Conference Agenda

We address the problem that classical risk measures may not detect the tail risk adequately. This can occur for instance due to the averaging process when computing Expected Shortfall. The current literature proposes a solution, the so-called adjusted Expected Shortfall. This risk measure is the supremum of Expected Shortfalls for all possible levels, adjusted with a function $g$, the so-called target risk profile. We generalize this idea by using other risk measures instead of Expected Shortfall. Therefore, we introduce the concept of general adjusted risk measures. For these the realization of the adjusted risk measure quantifies the minimal amount of capital that has to be raised and injected in a financial position $X$ to ensure that the risk measure is always smaller or equal to the adjustment function $g(p)$ for all levels $p\in[0,1]$. We discuss a variety of assumptions such that desirable properties for risk measures are satisfied in this setup. From a theoretical point of view, our main contribution is the analysis of equivalent assumptions such that a general adjusted risk measure is positive homogeneous and subadditive. Furthermore, we show that these conditions hold for a bunch of new risk measures, beyond the adjusted Expected Shortfall. For these risk measures, we derive their dual representations. Finally, we test the performance of these new risk measures in a case study based on the S$\&$P $500$.

In this talk, we revisit the recently introduced concept of return risk measures (RRMs). We extend it by allowing risk management via multiple so-called eligible assets. The resulting new risk measures are called multi-asset return risk measures (MARRMs). We analyze properties of these risk measures. In particular, we prove that a positively homogeneous MARRM is quasi-convex if and only if it is convex. Furthermore, we state conditions to avoid inconsistent risk evaluations Then, we point out the connection between MARRMs and the well-known concept of multi-asset risk measures (MARMs). This is used to obtain dual representations of MARRMs. Moreover, we compare RRMs, MARMs, and MARRMs in numerous case studies. Using typical continuous-time financial markets and different notions of acceptability of losses, we compare MARRMs and MARMs and draw conclusions about the cost of risk mitigation. In a real-world example, we compare the relative difference between RRMs and MARRMs in times of crisis.

The basic principle of any version of insurance is the paradigm that exchanging risk by sharing it in a pool is beneficial to all participants. In case of independent risks with a finite mean this is typically the case for risk averse decision makers due to the law of large numbers. The situation may be very different in case of infinite mean models. In that case risk sharing may have a negative effect. For the case of stable distributions this has been described by Ibragimov et al. (2009) where this effect is called the nondiversification trap. In a series of recent papers this has been studied further by Chen, Wang and coauthors, who obtained similar results for infinite mean Pareto and Fr\'echet distributions. We further investigate this property by showing that many of these results can be obtained as special cases of a simple result demonstrating that this holds for any distribution that is more skewed than a Cauchy distribution. We also relate this to the situation of deadly catastrophic risks, where we assume the possibility of a positive probability for an infinite damage. That case give a very simple intuition why this phenomenon can occur for catastrophic risks.

We study optimal stopping problems for two-dimensional geometric Brownian motions driven by constantly correlated standard Brownian motions on an infinite time interval. These problems are related to the pricing of perpetual American options such as basket options (with an additive payoff structure) and traffic-light options (with a multiplicative payoff structure) in a two-dimensional Black-Merton-Scholes model. We find closed formulas for the value functions expressed in terms of the optimal stopping boundaries which in turn are shown to be unique solutions to the appropriate nonlinear Fredholm integral equations. A key argument in the existence proof is played by a pointwise maximisation of the expressions obtained by the change-of-measure arguments. This provides tight bounds on the optimal stopping boundaries describing its asymptotic behaviour for marginal coordinate values.