Marius Hofert

Sampling Archimedean copulas

The challenge of efficiently sampling exchangeable and nested Archimedean copulas is addressed. Specific focus is put on large dimensions, where methods involving generator derivatives are not applicable. Additionally, new conditions under which Archimedean copulas can be mixed to construct nested Archimedean copulas are presented. Moreover, for some Archimedean families, direct sampling algorithms are given. For other families, sampling algorithms based on numerical inversion of Laplace transforms are suggested. For this purpose, the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm are compared in terms of precision and runtime. Examples are given, including both exchangeable and nested Archimedean copulas.

A note on generalized inverses

Motivated by too restrictive or even incorrect statements about generalized inverses in the literature, properties about these functions are investigated and proven. Examples and counterexamples show the importance of generalized inverses in mathematical theory and its applications.

Likelihood inference for Archimedean copulas in high dimensions under known margins

Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, and the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Availability of the generator derivatives in a form that permits fast and accurate computation makes maximum-likelihood estimation for Archimedean copulas feasible, even in large dimensions. It is shown, by large scale simulation of the performance of maximum likelihood estimators under known margins, that the root mean squared error actually decreases with both dimension and sample size at a similar rate. Confidence intervals for the parameter vector are derived under known margins. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the R package nacopula and can thus be studied in detail.

CDO pricing with nested Archimedean copulas

Companies in the same industry sector are usually more correlated than firms in different sectors, as they are similarly affected by macroeconomic effects, political decisions, and consumer trends. Despite the many stock return models taking this fact into account, there are only a few credit default models that take it into consideration. In this paper we present a default model based on nested Archimedean copulas that is able to capture hierarchical dependence structures among the obligors in a credit portfolio. Nested Archimedean copulas have a surprisingly simple and intuitive interpretation. The dependence among all companies in the same sector is described by an inner copula and the sectors are then coupled via an outer copula. Consequently, our model implies a larger default correlation for companies in the same industry sector than for companies in different sectors. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. This portfolio consists of CDS on 125 companies from six different industry sectors and is therefore an excellent portfolio for a comparison of our generalized model with a traditional copula model of the same family that does not take different sectors into account.

Efficiently sampling nested Archimedean copulas

Efficient sampling algorithms for both Archimedean and nested Archimedean copulas are presented. First, efficient sampling algorithms for the nested Archimedean families of Ali-Mikhail-Haq, Frank, and Joe are introduced. Second, a general strategy how to build a nested Archimedean copula from a given Archimedean generator is presented. Sampling this copula involves sampling an exponentially tilted stable distribution. A fast rejection algorithm is developed for the more general class of tilted Archimedean generators. It is proven that this algorithm reduces the complexity of the standard rejection algorithm to logarithmic complexity. As an application it is shown that the fast rejection algorithm outperforms existing algorithms for sampling exponentially tilted stable distributions involved, e.g., in nested Clayton copulas. Third, with the additional help of randomization of generator parameters, explicit sampling algorithms for several nested Archimedean copulas based on different Archimedean families are found. Additional results include approximations and some dependence properties, such as Kendalls tau and tail dependence parameters. The presented ideas may also apply in the more general context of sampling distributions given by their Laplace-Stieltjes transforms.

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Levy subordinators is given. Independent exponential random variables are divided by group-specific Levy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.

An Extreme Value Approach for Modeling Operational Risk Losses Depending on Covariates

A general methodology for modeling loss data depending on covariates is developed. The parameters of the frequency and severity distributions of the losses may depend on covariates. The loss frequency over time is modeled with a nonhomogeneous Poisson process with rate function depending on the covariates. This corresponds to a generalized additive model, which can be estimated with spline smoothing via penalized maximum likelihood estimation. The loss severity over time is modeled with a nonstationary generalized Pareto distribution (alternatively, a generalized extreme value distribution) depending on the covariates. Since spline smoothing cannot directly be applied in this case, an efficient algorithm based on orthogonal parameters is suggested. The methodology is applied both to simulated loss data and a database of operational risk losses collected from public media. Estimates, including confidence intervals, for risk measures such as Value‐at‐Risk as required by the Basel II/III framework are computed. Furthermore, an implementation of the statistical methodology in R is provided.

A stochastic representation and sampling algorithm for nested Archimedean copulas

A general sampling algorithm for nested Archimedean copulas was recently suggested. It is given in two different forms, a recursive or an explicit one. The explicit form allows for a simpler version of the algorithm which is numerically more stable and faster since less function evaluations are required. The algorithm can also be given in general form, not being restricted to a particular nesting such as fully nested Archimedean copulas. Further, several examples are given.

Construction and Sampling of Nested Archimedean Copulas

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.

Densities of nested Archimedean copulas

Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference it is important to have the density. The present work fills this gap. A general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas in arbitrary dimensions if the number of nesting levels is not too large. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.