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Credit models and the crisis, or: how I learned to stop worrying and love the CDOs

Brigo, Damiano; Pallavicini, Andrea; Torresetti, Roberto
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
We follow a long path for Credit Derivatives and Collateralized Debt Obligations (CDOs) in particular, from the introduction of the Gaussian copula model and the related implied correlations to the introduction of arbitrage-free dynamic loss models capable of calibrating all the tranches for all the maturities at the same time. En passant, we also illustrate the implied copula, a method that can consistently account for CDOs with different attachment and detachment points but not for different maturities. The discussion is abundantly supported by market examples through history. The dangers and critics we present to the use of the Gaussian copula and of implied correlation had all been published by us, among others, in 2006, showing that the quantitative community was aware of the model limitations before the crisis. We also explain why the Gaussian copula model is still used in its base correlation formulation, although under some possible extensions such as random recovery. Overall we conclude that the modeling effort in this area of the derivatives market is unfinished, partly for the lack of an operationally attractive single-name consistent dynamic loss model, and partly because of the diminished investment in this research area.; Comment: A vastly extended and updated version of this paper will appear as a book: "Credit models and the crisis: a journey into CDOs...

Rank-based inference for bivariate extreme-value copulas

Genest, Christian; Segers, Johan
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and Cap\'{e}ra\{a}, Foug\{e}res and Genest [Biometrika 84 (1997) 567--577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of $X$ and $Y$ are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.; Comment: Published in at http://dx.doi.org/10.1214/08-AOS672 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Algorithms for Finding Copulas Minimizing Convex Functions of Sums

Bernard, Carole; McLeish, Don
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
We develop improved rearrangement algorithms to find the dependence structure that minimizes a convex function of the sum of dependent variables with given margins. We propose a new multivariate dependence measure, which can assess the convergence of the rearrangement algorithms and can be used as a stopping rule. We show how to apply these algorithms for example to finding the dependence among variables for which the marginal distributions and the distribution of the sum or the difference are known. As an example, we can find the dependence between two uniformly distributed variables that makes the distribution of the sum of two uniform variables indistinguishable from a normal distribution. Using MCMC techniques, we design an algorithm that converges to the global optimum.

Mixed Cumulative Distribution Networks

Silva, Ricardo; Blundell, Charles; Teh, Yee Whye
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
Directed acyclic graphs (DAGs) are a popular framework to express multivariate probability distributions. Acyclic directed mixed graphs (ADMGs) are generalizations of DAGs that can succinctly capture much richer sets of conditional independencies, and are especially useful in modeling the effects of latent variables implicitly. Unfortunately there are currently no good parameterizations of general ADMGs. In this paper, we apply recent work on cumulative distribution networks and copulas to propose one one general construction for ADMG models. We consider a simple parameter estimation approach, and report some encouraging experimental results.; Comment: 11 pages, 4 figures

A Directional Multivariate Value at Risk

Torres, Raúl; Lillo, Rosa E.; Laniado, Henry
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
In economics, insurance and finance, value at risk (VaR) is a widely used measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, time horizon, and probability $\alpha$, the $100\alpha\%$ VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is $\alpha$. That is to say, it is a quantile of the distribution of the losses, which has both good analytic properties and easy interpretation as a risk measure. However, its extension to the multivariate framework is not unique because a unique definition of multivariate quantile does not exist. In the current literature, the multivariate quantiles are related to a specific partial order considered in $\mathbb{R}^{n}$, or to a property of the univariate quantile that is desirable to be extended to $\mathbb{R}^{n}$. In this work, we introduce a multivariate value at risk as a vector-valued directional risk measure, based on a directional multivariate quantile, which has recently been introduced in the literature. The directional approach allows the manager to consider external information or risk preferences in her/his analysis. We have derived some properties of the risk measure and we have compared the univariate \textit{VaR} over the marginals with the components of the directional multivariate VaR. We have also analyzed the relationship between some families of copulas...

Maximum entropy distribution of order statistics with given marginals

Butucea, Cristina; Delmas, Jean-François; Dutfoy, Anne; Fischer, Richard
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.; Comment: 35 pages, overview of the notations at page 33

Extremes of multivariate ARMAX processes

Ferreira, Marta; Ferreira, Helena
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
We define a new multivariate time series model by generalizing the ARMAX process in a multivariate way. We give conditions on stationarity and analyze local dependence and domains of attraction. As a consequence of the obtained result, we derive a new method of construction of multivariate extreme value copulas. We characterize the extremal dependence by computing the multivariate extremal index and bivariate upper tail dependence coefficients. An estimation procedure for the multivariate extremal index shall be presented. We also address the marginal estimation and propose a new estimator for the ARMAX autoregressive parameter.

