine learning, such as dimensionality reduction, support vector machines, decision trees, random forests, and clustering. Although the underlying theoretical ideas are clearly explained, this module is very hands-on, and you will implement various applications using Python in the weekly coursework assignments.

MTH762P Continuous-time Models in Finance

This module explains how we can price financial derivatives in a consistent manner, in the realistic case where the price of the underlying asset changes continuously in time. To do this, we first introduce the key ideas of stochastic calculus in a mathematically rigorous, but still accessible, way. Then, using the Black-Scholes model, we show how we can price a wide range of derivatives, using both the PDE approach and the alternative martingale approach. Finally we look at several more recent models that attempt to rectify some of the known deficiencies of the Black-Scholes model.

MTH761P Financial Instruments and Markets

This module first introduces you to various types of financial instruments, such as bonds and equities, and the markets in which they are traded. We then explain in detail what financial derivatives are, and how they can be used for hedging and speculation. We also look at how investors can construct optimal portfolios of assets by balancing risk and return in an appropriate way. This module will give you the practical knowledge that is essential for a career in investment banking or financial markets.

MTH763P Financial Mathematics Project and Dissertation

The project component of the MSc programme will give you the opportunity to undertake some significant and advanced study in an area of interest, under the guidance of an expert in that field. Many projects involve a substantial amount of programming and analysis. Your project will be assessed by a written dissertation (of up to 60 pages) which you will submit in early September.

Possible project topics may include:

  • The application of a 3-factor HJM model for pricing inflation-linked bonds
  • Credit valuation adjustment (CVA) for interest rate swaps: Investigation of wrong-way risk using Monte Carlo / OpenCL
  • The Heston model and its numerical implementation on a GPU using CUDA C/C++
  • Jump-diffusion models for equity prices
  • The LIBOR market model for interest rate derivatives
  • Option pricing using finite-difference methods on CPUs and GPUs
  • Parallelism in the Alternate Direction Implicit (ADI) method for solving PDEs for stochastic volatility models
  • Pricing passport options
  • The pricing and risk-management of basket credit derivatives (NTDs and CDOs) using Gaussian copula models
  • The SABR stochastic volatility model

MTH771P Foundations of Mathematical Modelling in Finance

This module introduces you to all of the fundamental concepts needed for your future studies in financial mathematics. After reviewing some key ideas from probability theory, we give an overview of some of the most important financial instruments, including shares, forward contracts and options. We next explain how derivative securities can be priced using the principle of no arbitrage. Various models for pricing options are then considered in detail, including the discrete-time binomial model and the continuous-time Black-Scholes model.

Topics include:

  • Review of key concepts in probability theory
  • Introduction to financial markets
  • Pricing derivatives by no-arbitrage arguments
  • Discrete-time option pricing models
  • Introduction to continuous-time stochastic processes and the Black-Scholes model

MTH786P Machine Learning with Python

This module will introduce you to some of the most widely-used techniques in machine learning (ML). After reviewing the necessary background mathematics, we will investigate various ML methods, such as linear regression, polynomial regression and classification with logistic regression. The module covers a very wide range of practical applications, with an emphasis on hands-on numerical work using Python. At the end of the module, you will be able to formalise a ML task, choose the appropriate method to process it numerically, implement the ML algorithm in Python, and assess the method’s performance.

MTHM007 Measure Theory and Probability

This is an introductory module on the Lebesgue theory of measure and integral with application to probability. You are expected to know the theory of Riemann integration. Measure in the line and plane, outer measure, measurable sets, Lebesgue measure, non-measurable sets. Sigma-algebras, measures, probability measures, measurable functions, random variables. Simple functions, Lebesgue integration, integration with respect to general measures. Expectation of random variables. Monotone and dominated convergence theorems, and applications. Absolute continuity and singularity, Radon-Nikodym theorem, probability densities. Possible further topics: product spaces, Fubini's theorem.

Topics include:

  • Measure in the line and plane, outer measure, measurable sets, Lebesgue measure, non-measurable sets.
  • Sigma-algebras, measures, probability measures, measurable functions, random variables.
  • Simple functions, Lebesgue integration, integration with respect to general measures. Expectation of random variables. Monotone and dominated convergence theorems, and applications.
  • Absolute continuity and singularity, Radon-Nikodym theorem, probability densities.
  • Possible further topics: product spaces, Fubini's theorem.

MTH790P Programming in C++ for Finance

This module will provide you with the necessary skills and techniques needed to investigate a variety of practical problems in mathematical finance. It is based on C++, the programming language of choice for many practitioners in the finance industry. You will learn about the basic concepts of the procedural part of C++ (inherited from the earlier C language), before being introduced to the fundamental ideas of object-oriented programming. The module is very ‘hands on’, with weekly sessions in the computer laboratory where you can put your theoretical knowledge into practice with a series of interesting and useful assignments.

Topics include:

  • Overview of technology in finance
  • Introduction to the Microsoft Visual Studio C++ development environment
  • Concepts in C++ such as data types, variables, arithmetic operations and arrays
  • Procedural programming, including branching statements, loops and functions
  • Introduction to object-oriented programming: Objects and classes
  • Examples from finance including bond pricing, histogramming historical price data, option pricing and risk management within the Black-Scholes framework

MTH712P Topics in Probability and Stochastic Processes

This module aims to present some advanced probabilistic concepts and demonstrate their application to stochastic modelling of real-world situations. The topics covered include renewal theory, continuous-time Markov processes and Brownian motion. In addition to exposure to proofs and theoretical material, students develop practical skills through a large number of problems and worked examples. A stochastic process is a system evolving in time in a random way. Besides being of fundamental interest to probability theory, stochastic processes have applications in diverse fields such as financial mathematics, operations research and mathematical biology. This module is an introduction to stochastic processes and related probabilistic concepts. The focus is on continuous time stochastic processes, for which time is an element of the real numbers.

Topics include:

  • Renewal processes: the renewal function; limiting behaviour; current and excess life; characterisation of the Poisson process as a renewal process; the renewal rewards theorem.
  • Continuous time Markov chains: description in terms of the sojourn times and jump chain; definition of the generator; the equation for the transition probability matrix in terms of the generator; use of the generator in ?nding stationary distributions.
  • Brownian motion: the invariance and re?ection principles; hitting times.