DFG-Forschungszentrum "Mathematik für Schlüsseltechnologien - MATHEON": Beyond Value at Risk - Quantifying and Hedging the Downside Risk (Projekt E 4)

Hedging financial risk has become a major topic in academia as well as in the financial industry. In the context of the mathematical theory of incomplete financial markets, the key idea is to find a dynamic investment strategy which is optimal in terms of some criterion of risk-minimization. For criteria in terms of quadratic measures of risk, this problem has been investigated in depth and Berlin has played a leading role in this development. Recently, there is a growing emphasis on quantitative measures of the downside risk which reflect the concerns of a supervising agency. Such a measure of risk should determine the minimal capital requirement which, if added to a given financial position, makes the position acceptable from the point of view of risk management. "Value at Risk", the current industry standard, has serious deficiencies, both on the practical and on the theoretical level. In particular, it does not satisfy some natural requirements of consistency. This has triggered a systematic investigation of the general structure of reasonable measures of risk, leading to the notions of coherent risk measures and of convex risk measures.