Financial Mathematics MSc

Course Content Programme structure

The programme offers five "compulsory" modules, taken by all candidates, along with a variety of elective modules from which students can pick and choose. There are lectures, examinations and coursework in eight modules altogether, including the five compulsory modules. Additionally, all students complete an individual research project on a selected topic in financial mathematics, leading to the submission of a dissertation.

Compulsory modules

Probability and stochastics. This course provides the basics of the probabilistic ideas and mathematical language needed to fully appreciate the modern mathematical theory of finance and its applications. Topics include: measurable spaces, sigma-algebras, filtrations, probability spaces, martingales, continuous-time stochastic processes, Poisson processes, Brownian motion, stochastic integration, Ito calculus, log-normal processes, stochastic differential equations, the Ornstein-Uhlenbeck process.

Financial markets. This course is designed to cover basic ideas about financial markets, including market terminology and conventions. Topics include: theory of interest, present value, future value, fixed-income securities, term structure of interest rates, elements of probability theory, mean-variance portfolio theory, the Markowitz model, capital asset pricing model (CAPM), portfolio performance, risk and utility, portfolio choice theorem, risk-neutral pricing, derivatives pricing theory, Cox-Ross-Rubinstein formula for option pricing.

Option pricing theory. The key ideas leading to the valuation of options and other important derivatives will be introduced. Topics include: risk-free asset, risky assets, single-period binomial model, option pricing on binomial trees, dynamical equations for price processes in continuous time, Radon-Nikodym process, equivalent martingale measures, Girsanov's theorem, change of measure, martingale representation theorem, self-financing strategy, market completeness, hedge portfolios, replication strategy, option pricing, Black-Scholes formula.

Interest rate theory. An in-depth analysis of interest-rate modelling and derivative pricing will be presented. Topics include: interest rate markets, discount bonds, the short rate, forward rates, swap rates, yields, the Vasicek model, the Hull-White model, the Heath-Jarrow-Merton formalism, the market model, bond option pricing in the Vasicek model, the positive interest framework, option and swaption pricing in the Flesaker-Hughston model.

Financial computing I. The idea of this course is to enable students to learn how the theory of pricing and hedging can be implemented numerically. Topics include: (i) The Unix/Linux environment, C/C++ programming: types, decisions, loops, functions, arrays, pointers, strings, files, dynamic memory, preprocessor(ii) data structures: lists and trees(iii) introduction to parallel (multi-core, shared memory) computing: open MP constructsapplications to matrix arithmetic, finite difference methods, Monte Carlo option pricing.

Elective modules

Portfolio theory. The general theory of financial portfolio based on utility theory will be introduced in this module. Topics include: utility functions, risk aversion, the St Petersburg paradox, convex dual functions, dynamic asset pricing, expectation, forecast and valuation, portfolio optimisation under budget constraints, wealth consumption, growth versus income.

Information in finance with application to credit risk management. An innovative and intuitive approach to asset pricing, based on the modelling of the flow of information in financial markets, will be introduced in this module. Topics include: information-based asset pricing – a new paradigm for financial risk managementmodelling frameworks for cash flows and market informationapplications to credit risk modelling, defaultable discount bond dynamics, the pricing and hedging of credit-risky derivatives such as credit default swaps (CDS), asset dependencies and correlation modelling, and the origin of stochastic volatility.

Mathematical theory of dynamic asset pricing. Financial modelling and risk management involve not only the valuation and hedging of various assets and their positions, but also the problem of asset allocation. The traditional approach of risk-neutral valuation treats the problem of valuation and hedging, but is limited when it comes to understanding asset returns and the behaviour of asset prices in the real-world 'physical' probability measure. The pricing kernel approach, however, treats these different aspects of financial modelling in a unified and coherent manner. This module introduces in detail the techniques of pricing kernel methodologies, and its applications to interest-rete modelling, foreign exchange market, and inflation-linked products. Another application concerns the modelling of financial markets where prices admit jumps. In this case, the relation between risk, risk aversion, and return is obscured in traditional approaches, but is made clear in the pricing kernel method. The module also covers the introduction to the theory of Lévy processes for jumps and its applications to dynamic asset pricing in the modern setting.

Financial computing II: In this parallel-computing module students will learn how to harness the power of a multi-core computer and Open MP to speed up a task by running it in parallel. Topics include: shared and distributed memory conceptsMessage Passing and introduction to MPI constructscommunications models, applications and pitfallsopen MP within MPIintroduction to Graphics ProcessorsGPU computing and the CUDA programming modelCUDA within MPIapplications to matrix arithmetic, finite difference methods, Monte Carlo option pricing.

Statistics for Finance. This module includes:

DISTRIBUTIONS OF RETURNS: return and loss distributionsstatistical properties of return distributions: mean, variance, skewness and kurtosistests for normality and QQ plotsheavy-tail distributions.

RISK MEASURES: definition of risk and risk measuresValue-at-RiskConditional Value-at Risk / Expected shortfallnon-parametric, parametric and semi-parametric approaches for the estimation of risk measurescoherent risk measuresforecasting risk measuresbacktesting methods: conditional and unconditional tests.

TIME SERIES MODELS: basics of time series modelling: mean, autocovariance, autocorrelation, stationarity, parameter estimation, model selection and forecastingwhite noise process, autoregressive and moving average (ARMA) process, generalised autoregressive conditional heteroscedasticity (GARCH) processforecasting risk measures using ARMA-GARCH processes. Financial Mathematics Dissertation

Towards the end of the Spring Term, students will choose a topic for an individual research project, which will lead to the preparation and submission of an MSc dissertation. The project supervisor will usually be a member of the Brunel financial mathematics group. In some cases the project may be overseen by an external supervisor based at a financial institution or another academic institution.

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