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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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...

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## Rank-based inference for bivariate extreme-value copulas

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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)

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## Algorithms for Finding Copulas Minimizing Convex Functions of Sums

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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.

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## Mixed Cumulative Distribution Networks

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2010
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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

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## A Directional Multivariate Value at Risk

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/02/2015
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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...

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## Maximum entropy distribution of order statistics with given marginals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/09/2015
Português

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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

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## Extremes of multivariate ARMAX processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2012
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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.

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## A test for Archimedeanity in bivariate copula models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/09/2011
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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

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## Block-Maxima of Vines

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/04/2015
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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

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## Exponent dependence measures of survival functions and correlated frailty models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/09/2014
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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

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## A Dynamic Correlation Modelling Framework with Consistent Stochastic Recovery

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/04/2010
Português

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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...

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## Relations Between Stochastic Orderings and generalized Stochastic Precedence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Information bounds for Gaussian copulas

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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)

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## Bayes Multiple Decision Functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/09/2011
Português

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/02/2012
Português

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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).

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## Poisson limits for empirical point processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/05/2006
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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

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## Statistical Testing for Conditional Copulas

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/04/2012
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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.

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## Testing the Gaussian Copula Hypothesis for Financial Assets Dependences

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/11/2001
Português

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#Condensed Matter - Statistical Mechanics#Nonlinear Sciences - Adaptation and Self-Organizing Systems#Quantitative Finance - Statistical Finance

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

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## A nested factor model for non-linear dependences in stock returns

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/09/2013
Português

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#Quantitative Finance - Risk Management#Quantitative Finance - Portfolio Management#Quantitative Finance - Statistical Finance

The aim of our work is to propose a natural framework to account for all the
empirically known properties of the multivariate distribution of stock returns.
We define and study a "nested factor model", where the linear factors part is
standard, but where the log-volatility of the linear factors and of the
residuals are themselves endowed with a factor structure and residuals. We
propose a calibration procedure to estimate these log-vol factors and the
residuals. We find that whereas the number of relevant linear factors is
relatively large (10 or more), only two or three log-vol factors emerge in our
analysis of the data. In fact, a minimal model where only one log-vol factor is
considered is already very satisfactory, as it accurately reproduces the
properties of bivariate copulas, in particular the dependence of the
medial-point on the linear correlation coefficient, as reported in
Chicheportiche and Bouchaud (2012). We have tested the ability of the model to
predict Out-of-Sample the risk of non-linear portfolios, and found that it
performs significantly better than other schemes.

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