Multivariate Measures of Concordance for Copulas and their Marginals

Building upon earlier work in which axioms were formulated for multivariate measures of concordance, we examine properties of such measures. In particular, we examine the relations between the measure of concordance of an $n$-copula and the measures of concordance of the copula's marginals

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

How Does Systematic Risk Impact US Credit Spreads? A Copula Study

It is well known that some relationship between systematic risk and credit risk prevails in financial markets. In our study, S&P 500 stock index return is our market risk proxy whereas credit spreads represent our credit risk proxy as a function of maturity, rating and economic sector. We address the problem of studying the joint distributions and evolutions of S&P 500 return and credit spreads. Graphical and non parametric statistical analysis (i.e.: Kendall’s tau and Spearman’s rho) show that such bivariate distributions are asymmetric with some negative relationship between S&P 500 return and credit spreads. In-deed, credit spreads widen when S&P 500 return decreases or drops under some given level. We investigate then this stylized fact using copula functions to characterize observed dependence structures between S&P 500 return and credit spreads. We focus at least on one parameter copulas and at most on one parameter Archimedean copulas, namely Gumbel, FGM, Frank and Clayton copula functions. Starting from empirical Kendall’s tau observed for each bivariate dependence structure, we induce parameter values for each copula type function belonging to our copulas set. Finally, we exhibit optimal characterizations for such dependence structures and use the optimal selected copulas to achieve a scenario analysis among which stress testing.systematic risk credit risk copulas Archimedean copulas stress testing

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing.Dependence structure, Extremal values, Copula modeling, Copula review

Matrix compatibility and correlation mixture representation of generalized Gini's gamma

Representations of measures of concordance in terms of Pearson' s correlation coefficient are studied. All transforms of random variables are characterized such that the correlation coefficient of the transformed random variables is a measure of concordance. Next, Gini' s gamma is generalized and it is shown that the resulting generalized Gini' s gamma can be represented as a mixture of measures of concordance that are Pearson' s correlation coefficients of transformed random variables. As an application of this correlation mixture representation of generalized Gini' s gamma, lower and upper bounds of the compatible set of generalized Gini' s gamma, which is the collection of all possible square matrices whose entries are pairwise bivariate generalized Gini' s gammas, are derived.Comment: 15 page

Dependence Among Order Statistics for Time-transformed Exponential Models

Let (X1, . . . ,Xn) be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions and with identical marginals. Let for 1 ≤ i ≤ n, Xi:n denote the corresponding ith order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is proved that for m = 2, . . . , n, the dependence of X2:m on X1:m is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that (X1:m,X2:m) is more concordant than (X1,X2). It will be interesting to examine whether these results can be extended to other exchangeable models

Dependence Among Order Statistics for time-transformed exponential models

Let X1, ..., Xn be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for 1 i n, Xi:n denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is in particular proved that for m = 2, ..., n, the dependence of X2:m on X1:m is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that X1:m, X2:mº is more concordant than X1, X2. It will be interesting to examine whether these results can be extended to other exchangeable models

Tensor approximation of generalized correlated diffusions and applications

This thesis documents my research activity conducted in the past three years at the Department of Statistical Science at the University College London. My investigation is focused on functional-analytic methods applied to the characterization of generalized correlated Markov processes. The main objective of the research is to formalize the properties of such a class of stochastic processes when approximated in a tensor space. This lead to the development of a new interpretation of the correlation among processes that is exploited for the analysis of copula functions and their statistical properties