The main tools and cocepts of financial and actuarial theory are designed to handle standards, or even small risk. The aim of this paper is to reconsider some selected financial problems, in a setup including infrequent extreme risks. We first consider investors maximizing the expected utility...

In discrete time, every time-consistent dynamic monetary risk measure can be written as a composition of one-step risk measures. We exploit this structure to give new dual representation results for time-consistent convex monetary risk measures in terms of one-step penalty functions. We first...

We consider a class of law-invariant convex risk measures which have a.robust representation of the form ρ(X ) = ...... . The supremum is taken over the set of all Radon Nikodym derivatives corresponding to the set of all probability measures on B(0,1] which are absolutely continuous with...

A new class of risk measures called cash sub-additive risk measures is introduced to assess the risk of future financial, non-financial and insurance positions. The debated cash additive axiom is relaxed into the cash sub-additive axiom to preserve the original difference between the numeraire...

In this paper, we show that coherent upper and lower previsions as well as coherent risk measures are only meaningful under the assumption that one starts with initial wealth being constantly 0. This implies at least for coherent upper and lower previsions a correction of their interpretation,...

A class of contribution values for pairs of random variables is introduced as a technical tool for the problem how the risk capital needed for a portfolio of random activities should be allocated to its components. The well known allocation model with expected shortfall as corresponding risk...

It is common practice to describe the future evolution of a financial profit by a continuous-time stochastic model. A risk measure can then be viewed as a functional on a space of continuous-time stochastic processes. We extend the notions of coherent and convex risk measures to the space of...

In empirical modeling, there have been two strands for pricing in the options literature, namely the parametric and nonparametric models. Often, the support for the nonparametric methods is based on a benchmark such as theBlack-Scholes model with constant volatility. In this paper, we examine...