Risk is present in virtually every aspect of human activity and affects the individual and family, the small enterprise, the corporation, the public sector and government. Stochastics, which covers the mathematical theories of probability, statistics and decision-making under uncertainty, is the core discipline for the measurement and management of risk.
The MSc in Risk and Stochastics, conducted by internationally renowned experts, offers in-depth instruction in advanced mathematical risk theory and its ramifications in insurance and finance. The course, launched in 2004, builds on the successful BSc in Actuarial Science within the Department of Statistics at the London School of Economics and Political Science. It is well supported by the School in terms of staff and administration and has been allocated the highest priority by the Department. It draws on world-class research in modern actuarial and financial mathematics within the Department. The programme is LSE's timely response to the strong developments in the interface of insurance and finance, which is manifest in mergers across the industries, in countless novel products, and in the strong impact of modern financial mathematics on insurance mathematics.
The normal entry requirement is a good BSc degree (first class honours but upper second class honours will be considered in exceptional circumstances) in actuarial science, statistics, mathematical economics or mathematics. It should include training in analysis and linear algebra, with rigorous proofs, and probability theory at the level of our third year undergraduate course ST302, a description of which can be found online in the LSE Calendar.
Our taught postgraduate courses are based around lectures, with problem classes and computer workshops. Most courses are assessed by a two-hour exam in the summer term although some contain an element of course work.
Students must take courses to the value of four full units. The courses in the programme are divided into two categories: compulsory courses and options.
Students must take the following five compulsory courses (all half units):
Plus three chosen from these two sets of option courses:
At least one of the following half-unit options:
Up to two of the following half-unit options:
Note: *Students taking FM442 can apply for a place on FM457 MATLAB for MSc students, a non-assessed computer course.
ST409 Stochastic Processes (half-unit)
Core syllabus:A broad introduction to stochastic processes for postgraduates with an emphasis on financial and actuarial applications.
Content: Martingales, Markov Chains, Poisson Processes, Brownian motion, stochastic differential equations and diffusion processes. Applications in Finance. Actuarial applications.
Course guide ST409
ST427 Insurance Mathematics (half-unit)
Core syllabus: This course is a self-contained introduction to life and non-life insurance mathematics.
Content: For life insurance, mortality laws are analysed from a probabilistic and statistical point of view; actuarial analysis of basic insurance products: pure endowment, life insurance/assurance and annuity; prospective/retrospective reserves of these products. Extension to general multi-states policy is studied via Markov chains. Pricing embedded interest and mortality guarantees is introduced. For non-life insurance, axiomatic approach to ordering of risks is presented, optimal forms of insurance from the insured's and from the insurer's point of view are analysed and Pareto-optimal risk exchnages are introduced; standard schemes of reinsurance are introduced and analysed; ruin probability of an insurance company and capital requirement are studied; heavy tail distributions and the extreme value theory are introduced. Case studies on current developments in life and non-life insurance industry are also presented.
Course guide ST427
ST433 Computational Methods in Finance and Insurance (half-unit)
Core syllabus: The purpose of this course is to (a) develop the students' computational skills, (b) introduce a range of numerical techniques of importance in actuarial and financial engineering, and (c) develop the ability of the students to apply the theory from the taught courses to practical problems, work out solutions including numerical work, and to present the results in a written report.
Content: Binomial and trinomial trees. Random number generation, the fundamentals of Monte Carlo simulation and a number of related issues. Finite difference schemes for the solution of ordinary and partial differential equations arising in insurance and finance. Numerical solutions to stochastic differential equations and their implementation.The course ends with an introduction to guidelines for writing a scholarly report/thesis.
Course guide ST433
ST439 Stochastics for Derivatives Modelling (half-unit)
Core syllabus: Valuation and hedging of derivative securities: General principles of mathematical finance.
Content: Asset price models. Option pricing by bilateral Laplace transforms as well as integro-partial differential equations. Utility indifference valuation. Minimal entropy martingale measures. Foellmer-Sondermann optimal hedging. Entropic hedging.
Course guide ST439
ST440 Recent Developments in Finance and Insurance (half-unit)
Core syllabus: Recent developments in the theory of stochastic processes and applications in finance and insurance and their interface.
Content: A variety of topics will be chosen from modelling with Lévy processes; securitisation; energy and commodity markets; actuarial and financial aspects of climate change.
Course guide ST440
ST422 Time Series (half-unit)
Core syllabus: A broad introduction to statistical time series for postgraduates.
