Archimedean Copulae Research Articles

Common Factor Cause-Specific Mortality Model

Recent pension reforms in Europe have implemented a link between retirement age and life expectancy. The accurate forecast of life tables and life expectancy is hence paramount for governmental policy and financial institutions. We developed a multi-population mortality model which includes a cause-specific environment using Archimedean copulae to model dependence between various groups of causes of death. For this, Dutch data on cause-of-death mortality and cause-specific mortality data from 14 comparable European countries were used. We find that the inclusion of a common factor to a cause-specific mortality context increases the robustness of the forecast and we underline that cause-specific mortality forecasts foresee a more pessimistic mortality future than general mortality models. Overall, we find that this non-trivial extension is robust to the copula specification for commonly chosen dependence parameters.

Hedging cryptos with Bitcoin futures

The introduction of derivatives on Bitcoin enables investors to hedge risk exposures in cryptocurrencies. Because of volatility swings and jumps in cryptocurrency prices, the traditional variance-based approach to obtain hedge ratios may not be suitable for hedgers. In this work, we consider two extensions of the traditional approach: first, different dependence structures are modelled by different copulae, such as the Gaussian, Student-t, Normal Inverse Gaussian and Archimedean copulae; second, different risk measures, such as value-at-risk, expected shortfall and spectral risk measures are employed to find the optimal hedge ratio. Extensive out-of-sample tests using the data from the time period December 2017 until May 2021 give insights in the practice of hedging various cryptos and crypto indices, including Bitcoin, Ethereum, Cardano, the CRIX index and a number of crypto-portfolios. Evidence shows that BTC futures can effectively hedge BTC and BTC-involved indices. This promising result is consistent across different risk measures and copulae except for the Frank copula. On the other hand, we observe complex and diverse dependence structures between non-BTC-related cryptocurrencies and the BTC futures. As a consequence, the hedge performance of non-BTC-related cryptocurrencies is mixed and even suitable for some assets.

CDO and HAC

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on copula functions with up to three parameters, with default intensities estimated from market data and with a random loss given default that is correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. We apply hierarchical Archimedean copulae (HAC) whose construction allows for the fact that the risky assets of the CDO pool are chosen from six different industry sectors. The dependence among the assets from the same group is specified with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. The copula with two and three parameters models the relation between the loss given default and the default times. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution.

HMM in Dynamic HAC Models

Understanding the dynamics of high dimensional non-normal dependency structure is a challenging task. This research aims at attacking this problem by building up a hidden Markov model (HMM) for Hierarchical Archimedean Copulae (HAC), where the HAC represent a wide class of models for high dimensional dependency, and HMM is a statistical technique to describe time varying dynamics. HMM applied to HAC provide flexible modeling for high dimensional non Gaussian time series. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application, where the model’s performance is compared with other dynamic models, and in the second application we simulate rainfall process.

Heterogeneous Archimedean Copula

Archimedean copulae build a large family of copulae exhibiting tail-dependency in many cases. We extend the classical homogeneous (exchangeable) Archimedean copula to the heterogeneous case. This will extend the use of this copula family to multivariate random variable with pairwise different dependencies.

HIDDEN MARKOV STRUCTURES FOR DYNAMIC COPULAE

Understanding the time series dynamics of a multi-dimensional dependency structure is a challenging task. Multivariate covariance driven Gaussian or mixed normal time varying models have only a limited ability to capture important features of the data such as heavy tails, asymmetry, and nonlinear dependencies. The present paper tackles this problem by proposing and analyzing a hidden Markov model (HMM) for hierarchical Archimedean copulae (HAC). The HAC constitute a wide class of models for multi-dimensional dependencies, and HMM is a statistical technique for describing regime switching dynamics. HMM applied to HAC flexibly models multivariate dimensional non-Gaussian time series.We apply the expectation maximization (EM) algorithm for parameter estimation. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application. This example is motivated by a local adaptive analysis that yields a time varying HAC model. We compare its forecasting performance with that of other classical dynamic models. In another, second, application, we model a rainfall process. This task is of particular theoretical and practical interest because of the specific structure and required untypical treatment of precipitation data.

Modelling the general dependence between commodity forward curves

This study proposes a novel framework for the joint modelling of commodity forward curves. Its key contribution is twofold. First, we introduce a family of dynamic conditional correlation models based on hierarchical Archimedean copulae (HAC-DCC), which are flexible but parsimonious instruments that capture a wide range of dynamic dependencies. Second, we apply these models in the context of commodity forward curves as part of the framework. An extensive Value-at-Risk analysis shows that certain HAC-DCC models consistently outperform other introduced benchmarks in terms of the preciseness of their out-of-sample distribution forecasts of the returns of various commodity futures portfolios. This shows that the proposed modelling framework, as one of its possible applications, can be a useful and convenient risk management tool.

False discovery rate control under Archimedean copula

We are concerned with the false discovery rate (FDR) of the linear step-up test $\varphi^{LSU}$ considered by Benjamini and Hochberg (1995). It is well known that $\varphi^{LSU}$ controls the FDR at level $m_{0}q/m$ if the joint distribution of $p$-values is multivariate totally positive of order $2$. In this, $m$ denotes the total number of hypotheses, $m_{0}$ the number of true null hypotheses, and $q$ the nominal FDR level. Under the assumption of an Archimedean $p$-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of $\varphi^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $\varphi^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $\varphi^{LSU}$ is exactly equal to $m_{0}q/m$, as in the case of stochastically independent $p$-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable $p$-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel-Hougaard copulae.