A test for Archimedeanity in bivariate copula models

Bücher, Axel; Dette, Holger; Volgushev, Stanislav
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
We propose a new test for the hypothesis that a bivariate copula is an Archimedean copula. The test statistic is based on a combination of two measures resulting from the characterization of Archimedean copulas by the property of associativity and by a strict upper bound on the diagonal by the Fr\'echet-upper bound. We prove weak convergence of this statistic and show that the critical values of the corresponding test can be determined by the multiplier bootstrap method. The test is shown to be consistent against all departures from Archimedeanity if the copula satisfies weak smoothness assumptions. A simulation study is presented which illustrates the finite sample properties of the new test.; Comment: 18 pages, 2 figures

Block-Maxima of Vines

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
We examine the dependence structure of finite block-maxima of multivariate distributions. We provide a closed form expression for the copula density of the vector of the block-maxima. Further, we show how partial derivatives of three-dimensional vine copulas can be obtained by only one-dimensional integration. Combining these results allows the numerical treatment of the block-maxima of any three-dimensional vine copula for finite block-sizes. We look at certain vine copula specifications and examine how the density of the block-maxima behaves for different block-sizes. Additionally, a real data example from hydrology is considered. In extreme-value theory for multivariate normal distributions, a certain scaling of each variable and the correlation matrix is necessary to obtain a non-trivial limiting distribution when the block-size goes to infinity. This scaling is applied to different three-dimensional vine copula specifications.; Comment: To appear in Extreme Value Modelling and Risk Analysis: Methods and Applications. Eds. D. Dey and J. Yan. Chapman & Hall/CRC Press

Exponent dependence measures of survival functions and correlated frailty models

Bendel, Jens; Dobler, Dennis; Janssen, Arnold
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
The present article studies survival analytic aspects of semiparametric copula dependence models with arbitrary univariate marginals. The underlying survival functions admit a representation via exponent measures which have an interpretation within the context of hazard functions. In particular, correlated frailty survival models are linked to copulas. Additionally, the relation to exponent measures of minumum-infinitely divisible distributions as well as to the L\'evy measure of the L\'evy-Khintchine formula is pointed out. The semiparametric character of the current analyses and the construction of survival times with dependencies of higher order are carried out in detail. Many examples including graphics give multifarious illustrations.; Comment: 40 pages, 25 figures

A Dynamic Correlation Modelling Framework with Consistent Stochastic Recovery

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
This paper describes a flexible and tractable bottom-up dynamic correlation modelling framework with a consistent stochastic recovery specification. The stochastic recovery specification only models the first two moments of the spot recovery rate as its higher moments have almost no contribution to the loss distribution and CDO tranche pricing. Observing that only the joint distribution of default indicators is needed to build the portfolio loss distribution, we propose a generic class of default indicator copulas to model CDO tranches, which can be easily calibrated to index tranche prices across multiple maturities. This correlation modelling framework has the unique advantage that the joint distribution of default time and other dynamic properties of the model can be changed separately from the loss distribution and tranche prices. After calibrating the model to index tranche prices, existing top-down methods can be applied to the common factor process to construct very flexible systemic dynamics without changing the already calibrated tranche prices. This modelling framework therefore combines the best features of the bottom-up and top-down models: it is fully consistent with all the single name market information and it admits very rich and flexible spread dynamics. Numerical results from a non-parametric implementation of this modelling framework are also presented. The non-parametric implementation achieved fast and accurate calibration to the index tranches across multiple maturities even under extreme market conditions. A conditional Markov chain method is also proposed to construct the systemic dynamics...

Relations Between Stochastic Orderings and generalized Stochastic Precedence

De Santis, Emilio; Fantozzi, Fabio; Spizzichino, Fabio
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
The concept of "stochastic precedence" between two real-valued random variables has often emerged in different applied frameworks. In this paper we consider a slightly more general, and completely natural, concept of stochastic precedence and analyze its relations with the notions of stochastic ordering. Such a study leads us to introducing some special classes of bivariate copulas. Motivations for our study can arise from different fields. In particular we consider the frame of Target-Based Approach in decisions under risk. This approach has been mainly developed under the assumption of stochastic independence between "Prospects" and "Targets". Our analysis concerns the case of stochastic dependence.; Comment: 13 pages, 6 figures

Information bounds for Gaussian copulas

Hoff, Peter D.; Niu, Xiaoyue; Wellner, Jon A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, that is, the probability of the multivariate ranks. In this article, we obtain limiting normal distributions of the rank likelihood for Gaussian copula models. Our results cover models with structured correlation matrices, such as exchangeable or circular correlation models, as well as unstructured correlation matrices. For all Gaussian copula models, the limiting distribution of the rank likelihood ratio is shown to be equal to that of a parametric likelihood ratio for an appropriately chosen multivariate normal model. This implies that the semiparametric information bounds for rank-based estimators are the same as the information bounds for estimators based on the full data, and that the multivariate normal distributions are least favorable.; Comment: Published in at http://dx.doi.org/10.3150/12-BEJ499 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Bayes Multiple Decision Functions