Content: Stationarity, Autocorrelation, ARIMA models, identification, estimation, diagnostic checking and linear prediction. Non-stationarity and differencing. Spectral analysis.
Course guide ST422
ST426 Applied Stochastic Processes (half-unit)
Core syllabus: This course builds on material discussed in ST409 Stocastic Processes.
Content: In particular, elements of the general theory of semimartingales will be covered and emphasis will be given on presenting a variety of models involving processes with general dynamics, including jumps. The teory will be applied to a range of topics in mathematical finance and insurance, as well as financial economics.
Course guide ST422
ST429 Probabilistic Methods in Risk Management and Insurance (half-unit)
Core syllabus: A self-contained introduction to probabilistic methods in risk management and insurance.
Content: This course starts with risk factors models and loss distributions, which are illustrated via examples in stocks, derivatives, and bonds portfolios. Value at risk and other risk measures are introduced and analysed. Several methods to model market risk are introduced: variance-covariance method, historical simulation, Monte Carlo simulation and back testing. Multivariate factor models are introduced and analysed: covariance and correlation estimations, multivariate normal distributions and their testing, normal mixture distributions. The theory of copulas is introduced: meta distributions, tail dependence, fitting copulas to data. Some limitations of current approaches are also discussed. Students will be exposed to financial data via sets of computer-based classes and exercises.
Course guide ST429
ST435 Advanced Probability Theory (half-unit)
Core syllabus: The course covers core topics in measure theoretic probability and modern stochastic calculus, thus laying a rigorous foundation for studies in statistics, actuarial science, financial mathematics, economics, and other areas where uncertainty is essential and needs to be described with advanced probability models. Emphasis is on probability theory as such rather than on special models occurring in its applications.
Content: Brief revision of mathematical tools: set theory, logics, techniques of proof, real and complex numbers, sequences, functions, metric spaces, notions of limits and convergence, continuity, differentiation and integration. Brief review of basic probability concepts in a measure theoretic setting: probability spaces, random variables, expected value, conditional probability and expectation, independence. Construction of probability spaces with emphasis on stochastic processes. Operator methods in probability: generating functions, moment generating functions, Laplace transforms, and characteristic functions.
Course guide ST435
ST436 Financial Statistics (half-unit)
Core syllabus: The course covers key statistical methods and data analytic techniques most relevant to finance. Hands-on experience in analysing financial data in the "R" environment is an essential part of the course.
Content: The course includes a selection of the following topics: time series modelling for asset returns and their stylized facts, conditional heteroscedastic models, efficient portfolios and CAPM, multifactor pricing models, intertemporal equilibrium models, present-value models, simulation methods for derivative pricing, statistical inference for high-frequency data, forecast and management of market risks.
Course guide ST436
ST441 Introduction to Markov Processes and their Applications (half-unit)
Core syllabus: A broad introduction to the theory of Markov processes and their use in finance, economics and statistical inference.
Content: Markov property and transition functions. Feller processes. Strong Markov property. Martingale problem and stochastic differential equations, relation with partial differential equations. Diffusion processes. Affine processes. Piecewise deterministic Markov processes.
Course guide ST441
FM404 Forecasting Financial Time Series (half-unit) (Department of Finance)
Core syllabus: This course will examine the techniques involved with forecasting key variables in finance, and how to incorporate model uncertainty into financial forecasts. Students will learn both the theory and the practice of forecasting in finance.
Content: The following topics will be covered: introduction to time series analysis; Maximum Likelihood Estimation (MLE) with time series data, and MLE based model selection; Bayesian inference, posterior probabilities, and Bayesian Model Averaging; Markov Chain Monte Carlo methods; present value regressions, vector autoregressios, causality, and cointegration; asset pricing and the Generalized Method of Moments (GMM); frequentist and Bayesian information theoretic alternatives to GMM.
Course guide FM404
FM441 Derivatives (half-unit) (Department of Finance)
Core syllabus: Provides a thorough grounding in the theory of derivatives pricing and hedging.
Content: This course develops the theories of no-arbitrage asset pricing. Particular emphasis is placed on pricing within a multi-period, mostly continuous-time, framework. A special feature of the course is its coverage of the modern theory of contingent claims valuation by PDE and martingale methods. These asset pricing methods are applied to the pricing of vanilla and exotic options and corporate liabilities, forwards, futures, as well as fixed income derivatives. The uses of derivatives in hedging and risk-management are discussed as well.
Course guide FM441
FM442 Quantitative Methods for Finance and Risk Analysis (half-unit) (Department of Finance)
Core syllabus: A graduate level course on the quantitative and statistical tools that are important in applied finance. Students will be exposed to application of these tools and the key properties of financial data through a set of computer-based classes and exercises.
