The Mathematics and Statistics of Quantitative Risk Management

Abstract

The Mathematics and Statistics of Quantitative Risk Management Workshop , organized by Thomas Mikosch (Copenhagen), Richard A. Davis (New York), and Paul Embrechts (Zürich), was held March 16th–March 22nd, 2008. This meeting was well attended with over 40 participants from four continents. This workshop was a blend of researchers with various backgrounds in mathematical finance, statistics, econometrics, extreme value theory, applied probability, and insurance. Modern quantitative risk management integrates a wide range of sophisticated mathematical techniques and tools. An overview from the statistical side is given in the recent monograph by McNeil, Frey, Embrechts. Relevant areas of research include the theory of high-dimensional data structures; rare event simulation; theory of risk measures; (multivariate) time series analysis; extreme event modeling and extreme value statistics; optimization; and linear, quadratic, and convex programming. Recent questions related to multi-period risk measures involve deep results from a variety of fields. Functional data analysis is instrumental for designing and analyzing risk measures, a geometric theory of extremes is useful for the analysis of generalized risk scenarios, Malliavin calculus has become important for the calculation of risk measure sensitivities, functional regular variation is a relevant concept for analyzing stochastic processes exhibiting extreme behavior, advanced rare event simulation techniques, numerical and optimization methods, Lévy processes and more general diffusions are the building blocks for constructing dynamic stochastic models in finance and econometrics. As evidenced by the recent upheavals in the markets and financial institutions, there is a pressing and critical need to develop and refine tools and methods in quantitative risk management. Expanding on the theory in quantitative risk management should have immediate impact for the financial and insurance industries as well as for supervisory authorities. The objective is to design mathematically tractable, practically relevant and statistically estimable risk measures. An advanced theory also allows one to critically study the present use of tools and methods in quantitative risk management. Risks in insurance and finance are described by mathematical and probabilistic models such as partial differential equations and stochastic differential equations describing the evolution of prices of risky assets – price of stock, composite stock indices, interest rates, foreign exchange rates, commodity prices – or difference equations describing the evolution of financial returns. The 2003 Nobel prize winning ARCH model is an outstanding example. Applications of these models require advanced simulation and numerical methods and statistics plays a vital role in the estimation of unknown parameters (possibly infinite dimensional) from historical data. Due to their complexity, problems of quantitative risk management require multidisciplinary solutions. They involve functional analysts who design and analyze risk measures, probabilists who model with stochastic differential equations and time series, applied probabilists who solve the simulation problems, numerical analysts who deal with high-dimensional integration and optimization problems, and statisticians who fit stochastic models to the data and predict future values of risky assets. Among the challenging problems which were discussed at the meeting are: Some of the main objectives of the workshop are summarized here: