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Volume 3 Issue 1

January 2015

Issue of Dependence Modeling

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Contents
Journal Overview

Open Access April 16, 2015

Cost-efficiency in multivariate Lévy models

Ludger Rüschendorf, Viktor Wolf

Article number: 000010151520150001

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Abstract

In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.

Open Access May 21, 2015

Building bridges between Mathematics, Insurance and Finance

An interview with Paul Embrechts

Fabrizio Durante, Giovanni Puccetti, Matthias Scherer

Article number: 000010151520150002

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Abstract

Paul Embrechts is Professor of Mathematics at the ETH Zurich specializing in Actuarial Mathematics and Quantitative Risk Management. Previous academic positions include the Universities of Leuven, Limburg and London (Imperial College). Dr. Embrechts has held visiting professorships at several universities, including the Scuola Normale in Pisa (Cattedra Galileiana), the London School of Economics (Centennial Professor of Finance), the University of Vienna, Paris 1 (Panthéon-Sorbonne), theNationalUniversity of Singapore, KyotoUniversity,was Visiting Man Chair 2014 at the Oxford-Man Institute of Oxford University and has an Honorary Doctorate from the University of Waterloo, Heriot-Watt University, Edinburgh, and the Université Catholique de Louvain. He is an Elected Fellow of the Institute of Mathematical Statistics and the American Statistical Association, Honorary Fellow of the Institute and the Faculty of Actuaries, Actuary-SAA, Member Honoris Causa of the Belgian Institute of Actuaries and is on the editorial board of numerous scientific journals.He belongs to various national and international research and academic advisory committees. He co-authored the influential books Modelling of Extremal Events for Insurance and Finance, Springer, 1997 [8] andQuantitative RiskManagement: Concepts, Techniques and Tools, Princeton UP, 2005, 2015 [14] and published over 180 scientific papers. Dr. Embrechts consults on issues in Quantitative Risk Management for financial institutions, insurance companies and international regulatory authorities.

Open Access May 22, 2015

On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions

German Bernhart, Jan-Frederik Mai, Matthias Scherer

Article number: 000010151520150003

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Abstract

Min-stable multivariate exponential (MSMVE) distributions constitute an important family of distributions, among others due to their relation to extreme-value distributions. Being true multivariate exponential models, they also represent a natural choicewhen modeling default times in credit portfolios. Despite being well-studied on an abstract level, the number of known parametric families is small. Furthermore, for most families only implicit stochastic representations are known. The present paper develops new parametric families of MSMVE distributions in arbitrary dimensions. Furthermore, a convenient stochastic representation is stated for such models, which is helpful with regard to sampling strategies.

Open Access May 25, 2015

Equivalent or absolutely continuous probability measures with given marginals

Patrizia Berti, Luca Pratelli, Pietro Rigo, Fabio Spizzichino

Article number: 000010151520150004

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Abstract

Let (X,A) and (Y,B) be measurable spaces. Supposewe are given a probability α on A, a probability β on B and a probability μ on the product σ-field A ⊗ B. Is there a probability ν on A⊗B, with marginals α and β, such that ν ≪ μ or ν ~ μ ? Such a ν, provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of ν are provided, distinguishing ν ≪ μ from ν ~ μ.

Open Access May 28, 2015

Forecasting time series with multivariate copulas

Clarence Simard, Bruno Rémillard

Article number: 000010151520150005

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Abstract

In this paper we present a forecasting method for time series using copula-based models for multivariate time series. We study how the performance of the predictions evolves when changing the strength of the different possible dependencies, as well as the structure of the dependence. We also look at the impact of the marginal distributions. The impact of estimation errors on the performance of the predictions is also considered. In all the experiments, we compare predictions from our multivariate method with predictions from the univariate version which has been introduced in the literature recently. To simplify implementation, a test of independence between univariate Markovian time series is proposed. Finally, we illustrate the methodology by a practical implementation with financial data.

