Hierarchies of Archimedean copulas

We present a flexible class of hierarchical copulas capable of modelling multidimensional joint distributions of asset returns with a richer rank correlation structure than existing models. We derive estimators and simulation techniques. The methods are applied to an illustrative portfolio consisting of a subset of DAX stocks.     

Sampling from Archimedean copulas

We develop sampling algorithms for multivariate Archimedean copulas. For exchangeable copulas, where there is only one generating function, we first analyse the distribution of the copula itself, deriving a number of integral representations and a generating function representation. One of the integral representations is related, by a form of convolution, to the distribution whose Laplace transform yields the copula generating function. In the infinite-dimensional limit there is a direct connection between the distribution of the copula value and the inverse Laplace transform. Armed with these results, we present three sampling algorithms, all of which entail drawing from a one-dimensional distribution and then scaling the result to create random deviates distributed according to the copula. We implement and compare the various methods. For more general cases, in which an N-dimensional Archimedean copula is given by N?1 nested generating functions, we present algorithms in which each new variate is drawn conditional only on the value of the copula of the previously drawn variates. We also discuss the use of composite nested and exchangeable copulas for modelling random variates with a natural hierarchical structure, such as ratings and sectors for obligors in credit baskets.     

Smooth nonparametric Bernstein vine copulas

We consider the problem of accurately modelling the distribution of the market risk of a multivariate financial portfolio. We employ a multivariate GARCH model in which the dependence structure between the assets is modelled via a vine copula. We address the problem of how the parametric pair-copulas in a vine copula should be chosen by proposing to use nonparametric Bernstein copulas as bivariate pair-copulas. An extensive simulation study illustrates that our smooth nonparametric vine copula model is able to match the results of a competing parametric vine model calibrated via Akaike’s Information Criterion while at the same time significantly reducing model risk. Our empirical analysis of financial market data demonstrates that our proposed model yields Value-at-Risk forecasts that are significantly more accurate than those of a benchmark parametric model.     

Selecting copulas for risk management

Copulas offer financial risk managers a powerful tool to model the dependence between the different elements of a portfolio and are preferable to the traditional, correlation-based approach. In this paper, we show the importance of selecting an accurate copula for risk management. We extend standard goodness-of-fit tests to copulas. Contrary to existing, indirect tests, these tests can be applied to any copula of any dimension and are based on a direct comparison of a given copula with observed data. For a portfolio consisting of stocks, bonds and real estate, these tests provide clear evidence in favor of the Student’s t copula, and reject both the correlation-based Gaussian copula and the extreme value-based Gumbel copula. In comparison with the Student’s t copula, we find that the Gaussian copula underestimates the probability of joint extreme downward movements, while the Gumbel copula overestimates this risk. Similarly we establish that the Gaussian copula is too optimistic on diversification benefits, while the Gumbel copula is too pessimistic. Moreover, these differences are significant.     

Measuring financial risks with copulas

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the concept of copulas. We select some special copulas and identify the type of dependency captured by each one. We fit copulas to daily returns and simulate from the fitted models. We compare the effect of the choice of copula on risk measures and assess the variability of one-step-ahead predictions of portfolio losses. We analyze extreme scenarios and fit extreme value copulas to the block maxima and minima from daily returns. The stress scenarios constructed are compared to those obtained using models from the extreme value theory. We illustrate the usefulness of the copula approach using two stock market indexes.     

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.     

Jackknife empirical likelihood for parametric copulas

For fitting a parametric copula to multivariate data, a popular way is to employ the so-called pseudo maximum likelihood estimation proposed by Genest, Ghoudi, and Rivest. Although interval estimation can be obtained via estimating the asymptotic covariance of the pseudo maximum likelihood estimation, we propose a jackknife empirical likelihood method to construct confidence regions for the parameters without estimating any additional quantities such as the asymptotic covariance. A simulation study shows the advantages of the new method in case of strong dependence or having more than one parameter involved.     

Modeling duration clusters with dynamic copulas

This paper suggests a dynamic copula approach that allows more flexibility in capturing duration clusters of ultra-high frequent order book data. The proposed framework involves a time-varying mixing parameter and does not only model (a) the degree of dependence of consecutive durations, but also (b) the structure of (temporal) dependence of the duration process.     

Forecasting liquidity-adjusted intraday Value-at-Risk with vine copulas

We propose to model the joint distribution of bid-ask spreads and log returns of a stock portfolio by using Autoregressive Conditional Double Poisson and GARCH processes for the marginals and vine copulas for the dependence structure. By estimating the joint multivariate distribution of both returns and bid-ask spreads from intraday data, we incorporate the measurement of commonalities in liquidity and comovements of stocks and bid-ask spreads into the forecasting of three types of liquidity-adjusted intraday Value-at-Risk (L-IVaR). In a preliminary analysis, we document strong extreme comovements in liquidity and strong tail dependence between bid-ask spreads and log returns across the firms in our sample thus motivating our use of a vine copula model. Furthermore, the backtesting results for the L-IVaR of a portfolio consisting of five stocks listed on the NASDAQ show that the proposed models perform well in forecasting liquidity-adjusted intraday portfolio profits and losses.     

Modeling the leverage effect with copulas and realized volatility

In this paper, we propose the use of static and dynamic copulas to study the leverage effect in the S&P 500 index. Copula models can conveniently separate the leverage effect from the marginal distributions of the return and its volatility. Daily volatility is proxied by a measure of realized volatility, which is constructed from high-frequency data. We uncover a significant leverage effect in the S&P 500 index, and this leverage effect is found to be changing over time in a highly persistent manner. Moreover the dynamic copula models are shown to outperform the static counterparts.     

