Alfred Müller

A spot market model for pricing derivatives in electricity markets

Abstract In this paper, we analyse the evolution of prices in deregulated electricity markets. We present a general model that simultaneously takes into account the following features: seasonal patterns, price spikes, mean reversion, price dependent volatilities and long term non-stationarity. We estimate the parameters of the model using historical data from the European Energy Exchange. Finally, we demonstrate how it can be used for pricing derivatives via Monte Carlo simulation.

Stop-loss order for portfolios of dependent risks

Abstract The paper considers the riskiness of portfolios of dependent risks. The supermodular stochastic order is used to compare the dependence of multivariate distributions with equal marginals. It is shown that supermodular ordering implies stop-loss order of the portfolios. Moreover, the riskiest portfolio under all portfolios with equal marginals is characterized. This extends the results of Dhaene and Goovaerts (1996, 1997).

Integral probability metrics and their generating classes of functions

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

Modeling and Comparing Dependencies in Multivariate Risk Portfolios

In this paper we investigate multivariate risk portfolios, where the risks are dependent. By providing some natural models for risk portfolios with the same marginal distributions we are able to compare two portfolios with different dependence structure with respect to their stop-loss premiums. In particular, some comparison results for portfolios with two-point distributions are obtained. The analysis is based on the concept of the so-called supermodular ordering. We also give some numerical results which indicate that dependencies in risk portfolios can have a severe impact on the stop-loss premium. In fact, we show that the effect of dependencies can grow beyond any given bound.

Stochastic Comparison of Random Vectors with a Common Copula

We consider two random vectorsX andY, such that the components of  X are dominated in the convex order by the corresponding components of  Y. We want to find conditions under which this implies that any positive linear combination of the components of  X is dominated in the convex order by the same positive linear combination of the components of  Y. This problem has a motivation in the comparison of portfolios in terms of risk. The conditions for the above dominance will concern the dependence structure of the two random vectorsX andY, namely, the two random vectors will have a common copula and will be conditionally increasing. This new concept of dependence is strictly related to the idea of conditionally increasing in sequence, but, in addition, it is invariant under permutation. We will actually prove that, under the above conditions,X will be dominated byY in the directionally convex order, which yields as a corollary the dominance for positive linear combinations. This result will be applied to a portfolio optimization problem.

STOCHASTIC ORDERS GENERATED BY INTEGRALS: A UNIFIED STUDY

We consider stochastic orders of the following type. Let t be a class of functions and let P and Q be probability measures. Then define Pg- Q, if ffdP- ffdQ for all fin t. Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

Stochastic Ordering of Multivariate Normal Distributions

We show an interesting identity for Ef(Y) − Ef(X), where X, Yare normally distributed random vectors and f is a function fulfilling some weak regularity condition. This identity will be used for a unified derivation of sufficient conditions for stochastic ordering results of multivariate normal distributions, some well known ones as well as some new ones. Moreover, we will show that many of these conditions are also necessary. As examples we will consider the usual stochastic order, convex order, upper orthant order, supermodular order and directionally convex order.

Evidence for long‐term memory in sea level

Detection and attribution of anthropogenic climate change signals in sea level rise (SLR) has experienced considerable attention during the last decades. Here we provide evidence that superimposed on any possible anthropogenic trend there is a significant amount of natural decadal and multidecadal variability. Using a set of 60 centennial tide gauge records and an ocean reanalysis, we find that sea levels exhibit long-term correlations on time scales up to several decades that are independent of any systematic rise. A large fraction of this long-term variability is related to the steric component of sea level, but we also find long-term correlations in current estimates of mass loss from glaciers and ice caps. These findings suggest that (i) recent attempts to detect a significant acceleration in regional SLR might underestimate the impact of natural variability and (ii) any future regional SLR threshold might be exceeded earlier/later than from anthropogenic change alone.

Asymptotic ruin probabilities for risk processes with dependent increments

In this paper, we derive a Lundberg type result for asymptotic ruin probabilities in the case of a risk process with dependent increments. We only assume that the probability generating functions exist, and that their logarithmic average converges. Under these assumptions we present an elementary proof of the Lundberg limiting result, which only uses simple exponential inequalities, and does not rely on results from large deviation theory. Moreover, we use dependence orderings to investigate, how dependencies between the claims affect the Lundberg coefficient. The results are illustrated by several examples, including Gaussian and AR(1)-processes, and a risk process with adapted premium rules.

Orderings of risks: A comparative study via stop-loss transforms

Abstract The relations between the following concepts for ordering risks are investigated: stochastic dominance, stop-loss order, convex order and being more dangerous. Using characterizations via stop-loss transforms, we give an elementary proof of the separation theorem for stop-loss order, and we correct a mistake in a result of van Heerwaarden (1991) about the connection between stop-loss order and being more dangerous. This is done by introducing a new notion of convergence for distributions. Moreover, we consider lattice properties of these orders.