Wu, Wensong; Peña, Edsel A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
This paper deals with the problem of simultaneously making many (M) binary decisions based on one realization of a random data matrix X. M is typically large and X will usually have M rows associated with each of the M decisions to make, but for each row the data may be low dimensional. A Bayesian decision-theoretic approach for this problem is implemented with the overall loss function being a cost-weighted linear combination of Type I and Type II loss functions. The class of loss functions considered allows for the use of the false discovery rate (FDR), false nondiscovery rate (FNR), and missed discovery rate (MDR) in assessing the decision. Through this Bayesian paradigm, the Bayes multiple decision function (BMDF) is derived and an efficient algorithm to obtain the optimal Bayes action is described. In contrast to many works in the literature where the rows of the matrix X are assumed to be stochastically independent, we allow in this paper a dependent data structure with the associations obtained through a class of frailty-induced Archimedean copulas. In particular, non-Gaussian dependent data structure, which is the norm rather than the exception when dealing with failure-time data, can be entertained. The numerical implementation of the determination of the Bayes optimal action is facilitated through sequential Monte Carlo techniques. The main theory developed could also be extended to the problem of multiple hypotheses testing...

Detecting regime switches in the dependence structure of high dimensional financial data

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
Misperceptions about extreme dependencies between different financial assets have been an im- portant element of the recent financial crisis. This paper studies inhomogeneity in dependence structures using Markov switching regular vine copulas. These account for asymmetric depen- dencies and tail dependencies in high dimensional data. We develop methods for fast maximum likelihood as well as Bayesian inference. Our algorithms are validated in simulations and applied to financial data. We find that regime switches are present in the dependence structure of various data sets and show that regime switching models could provide tools for the accurate description of inhomogeneity during times of crisis.

An analysis of the R\"uschendorf transform - with a view towards Sklar's Theorem

Oertel, Frank
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
In many applications including financial risk measurement, copulas have shown to be a powerful building block to reflect multivariate dependence between several random variables including the mapping of tail dependencies. A famous key result in this field is Sklar's Theorem. Meanwhile, there exist several approaches to prove Sklar's Theorem in its full generality. An elegant probabilistic proof was provided by L. R\"{u}schendorf. To this end he implemented a certain "distributional transform" which naturally transforms an arbitrary distribution function $F$ to a flexible parameter-dependent function which exhibits exactly the same jump size as $F$. By using some real analysis and measure theory only (without involving the use of a given probability measure) we expand into the underlying rich structure of the distributional transform. Based on derived results from this analysis (such as Proposition 2.5 and Theorem 2.12) including a strong and frequent use of the right quantile function, we revisit R\"{u}schendorf's proof of Sklar's theorem and provide some supplementing observations including a further characterisation of distribution functions (Remark 2.3) and a strict mathematical description of their "flat pieces" (Corollary 2.8 and Remark 2.9).

Poisson limits for empirical point processes

Dabrowski, André; Ivanoof, Gail; Kulik, Rafal
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
Define the scaled empirical point process on an independent and identically distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. We obtain limits at points where the density of $Y_i$ may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. These results also hold for multivariate $Y_i$ with little additional effort. Applications are provided both to nearest-neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas.; Comment: 15 pages

Statistical Testing for Conditional Copulas

Acar, Elif F.; Craiu, Radu V.; Yao, Fang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
In conditional copula models, the copula parameter is deterministically linked to a covariate via the calibration function. The latter is of central interest for inference and is usually estimated nonparametrically. However, when a parametric model for the calibration function is appropriate, the resulting estimator exhibits significant gains in statistical efficiency and requires smaller computational costs. We develop methodology for testing a parametric formulation of the calibration function against a general alternative and propose a generalized likelihood ratio-type test that enables conditional copula model diagnostics. We derive the asymptotic null distribution of the proposed test and study its finite sample performance using simulations. The method is applied to two data examples.

Testing the Gaussian Copula Hypothesis for Financial Assets Dependences

Malevergne, Y.; Sornette, D.
Tipo: Artigo de Revista Científica
Using one of the key property of copulas that they remain invariant under an arbitrary monotonous change of variable, we investigate the null hypothesis that the dependence between financial assets can be modeled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student's copula, cannot be rejected if it has sufficiently many degrees of freedom''. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the correlation coefficient between the pair of asset is too high as the tail dependence neglected by the Gaussian copula can be as large as 0.6, i.e., three out five extreme events which occur in unison are missed.; Comment: Latex document of 43 pages including 14 eps figures