Majority of the human activities is subject to risk. The following questions arise: What is risk ? How to measure it ? and finally how to control it ? (is it possible?).
Although primary motivation and typical examples come from mathematics of finance or theory of insurance, the problems of the school are interdisciplinary. A special attention will be devoted to application and computational aspects. It is expected that school shall gather a group of open minded PhD students and young researchers ready to learn, study and then to contribute to that new extremally important from application point of view area of applied mathematics.
Using terminology from insurance and finance the organizers would like to concentrate on the following aspects:
Risk sensitive portfolio. It was Harry Markowitz (future Nobel Price winner in economy in 1990), who started in 1952 to think about risk in financial investments. The economists were interested in maximization of profits so far do not taking into account potential risk corresponding to optimal investments. The first primitive measure of risk was the portfolio variance and mathematically the problem was to maximize portfolio growth (for one period) under the condition that the variance of the portfolio is below certain level or alternatively to minimize the portfolio variance with portfolio growth above certain level. Variance as a measure of risk was then replaced by semivariance and by shortfall, which measured deviations of the portfolio growth below the expected value. In particular risk sensitive cost functionals, which together with the expected portfolio growth measured variance and other moments with certain weights (so called negative risk factors) were introduced in the sequel.
Risk measures and management. The most commonly used in practice method of measuring financial risk exposure is the Value at-Risk (VaR). VaR is a simple (first order) quantile risk measure; this feature results in its limitations for the risk-aversion modelling. Recently, the second order quantile risk measures have been introduced in different ways by many authors. The measure, known as the Conditional Value-at-Risk (CVaR) or the Tail VaR, represents the mean shortfall at a prespecified confidence level. It leads to LP solvable portfolio optimization models in the case of discrete random variables represented by their realizations under the specified scenarios. Further analysis of risk led to axiomatic approach to risk measures introducing instead of dispersion measures so called coherent and convex measures of risk.
Ruin probability and operational risk. The recent increasing interplay between actuarial and financial mathematics has led to a surge of risk theoretic modelling. Especially actuarial ruin probabilities under fairly general conditions on the underlying risk process have become a focus of attention. In the wake of new regulations imposed on financial institutions by the Basel Committee on Banking Supervision, the quantitative modelling of operational losses becomes a key consideration. Typical examples of operational risks include losses resulting from system failure, fraud, litigation, and handling of transactions.
Volatility risk. In the classic approach to volatility risk, the stochastic spot volatility is modelled as an autonomous stochastic process. It is aimed to represent the volatility risk, which disappears when the volatility coefficient is non-random. Usually, both the underlying asset and the stochastic spot volatility are assumed to follow (possibly correlated) diffusion processes. The direct connection to the observed implied volatilities is lost in this approach, and a continual re-calibration is required in order to match the market prices of traded options. Liquid options are taken as primary assets in the recently developed market-based approach to uncertain volatility. By modelling directly the dynamics of the stochastic implied volatility surface, it is possible to achieve the perfect fit to prices of a large class of derivative assets.
Default risk. The major issue in stochastic modelling of default risk is the role of the information flow. Some recent studies are done using an initial enlargement of the filtration. The so-called progressive enlargements are used in default risk setting. The main objective of the school will the presentation of strict mathematical results concerning the following issues: optimization problems based on strategies adapted to the progressively enlarged filtrations, calibration of models with defaults and adequate Monte Carlo methods for simulations of such models, hedging of defaultable claims in the complete and incomplete setup.
Crash modelling. In contrast to the usual modelling of the occurrence of market crashes (see the classical approaches to explain large stock price moves, where assets’ prices are given as Levy processes or other types of stochastic processes with heavy-tailed distributions), models based on a worst-case approach where upper bounds on both the number of crashes until the time horizon and on the maximum height of a single crash are assumed to be known will be presented. Between the crashes, the stock price moves according to a geometric Brownian motion, but at crash times it experiences a sudden large fall. The investor is assumed to be focused on avoiding large losses in adverse situations by putting the worst-case bound on the utility of the terminal wealth at a high level. Such an approach of particular interest for insurance companies. It can also be used for other stochastic control problems where the controlled process might be affected by the so-called catastrophic events. The problem is closely related to high risk scenarios and extremes: the conditional behaviour of risk factors given that a large move on portfolios has been observed.
Innovative financial products. In recent years, the techniques and methodologies developed in the framework of traditional financial markets are used in new areas far beyond its original applications. The real option approach to investment valuation is an example of such extension of the options pricing methodology. From the abstract point of view, any investment process consists of a set of managerial decisions and an uncertain flow of profits and losses associated with each decision strategy. The number of factors which influence the outcome of a given decision strategy is very large, and the influence of a particular factor is hard to isolate. Therefore, it is natural to analyse the qualitative characteristics of the whole set of factors, and not to analyse each factor separately. We plan to discuss a new category of commodities on competitive markets. A special role of these commodities is due to their trade characteristics: their storage is very costly (natural gas) or almost impossible (electricity), and there are subject to two different sources of risk: price risk and volume risk. We shall concentrate on the analysis of a new type of financial products that appeared at the interface of finance and insurance. Their purpose is securitisation, that is, shifting of certain exterior risks from insurance companies to capital markets.
Quality of the research training
The school has a clear goal of the training and transfer of knowledge on stochastic analysis applied to complex financial systems under uncertainty in a large part of European research area. Lecturers and participants of the school will represent institutions from several Member States and other Associate Countries. We expect that the joint activity will create an obvious synergy effect by stimulating co-operation and interaction between participants of different professional backgrounds: mathematicians of unquestionable position in theoretical research, economists and finance specialists. We foresee that a broad variety of expertise of participants and their concentration will add substantially to the strengthening of the European research area. The activity of the school will also add to the development of a common language and better mutual understanding that will allow a more effective communication and transfer of knowledge.