Copula Theory and Its Applications

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.

Modeling of stochastic dependence is crucial to pricing and hedging of basket derivatives, as well as to pricing and hedging of some other financial products, such as rating-triggered corporate step-up bonds. The classical approach to modeling of dependence in finance via static copulae (and Sklar’s theorem) is inadequate for consistent valuation and hedging in time. In this survey we present recent developments in the area of modeling of dependence between stochastic processes with given marginal laws. Some of these results have already been successfully applied in finance in connection with the portfolio credit risk.

This chapter provides a survey of estimation methods for copula models. We review parametric, semiparametric and nonparametric approaches to inference on copulas for random samples with dependent components and copula-based time series. Among other topics, the survey discusses several problems of robust statistical analysis for copula models.

In this survey we introduce and discuss the pair-copula construction method to build flexible multivariate distributions. This class includes drawable (D), canonical (C) and regular vines developed in [5] and [4]. Estimation and model selection methods are studied both in a classical as well as in a Bayesian setting. This flexible class of multivariate copulas can be applied to model complex dependencies. Literature to applications in modeling financial data as well as Bayesian belief networks are provided. It closes with a section on open problems.

Quantitative Risk Management (QRM) often starts with a vector of oneperiodprofit-and-loss random variables \({\bf{X}} = (X_1 , \ldots ,X_d )'\) defined on some probability space \((\Omega ,\Im ,\mathbb P)\). Risk Aggregation concerns the study of the aggregate financial position \(\psi ({\bf{X}})\), for some measurable function \(\psi :\mathbb R^d \to \mathbb R\). A risk measure ρ then maps \(\psi ({\bf{X}})\) to \(\rho (\psi ({\bf{X}})) \in \mathbb R\), to be interpreted as the regulatory capital needed to be able to hold the aggregate position \(\psi ({\bf{X}})\) over a predetermined fixed time period. Risk Aggregation has often been studied within the framework when only the marginal distributions \(F_1 , \ldots ,F_d\) of the individual risks \(X_1 , \ldots ,X_d\) are available. Recently, especially in the management of operational risk, cases in which further dependence information is available have become relevant. We introduce a general mathematical framework which interpolates between marginal knowledge \((F_1 , \ldots ,F_d )\) and full knowledge of F X, the distribution of X. We illustrate the basic issues through some pedagogic examples of actuarial and financial interest. In particular, we study Risk Aggregation under different mathematical set-ups, for different aggregating functionals¬ and risk measures 〉 , focusing on Value-at-Risk. We show how the theory of Mass Transportations and tools originally developed to solve so-called Monge-Kantorovich problems turn out to be useful in this context. Finally, we introduce some new numerical integration techniques which solve some open aggregation problems and raise new interesting research issues.

Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.

Nested Archimedean copulas are explicit copulas which generalize Archimedean copulas to allow for asymmetries. Starting with completely monotone Archimedean generators, it is usually not clear when the corresponding Archimedean copulas can be nested to build indeed a proper copula. This article presents interesting results about the construction of nested Archimedean copulas. The presented construction principles are directly linked to sampling algorithms, which are also discussed in this work.

The study and modeling of interdependencies between extreme events is crucial for many applications of probability theory and statistics. Thanks to Sklar’s Theorem such tasks decompose into the study of the tail behaviour of the marginal univariate distributions and of the tail (i.e. corner) behaviour of the corresponding copulas. In this chapter we will deal with the second "subproblem". There are several approaches known in the literature.We shall deal with the one based on the tail expansion of copulas near the vertex (0, …, 0) of the unit multicube.We present the notions related to the tail expansion – leading parts, tail dependence functions and limiting invariant measures.We briefly discuss their properties and characterizations and provide several examples of the tail behaviour of copulas. Next we show relations between the tail expansion method and other approaches to the tail behaviour of copulas like the ones based on conditional copulas or associated extreme value copulas. At the end we present possible applications of the notion of tail expansions to quantitative finance, especially to risk measurement.

We discuss useful representations of lifetime distributions of coherent systems by means of convex combinations of marginal distributions of order statistics based on the lifetimes of exchangeable components. The representations are applied for characterizing distributions of system lifetimes composed of exchangeable units with given marginal distribution and joint absolutely continuous copula. The characterizations are used for calculating sharp bounds on the expectations and variancesof system lifetimes by means of respective parameters of single unit lifetime distribution.