Hidden Markov Structures for Dynamic Copulae

Understanding the time series dynamics of a multivariate dimensional dependency structure is a challenging task. A multivariate covariance driven Gaussian or mixed normal time varying models are limited in capturing important data features such as heavy tails, asymmetry, and nonlinear dependencies. This research aims at tackling this problem by proposing and analysing a hidden Markov model (HMM) for hierarchical Archimedean copulae (HAC). The HAC constitute a wide class of models for multivariate dimensional dependencies, and HMM is a statistical technique for describing regime switching dynamics. HMM applied to HAC flexibly models multivariate dimensional non-Gaussian time series.We apply the expectation maximization (EM) algorithm for parameter estimation. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application. This example is motivated by a local adaptive analysis that yields a time varying HAC model. We compare the forecasting performance with other classical dynamic models. In another, second, application we model a rainfall process. This task is of particular theoretical and practical interest because of the specific structure and required untypical treatment of precipitation data.

Hierarchical Archimedean Copulae: TheHACPackage

This paper presents the R package HAC, which provides user friendly methods for dealing with hierarchical Archimedean copulae (HAC). Computationally efficient estimation procedures allow to recover the structure and the parameters of HAC from data. In addition, arbitrary HAC can be constructed to sample random vectors and to compute the values of the corresponding cumulative distribution plus density functions. Accurate graphics of the HAC structure can be produced by the plot method implemented for these objects.

Dynamic structured copula models

Abstract There is an increasing demand for models of multivariate time-series with time-varying and non-Gaussian dependencies. The available models suffer from the curse of dimensionality or from restrictive assumptions on the parameters and distributions. A promising class of models is that of hierarchical Archimedean copulae (HAC), which allows for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change-point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. Both examples reveal periods with constant and turmoil dependencies. The economic significance of the suggested modelling is evaluated using the Value-at-Risk of a portfolio.

Valuation of collateralized debt obligations with hierarchical Archimedean copulae

Abstract Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on hierarchical Archimedean copulae (HAC) with up to three parameters, with default intensities calibrated to market data and with random loss given defaults that are correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution. We also obtain a flat correlation smile across tranches thereby solving the implied correlation puzzle.

HMM in dynamic HAC models

Understanding the dynamics of high dimensional non-normal dependency structure is a challenging task. This research aims at attacking this problem by building up a hidden Markov model (HMM) for Hierarchical Archimedean Copulae (HAC), where the HAC represent a wide class of models for high dimensional dependency, and HMM is a statistical technique to describe time varying dynamics. HMM applied to HAC provide flexible modeling for high dimensional non Gaussian time series. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application, where the model’s performance is compared with other dynamic models, and in the second application we simulate rainfall process.

Time varying Hierarchical Archimedean Copulae

There is increasing demand for models of time-varying and non-Gaussian dependencies for mul- tivariate time-series. Available models suffer from the curse of dimensionality or restrictive assumptions on the parameters and the distribution. A promising class of models are the hierarchical Archimedean copulae (HAC) that allow for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. We find that for stock indices the copula parameter changes dynam- ically but the hierarchical structure is constant over time. Interestingly in our exchange rate example both structure and parameters vary dynamically.

Time Varying Hierarchical Archimedean Copulae

There is increasing demand for models of time-varying and non-Gaussian dependencies for multivariate time-series. Available models suffer from the curse of dimensionality or restrictive assumptions on the parameters and the distribution. A promising class of models are the hierarchical Archimedean copulae (HAC) that allow for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. We find that for stock indices the copula parameter changes dynamically but the hierarchical structure is constant over time. Interestingly in our exchange rate example both structure and parameters vary dynamically.

CDO and HAC

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on copula functions with up to three parameters, with default intensities estimated from market data and with a random loss given default that is correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. We apply hierarchical Archimedean copulae (HAC) whose construction allows for the fact that the risky assets of the CDO pool are chosen from six dierent industry sectors. The dependence among the assets from the same group is specified with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. The copula with two and three parameters models the relation between the loss given default and the default times. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution.

Archimedean copulae for risk measurement

In this paper some Archimedean copula functions for bivariate financial returns are studied. The choice of this family is due to their ability to capture the tail dependence, which is an association measure we can detect in many bivariate financial time-series. A time-varying version of these copulae is also investigated. Finally, the Value-at-Risk is computed and its performance is compared across different copula specifications.

Archimedean copulae and positive dependence

In this paper, we consider different issues related to Archimedean copulae and positive dependence. In the first part, we characterize Archimedean copulae that possess positive dependence properties such as multivariate total positivity of order 2 (MTP 2) and conditionally increasingness in sequence. In the second part, we investigate conditions for exchangeable binary sequences to admit an Archimedean copula, and we show that they depend on the extendibility of the sequence, and therefore on its positive-dependence properties.

Archimedean copulae and positive dependence

In this paper, we consider different issues related to Archimedean copulae and positive dependence. In the first part, we characterize Archimedean copulae that possess positive dependence properties such as multivariate total positivity of order 2 (MTP 2 ) and conditionally increasingness in sequence. In the second part, we investigate conditions for exchangeable binary sequences to admit an Archimedean copula, and we show that they depend on the extendibility of the sequence, and therefore on its positive-dependence properties.

Tail Dependence from a Distributional Point of View

The dependence structure in the tails of bivariate random variables is studied by means of appropriate copulae. Weak convergence results show that these copulae are natural dependence structures for joint tail events. The results obtained apply to particular types of copulae such as archimedean copulae and the Gaussian copula. Further, connections to multivariate extreme value theory are investigated and a two-dimensional Pickands–Balkema–de Haan Theorem type is derived. Finally, a counterexample showing that the tail dependence coefficients do not completely determine the dependence structure of bivariate rare events is provided.