Content: The following topics will be covered; review of statistics and introduction to time-series econometrics; modelling financial returns; an introduction to the analysis of financial data using MATLAB; volatility models; modelling extreme portfolio returns and Value-at-Risk.
Course guide FM442
FM445 Portfolio Management (half-unit) (Department of Finance)
Core syllabus: This course aims to cover the main topics in equity portfolio management.
Content: Some of the topics covered in the course include: Portfolio optimization techniques; Multi-factor models and their applications; Trading strategies; International portfolio management and currency hedging; Trading costs; Portfolio performance measurement and attribution; Style analysis; Mutual funds; Hedge funds. The course is based on a number of empirical applications and case studies, so that students can gain a better understanding of implementation issues related to managing an equity portfolio.
Course guide FM445
MA409 Continuous Time Optimisation (half-unit) (Department of Mathematics)
Core syllabus: This is a course in optimisation theory using the methods of the Calculus of Variations. No specific knowledge of functional analysis will be assumed and the emphasis will be on examples. It introduces key methods of continuous time optimisation in a deterministic context, and later under uncertainty.
Content: Calculus of variations and the Euler-Lagrange Equations. Sufficiency conditions. Pontryagin Maximum Principle. Extremal controls. Transversality conditions. Linear time-invariant state equations. Bang-bang control and switching functions. Singular control. Dynamical programming. Control under uncertainty. Itô's Lemma. Hamilton-Jacobi-Bellman equation. Verification lemma. Applications to Economics and Finance: Economic Growth models, Consumption and investment, Optimal Abandonment. If time allows: Black-Scholes model.
Course guide MA409
MA411 Probability and Measure (half-unit) (Department of Mathematics)
Core syllabus: The purposes of this course are (a) to explain the formal basis of abstract probability theory, and the justification for basic results in the theory, (b) to explore those aspects of the theory most used in advanced analytical models in economics and finance.
Content: The approach taken will be formal. Probability spaces and probability measures. Random variables. Expectation and integration. Convergence of random variables. Conditional expectation. The Radon-Nikodym Theorem. Martingales. Stochastic processes. Brownian motion. The Ito integral.
Course guide MA411
MA415 The Mathematics of the Black & Scholes Theory (half-unit) (Department of Mathematics)
Core syllabus: This course is concerned with a mathematical development of the risk-neutral valuation theory.
Content: In the context of the binomial tree model for a risky asset, the course introduces the concepts of replication and martingale probability measures. The mathematics of the Black & Scholes methodology follow; in particular, the expression of European contingent claims as expectations with respect to the risk-neutral probability measure of the corresponding discounted payoffs, pricing formulae for European put and call options, and the Black & Scholes PDE are derived. A class of exotic options is then considered. In particular, pricing formulas for lookback and barrier options are derived using PDE techniques as well as the reflection property of the standard Brownian motion.
Course guide MA415
MA416 The Foundations of Interest Rate, Foreign Exchange, and Credit Risk Theory (half-unit) (Department of Mathematics)
Core syllabus: This course is concerned with the mathematical foundations of interest rate and foreign exchange theory.
Content: The course starts with a development of the multi-dimensional Black & Scholes theory with stochastic market data. This is then used to show how discount bond dynamics modelling can be approached by (a) the modelling of the short-rate process and the market price of risk, which underlies the family of short-rate models, or (b) the modelling of the market price of risk and the discount bond volatility structure, which gives rise to the Heath-Jarrow-Morton (HJM) framework. The course then expands on the theory of interest rate market models, foreign exchange dynamics, and credit risk.
Course guide MA416
MA420 Quantifying Risk Modelling and Alternative Markets (half-unit) (Department of Mathematics)
Core syllabus: This course is concerned with various issues arising in the context of investment risk specification as well as with the mathematical theory of so-called alternative markets, such as commodity and energy markets. In particular, the course considers the structural credit risk models and the quantification of risk by means of copulas and risk measures. Also, the course expands on the modeling of alternative markets and addresses the problem of valuation of investments in real assets.
Course guide MA420
Students who graduate from the MSc Risk and Stochastics degree programme are eligible to apply for exemption from the Institute of Actuaries subject 'ST0' on successful completion of the ST433 project.
Further details are available here
The application form can be accessed here
For general information on the MSc programme please email MSc Enquiries at firstname.lastname@example.org.
For advice on your application please refer to the Frequently Asked Questions section of the Graduate Admissions website.
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