# The Mathematics and Statistics of Quantitative Risk Management

### Thomas Mikosch

University of Copenhagen, Denmark### Paul Embrechts

ETH Zürich, Switzerland### Richard A. Davis

Columbia University, New York, USA

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## Abstract

\emph{The Mathematics and Statistics of Quantitative Risk Management Workshop}, organized by Thomas Mikosch (Copenhagen), Richard A. Davis (New York), and Paul Embrechts (Z\"urich), 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, L\'evy processes and more general diffusions are the building blocks for constructing dynamic stochastic models in finance and econometrics. \par 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. \par 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. \par 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. \par Among the challenging problems which were discussed at the meeting are: \begin{itemize} \item Risk problems are often high-dimensional: a portfolio typically consists of several hundred assets. Modern mathematics and statistics does not offer immediate solutions. For example, the number of historical observations is often smaller than the number of parameters in the model. Techniques from function data analysis (FDA) may prove useful in this context. FDA methods are designed to deal with panel data in which the number of panels, which consist of time series, can be large. \medskip \item Risks are dependent across the assets and through time. A key problem is the sensitivity of a particular modeling paradigm to model miss-specification of multivariate models. Robustness to parameter estimation does not quite fit the bill, since, for example, parameters coming from a particular copula (arising from a multivariate distribution) may be completely meaningless if the true model does not involve such quantities. Emphasizing this aspect of sensitivity to model miss-specification encompasses a number of the issues that were ultimately addressed at this workshop. \medskip \item Financial and insurance data are not stationary. They contain structural breaks due to changes in the economic or social environments. A relevant question is how such changes can be incorporated in theoretical models and in the corresponding statistical analysis of data. Given one accepts structural breaks, a natural questions arises as to the range of data on which one may conduct reliable inferences. \medskip \item Various popular models for risk management are based on statistical ideas and techniques (copulas, variance-covariance models, historical simulation,...). Although these methods are popular, their limitations have not been theoretically studied. For example, it is unclear what sense popular classes of copulas (Gaussian, student, Archimedean, etc.) achieve in a universe of multivariate distributions where the classes of distributions described by them are far from being dense in the class of all multivariate distributions. The discussions at the workshop did not solve the problem, but the talks given brought more theoretical clarity as regards the estimation of certain types of copulas such as Archimedean, extreme value, and Paretean copula. \medskip \item Modern risk management asks for the determination/estimation and aggregation of risk measures calculated at high quantiles (99.9\ and across different time periods, from ten days to one year. This requires careful statistical analysis. The discussions showed that multivariate extreme value theory comes close to its boundaries of applicability and techniques. Rare event simulation using importance sampling can be useful, but may break down when heavy-tailed risks are involved. \medskip \item It was also pointed out where mathematical theory reaches its limits. For instance, the non-existence of useful risk measures on spaces of random variables with infinite mean (as a consequence of results in functional analysis) was shown. The numerical calculation of risk measures and the solution of related optimization questions (capital allocation, calculation of worst case scenarios) leads to challenging mathematical problems which can be hard to solve. \medskip \item A natural topic of the workshop was the recent worldwide crisis of credit portfolios. In the past, mathematical models have been designed to avoid the present situation and they are implemented in the framework of the Basel II accord. But they obviously have not been used successfully. Both formal and informal reasons for the present situation were discussed. Although it would be inappropriate to blame a mathematical model for its failure, there is evidence that various models are too simplistic and do not incorporate market information sufficiently fast. Further, it appears that the statistical analysis of the data was not conducted with sufficient care. \end{itemize} \par \medskip Some of the main objectives of the workshop are summarized here: \begin{itemize} \item Theory and statistical practice of risk management bear a multitude of contradictory problems which were discussed in a rigorous way. \medskip \item The workshop emphasized some of the major problems in this area. The critique mainly concerns statistical problems although modeling problems (called ``model risk'' in practice) were given serious consideration. \medskip \item The workshop brought together some of the leading academic researchers to discuss successes, failures and limitations of present statistical technology in risk management. \medskip \item The mixture of researchers from different fields who often do not go to the same conferences, was viewed as a successful experiment by all participants. \medskip \item The workshop set the stage for future statistical and mathematicalresearch in the area of quantitative risk management. At present there seem to exist more problems than solutions. Therefore a future meeting (perhaps in 2011) to address these issues would be useful.