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.
Entry Requirements
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|.
Course Structure
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.
The courses in the programme are divided into two categories: compulsory courses and options.
Students must take courses to the value of four full units.
Compulsory courses:
Plus, two chosen from option courses:
Compulsory Courses
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.
Content
The course consists of two modules that are taught in parallel throughout the term:
ST431.1 Life Insurance
Core syllabus
A self-contained, comprehensive introduction to life and pensions insurance mathematics based on the theory of stochastic processes, notably counting processes and their associated counting martingales. A variety of insurance schemes are analysed, ranging from the traditional participating policy to modern index-linked insurance with benefits dependent on prices of traded securities or other market indices. An introduction to statistical life history analysis is part of the course.
Content
The time continuous non-homogeneous Markov chain model for life histories and developments of life insurance policies. Extension to semi-Markov chains. Diffusion and jump process models for financial markets. Actuarial analysis of basic insurance products: life endowment, life annuity, life assurance, and disability insurance. Extension to the general multi-state policy. The participating policy; surplus and bonus. Index-linked benefits (unit-linked, salary-dependent and others). Pricing of embedded interest and mortality guarantees. Defined benefits v defined contributions. Risk minimization in the framework of theory of incomplete financial markets. Portfolio analysis of combined insurance risk and financial risk; solvency and market value of insurance companies. Pension funding. Statistical life history analysis.
ST431.2 Non-life Insurance
Core syllabus
This module covers the core matter of mathematical risk theory. It gives precise content to the notion of (insurable) risk and presents theories for how to mitigate and possibly eliminate such risk through insurance schemes and, at the level of the insurance companies, through reinsurance. Emphasis is on principles for pricing of insurance products and on solvency control of insurers.
Content
Axiomatic approach to ordering of risks; expected utility; optimal forms of insurance from the insured's and from the insurer's point of view, Pareto-optimal risk exchanges. Premium principles and ordering of risks (stop-loss, convex, and other orders). Ruin theory in various model scenarios including catastrophe risk and investment risk, with application to the issue of optimal reinsurance. Evaluation of total claims distributions for risk portfolios. Value at Risk (VaR). Claims reserving in non-life insurance - a marked point process approach.
Content
The course consists of two modules that are taught in parallel throughout the term:
ST432.1 Derivatives
Core syllabus
Valuation and hedging of derivative securities.
Content
General principles of mathematical finance. Asset price models. Option pricing by bilateral Laplace transforms as well as integro-partial differential equations. Utlilty indifference valuation. Minimal entropy martingale measures. Foellmer-Sondermann optimal hedging. Entropic hedging.
ST432.2 Levy Finance
Core syllabus
Lévy processes with applications in finance.
Content
Lévy processes have stationary and independent increments. They form a surprisingly rich class of processes that comprises Brownian motion, compound Poisson processes, and a huge family of processes with omnipresent jumps. This course gives an introduction to the general theory of such processes and presents a number of special processes commonly used in modelling of financial data. The theory is applied to a range of topics in mathematical finance such as pricing and hedging of options, credit risk models, and interest rate models.
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.
Option courses
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.
Course 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.
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.
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.
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.
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.
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.
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.
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.
Further Information
For general information on the MSc programme and advice about academic requirements please email MSc Enquiries at mscstats@lse.ac.uk|.
For advice on your application please refer to the Frequently Asked Questions section of the Graduate Admissions| website.
New Check out the new MSc Risk and Stochastics brochure here|