Open Access June 18, 2015

Measuring association via lack of co-monotonicity: the LOC index and a problem of educational assessment

Danang Teguh Qoyyimi, Ricardas Zitikis

Article number: 000010151520150006

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Abstract

Measuring association, or the lack of it, between variables plays an important role in a variety of research areas, including education,which is of our primary interest in this paper. Given, for example, student marks on several study subjects, we may for a number of reasons be interested in measuring the lack of comonotonicity (LOC) between the marks, which rarely follow monotone, let alone linear, patterns. For this purpose, in this paperwe explore a novel approach based on a LOCindex,which is related to, yet substantially different from, Eckhard Liebscher’s recently suggested coefficient of monotonically increasing dependence. To illustrate the new technique,we analyze a data-set of student marks on mathematics, reading and spelling.

Open Access July 22, 2015

Dependence Measuring from Conditional Variances

Noppadon Kamnitui, Tippawan Santiwipanont, Songkiat Sumetkijakan

Article number: 000010151520150007

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Abstract

A conditional variance is an indicator of the level of independence between two random variables. We exploit this intuitive relationship and define a measure v which is almost a measure of mutual complete dependence. Unsurprisingly, the measure attains its minimum value for many pairs of non-independent ran- dom variables. Adjusting the measure so as to make it invariant under all Borel measurable injective trans- formations, we obtain a copula-based measure of dependence v* satisfying A. Rényi’s postulates. Finally, we observe that every nontrivial convex combination of v and v* is a measure of mutual complete dependence.

Open Access September 9, 2015

An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem

Frank Oertel

Article number: 000010151520150008

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Abstract

We revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.

Open Access October 16, 2015

Seven Proofs for the Subadditivity of Expected Shortfall

Paul Embrechts, Ruodu Wang

Article number: 000010151520150009

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Abstract

Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.

Open Access October 16, 2015

High level quantile approximations of sums of risks

A. Cuberos, E. Masiello, V. Maume-Deschamps

Article number: 000010151520150010

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Abstract

The approximation of a high level quantile or of the expectation over a high quantile (Value at Risk (VaR) or Tail Value at Risk (TVaR) in risk management) is crucial for the insurance industry.We propose a new method to estimate high level quantiles of sums of risks. It is based on the estimation of the ratio between the VaR (or TVaR) of the sum and the VaR (or TVaR) of the maximum of the risks. We show that using the distribution of the maximum to approximate the VaR is much better than using the marginal. Our method seems to work well in high dimension (100 and higher) and gives good results when approximating the VaR or TVaR in high levels on strongly dependent risks where at least one of the risks is heavy tailed.

Open Access October 26, 2015

Multivariate Markov Families of Copulas

Ludger Overbeck, Wolfgang M. Schmidt

Article number: 000010151520150011

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Abstract

For the Markov property of a multivariate process, a necessary and suficient condition on the multidimensional copula of the finite-dimensional distributions is given. This establishes that the Markov property is solely a property of the copula, i.e., of the dependence structure. This extends results by Darsow et al. [11] from dimension one to the multivariate case. In addition to the one-dimensional case also the spatial copula between the different dimensions has to be taken into account. Examples are also given.

Open Access October 27, 2015

Quantile of a Mixture with Application to Model Risk Assessment

Carole Bernard, Steven Vanduffel

Article number: 000010151520150012

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Abstract

We provide an explicit expression for the quantile of a mixture of two random variables. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial dependence information is available. This paper complements the work of [4].

Open Access October 29, 2015

A Journey from Statistics and Probability to Risk Theory An interview with Ludger Rüschendorf

Fabrizio Durante, Giovanni Puccetti, Matthias Scherer

Article number: 000010151520150013

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Open Access November 3, 2015

Bivariate copulas, norms and non-exchangeability

Pier Luigi Papini

Article number: 000010151520150014

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Abstract

The present paper is related to the study of asymmetry for copulas by introducing functionals based on different norms for continuous variables. In particular, we discuss some facts concerning asymmetry and we point out some flaws occurring in the recent literature dealing with this matter.

