Ludger Rüschendorf

An Academic Response to Basel 3.5

Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we highlight some of the underlying issues, both methodologically, as well as through examples. In particular, we frame this discussion in the context of two recent regulatory documents we refer to as Basel 3.5.

Minimax and minimal distance martingale measures and their relationship to portfolio optimization

Abstract. In this paper we give a characterization of minimal distance martingale measures with respect to f-divergence distances in a general semimartingale market model. We provide necessary and sufficient conditions for minimal distance martingale measures and determine them explicitly for exponential Lévy processes with respect to several classical distances. It is shown that the minimal distance martingale measures are equivalent to minimax martingale measures with respect to related utility functions and that optimal portfolios can be characterized by them. Related results in the context of continuous-time diffusion models were first obtained by He and Pearson (1991b) and Karatzas et al. (1991) and in a general semimartingale setting by Kramkov and Schachermayer (1999). Finally parts of the results are extended to utility-based hedging.

The contraction method for recursive algorithms

In this paper we give an introduction to the analysis of algorithms by the contraction method. By means of this method several interesting classes of recursions can be analyzed as particular cases of our general framework. We introduce the main steps of this technique which is based on contraction properties of the algorithm with respect to suitable probability metrics. Typically the limiting distribution is characterized as a fixed point of a limiting operator on the class of probability distributions. We explain this method in the context of several “divide and conquer” algorithms. In the second part of the paper we introduce a new quite general model for branching dynamical systems and explain that the contraction method can be applied in this model. This model includes many classical examples of random trees and gives a general frame for further applications.

A characterization of random variables with minimum L 2 -distance

A complete characterization of multivariate random variables with minimum L2 Wasserstein-distance is proved by means of duality theory and convex analysis. This characterization allows to determine explicitly the optimal couplings for several multivariate distributions. A partial solution of this problem has been found in recent papers by Knott and Smith.

On convex risk measures on L p -spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on Lp spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on Lp we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for Lp risks.

Solution of a statistical optimization problem by rearrangement methods

SummaryInequalities for the rearrangement of functions are applied to obtain a solution of a statistical optimization problem. This optimization problem arises in situations where one wants to describe the influence of stochastic dependence on a statistical problem.

Computation of sharp bounds on the distribution of a function of dependent risks

We propose a new algorithm to compute numerically sharp lower and upper bounds on the distribution of a function of d dependent random variables having fixed marginal distributions. Compared to the existing literature, the bounds are widely applicable, more accurate and more easily obtained.

The Wasserstein distance and approximation theorems

SummaryBy an extension of the idea of the multivariate quantile transform we obtain an explicit formula for the Wasserstein distance between multivariate distributions in certain cases. For the general case we use a modification of the definition of the Wasserstein distance and determine optimal ‘markov-constructions’. We give some applications to the problem of approximation of stochastic processes by simpler ones, as e.g. weakly dependent processes by independent sequences and, finally, determine the optimal martingale approximation to a given sequence of random variables; the Doob decomposition gives only the ‘one-step optimal’ approximation.

On c-optimal random variables

A characterization is proved for random variables which are optimal couplings w.r.t. a general function c. It turns out that on very general probability spaces optimal couplings can be characterized by generalized subgradients of c-convex functions. An interesting application of optimal couplings are minimal lp-metrics.

Inequalities for the expectation of Δ-monotone functions

SummaryFor some subsets of the set of all Δ-monotone functions on [0,1] n we characterize distribution functions F, G such that EFf≦EGf for all f within these subsets. Furthermore, we determine sharp upper and lower bounds of integrals of functions in these subsets w.r.t. all distributions with fixed marginals and give some applications of these results.