This chapter constitutes a survey on copula-based measures of multivariate association - i.e. association in a d-dimensional random vector \(X = (X_1 , \ldots ,X_d )\) where \(d \ge 2\). Some of the measures discussed are multivariate extensions of wellknown bivariate measures such as Spearman’s rho, Kendall’s tau, Blomqvist’s beta or Gini’s gamma. Others rely on information theory or are based on L p-distances of copulas. Various measures of multivariate tail dependence are derived by extending the coefficient of bivariate tail dependence. Nonparametric estimation of these measures based on the empirical copula is further addressed.

We aim at providing probabilistic explanations of equivalences, between conditions of positive dependence and of univariate ageing, that have been pointed out in the literature. To this purpose we consider bivariate survival functions \(\bar F(x,y)\) and properties of them that are respectively invariant under transformations of the type \(\bar F(\varphi (x),\,\,\varphi (y))\) and \(\psi (\bar F(x,y))\), for \(\varphi ,\,\,\psi :\,\,[0,1] \to [0,1]\) increasing bijections. Bivariate Schur-constant survival models will have a central role in our discussion.

In this contribution we provide a consistent pricing setting for multivariate equity derivatives. Consistently with the prescriptions of the Efficient Market Hypothesis and of the martingale pricing approach, we provide a model in which prices are martingales both with respect to their own filtration and to the enlarged multivariate filtration. We show that if the log-prices follow processes with independent increments and each one of them is not Granger caused by the others, the pricing procedure can be performed by simply: i) generating time series of each asset; ii) linking assets at each time with a prescribed copula function. We provide applications to multivariate digital options and spread options.

Assessment of rock formation permeability is a complicated and challenging problem that plays a key role in oil reservoir modeling, production forecast, and the optimal exploitation management. Generally, permeability evaluation is performed using porosity-permeability relationships obtained by integrated analysis of various petrophysical measurements taken from cores and wireline well logs. Dependence relationships between pairs of petrophysical variables, such as permeability and porosity, are usually nonlinear and complex, and therefore those statistical tools that rely on assumptions of linearity and/or normality and/or existence of moments are commonly not suitable in this case. But even expecting a single copula family to be able to model a complex bivariate dependency seems to be still too restrictive, at least for the petrophysical variables under consideration in this work. Therefore, we explore the use of the Bernstein copula, and we also look for an appropriate partition of the data into subsets for which the dependence strucure was simpler to model, and then a conditional gluing copula technique is applied to build the bivariate joint distribution for the whole data set.

In this chapter, we consider a non-parametric testing procedure for an extension of the Koziol-Green model under two types of informative censoring. For the first type of informative censoring, we allow the censoring time to depend on the lifetime through an Archimedean copula function. For the second type, we generalize the relationship between the marginal distributions of the censoring time and lifetime by means of another copula function on the observed time and censoring indicator. In addition, we describe a bootstrap procedure to approximate the null distribution of the test statistics and illustrate it on a practical data set on survival with malignant melanoma.

Copula theory has got a rapid development in recent years. Most used copulas are symmetric: Archimedean are symmetric by construction while other continuous multivariate copulas are usually constructed from elliptical distributions and therefore are symmetric. From skewed copulas we can refer only to a copula introduced in [5], which the authors called skew t-copula. The construction of it differs from our approach.We introduce a multivariate t-copula which is based on the skew t-distribution introduced in [1]. Parameters of the copula have been estimated by method of moments and a simulation rule is given. The behaviour of estimates of the shape parameter of the skewed t-distribution is illustrated by simulation. The skew t-copula is used for modelling real data.

While Archimedean copulas are parameterized by real-valued functions, exchangeable Marshall-Olkin copulas are defined via sequences of real numbers. From a probabilistic perspective, the models behind both families have a different motivation. Consequently, their statistical properties are also different. In this regard, their striking analytical similarities are even more surprising. Considering sequences as discretized functions, most statements about Archimedean copulas and their corresponding generator functions translate into equivalent statements about exchangeable Marshall-Olkin copulas and their parameterizing sequences. This chapter reviews classical and recent results on both families of copulas with a focus on completely monotone functions and sequences.

The (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while Morgenstern utility functions (applying to risk aversion decision makers) are nondecreasing and concave. In this presentation, relationships between generators and utility functions are established. For some well known Archimedean copula families, links between the generator and the corresponding utility function are demonstrated. Some new copula families are derived from classes of utility functions which appeared in the literature, and their properties are discussed. It is shown how dependence properties of an Archimedean copula translate into properties of the utility function from which they are constructed.