Evolution of multivariate copulas in continuous and discrete processes

There has been much interest in copulas, which are known to provide a flexible tool for analyzing the dependence structure among random variables. Dependence relations must be dynamic rather than static in nature. However, copulas are useful mainly for static matters. Thus we introduce evolving multivariate copulas, which transform through time autonomously governed by the multivariate heat equation. Our aims are to prove their existences and solutions to analyze their transitions. Moreover, we construct discrete type to apply empirical data analysis and investigate their properties, and prove that they converge to their original continuous type.     

Portfolio selection with commodities under conditional copulas and skew preferences

This article investigates the portfolio selection problem of an investor with three-moment preferences taking positions in commodity futures. To model the asset returns, we propose a conditional asymmetric t copula with skewed and fat-tailed marginal distributions, such that we can capture the impact on optimal portfolios of time-varying moments, state-dependent correlations, and tail and asymmetric dependence. In the empirical application with oil, gold and equity data from 1990 to 2010, the conditional t copulas portfolios achieve better performance than those based on more conventional strategies. The specification of higher moments in the marginal distributions and the type of tail dependence in the copula has significant implications for the out-of-sample portfolio performance.     

Multivariate autoregressive modeling of time series count data using copulas

We introduce the Multivariate Autoregressive Conditional Double Poisson model to deal with discreteness, overdispersion and both auto and cross-correlation, arising with multivariate counts. We model counts with a double Poisson and assume that conditionally on past observations the means follow a Vector Autoregression. We resort to copulas to introduce contemporaneous correlation. We apply it to the study of sector and stock-specific news related to the comovements in the number of trades per unit of time of the most important US department stores traded on the NYSE. We show that the market leaders inside a specific sector are related to their size measured by their market capitalization.     

Elliptical families and copulas: tilting and premium; capital allocation

Elliptical copula measures with symmetrical marginals are proposed as a natural generalization of the elliptical family, which preserves the symmetrical character of marginals, but is more flexible in the choice of their shape parameters. The properties of these copulas are investigated and the elliptical copula tilting and corresponding premium are proposed as a natural tool for portfolio capital allocation. For the case of the multivariate normal family, such a tilting and premium coincide with the Esscher transform and premium.     

Vine copulas and applications to the European Union sovereign debt analysis

European sovereign debt crisis has become a very popular topic since late 2009. In this paper, sovereign debt crisis is investigated by calculating the probabilities of the potential future crisis of 11 countries in the European Union. We use sovereign spreads of the European countries against Germany as targets and apply the GARCH based vine copula simulation technique. The methodology solves the difficulties of calculating the probabilities of rarely happening events and takes sovereign debt movement dependence, especially tail dependence, into consideration. Results indicate that Italy and Spain are the most likely next victims of the sovereign debt crisis, followed by Ireland, France and Belgium. The UK, Sweden and Denmark, which are outside the euro area, are the most financially stable countries in the sample.     

Recovering copulas from limited information and an application to asset allocation

This paper proposes an entropy-based method to construct a new class of copulas - the most entropic canonical copulas (MECC). Our empirical study focuses on an investment problem for an investor with a constant relative risk aversion (CRRA) utility function allocating wealth between the Dow Jones Large-Cap and Small-Cap indices, of which the contemporaneous dependence can be modeled by the MECC or other commonly-used copulas. Both the theoretical analysis of the method and the empirical study indicate the potential for enormous statistical and economic gains as a result of using the MECC.     

Empirical estimation of tail dependence using copulas: application to Asian markets

This paper introduces non-parametric estimators for upper and lower tail dependence whose confidence intervals are obtained with a bootstrap method. We call these estimators ‘naïve estimators’ as they represent a discretization of Joe's formulae linking copulas to tail dependence. We apply the methodology to an empirical data set composed of three composite indexes for the three Tigers (Thailand, Malaysia and Indonesia). The extremes show a dependence structure which is symmetric for the Thai and Malaysian markets and asymmetric for the Thai and Indonesian markets and for the Malaysian and the Indonesian markets. Using these results we estimate the copula (which belongs to the Student or Archimedean copula families) for each pair of markets by two methods. Finally, we provide risk measurements using the best copula associated with each pair of markets.     

Singular risk-neutral valuation equations

Many risk-neutral pricing problems proposed in the finance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding partial integro-differential equation. Often, these PIDEs have singular diffusion matrices and coefficients that are not Lipschitz-continuous up to the boundary. In addition, in general, boundary conditions are not specified. In this paper, we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDEs with all the above features, under a Lyapunov-type condition. Our results apply to European and Asian option pricing, in jump-diffusion stochastic volatility and path-dependent volatility models. We verify our Lyapunov-type condition in several examples, including the arithmetic Asian option in the Heston model.     

Analysing financial contagion and asymmetric market dependence with volatility indices via copulas

This paper explores the cross-market dependence between five popular equity indices (S&P 500, NASDAQ 100, DAX 30, FTSE 100, and Nikkei 225), and their corresponding volatility indices (VIX, VXN, VDAX, VFTSE, and VXJ). In particular, we propose a dynamic mixed copula approach which is able to capture the time-varying tail dependence coefficient (TDC). The findings indicate the existence of financial contagion and significant asymmetric TDCs for major international equity markets. In some situations, although contagion cannot be clearly detected by stock index movements, it can be captured by dependence between volatility indices. The results imply that contagion is not only reflected in the first moment of index returns, but also the second moment, i.e. the volatility. Results also show that dependence between volatility indices is more easily influenced by financial shocks and reflects the instantaneous information faster than the stock market indices.