Open Access November 19, 2015

On the tail dependence in bivariate hydrological frequency analysis

Alexandre Lekina, Fateh Chebana, Taha B. M. J. Ouarda

Article number: 000010151520150015

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Abstract

In Bivariate Frequency Analysis (BFA) of hydrological events, the study and quantification of the dependence between several variables of interest is commonly carried out through Pearson’s correlation (r), Kendall’s tau (τ) or Spearman’s rho (ρ). These measures provide an overall evaluation of the dependence. However, in BFA, the focus is on the extreme events which occur on the tail of the distribution. Therefore, these measures are not appropriate to quantify the dependence in the tail distribution. To quantify such a risk, in Extreme Value Analysis (EVA), a number of concepts and methods are available but are not appropriately employed in hydrological BFA. In the present paper, we study the tail dependence measures with their nonparametric estimations. In order to cover a wide range of possible cases, an application dealing with bivariate flood characteristics (peak flow, flood volume and event duration) is carried out on three gauging sites in Canada. Results show that r, τ and ρ are inadequate to quantify the extreme risk and to reflect the dependence characteristics in the tail. In addition, the upper tail dependence measure, commonly employed in hydrology, is shown not to be always appropriate especially when considered alone: it can lead to an overestimation or underestimation of the risk. Therefore, for an effective risk assessment, it is recommended to consider more than one tail dependence measure.

Open Access November 30, 2015

A theory for non-linear prediction approach in the presence of vague variables: with application to BMI monitoring

R. Pourmousa, M. Rezapour, M. Mashinchi

Article number: 000010151520150016

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Abstract

In the statistical literature, truncated distributions can be used for modeling real data. Due to error of measurement in truncated continuous data, choosing a crisp trimmed point caucuses a fault inference, so using fuzzy sets to define a threshold pointmay leads us more efficient results with respect to crisp thresholds. Arellano-Valle et al. [2] defined a selection distribution for analysis of truncated data with crisp threshold. In this paper, we define fuzzy multivariate selection distribution that is an extension of the selection distributions using fuzzy threshold. A practical data set with a fuzzy threshold point is considered to investigate the relationship between high blood pressure and BMI.

Open Access December 12, 2015

A classification method for binary predictors combining similarity measures and mixture models

Seydou N. Sylla, Stéphane Girard, Abdou Ka Diongue, Aldiouma Diallo, Cheikh Sokhna

Article number: 000010151520150017

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Abstract

In this paper, a new supervised classification method dedicated to binary predictors is proposed. Its originality is to combine a model-based classification rule with similarity measures thanks to the introduction of new family of exponential kernels. Some links are established between existing similarity measures when applied to binary predictors. A new family of measures is also introduced to unify some of the existing literature. The performance of the new classification method is illustrated on two real datasets (verbal autopsy data and handwritten digit data) using 76 similarity measures.

About this journal

The journal Dependence Modeling aims at providing a medium for exchanging results and ideas in the area of multivariate dependence modeling.

It is an open access fully peer-reviewed journal providing the readers with free, instant, and permanent access to all content worldwide.

Dependence Modeling is listed by Web of Science (Emerging Sources Citation Index), Scopus, MathSciNet and Zentralblatt Math.

The journal presents different types of articles:

"Research Articles" on fundamental theoretical aspects, as well as on significant applications in science, engineering, economics, finance, insurance and other fields.

"Review Articles" which present the existing literature on the specific topic from new perspectives.

"Interview articles" limited to two papers per year, covering interviews with milestone personalities in the field of Dependence Modeling.

Aims and Scope

The journal topics include (but are not limited to):　

Copula methods

Multivariate distributions

Estimation and goodness-of-fit tests

Measures of association

Quantitative risk management

Risk measures and stochastic orders

Time series

Environmental sciences

Computational methods and software

Extreme-value theory

Limit laws

Mass Transportations