Copula Correlation Research Articles

Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach

International equity markets are characterized by nonlinear dependence and asymmetries. We propose a new dynamic asymmetric copula model to capture long-run and short-run dependence, multivariate nonnormality, and asymmetries in large cross-sections. We find that copula correlations have increased markedly in both developed markets (DMs) and emerging markets (EMs), but they are much lower for EMs than for DMs. Tail dependence has also increased but its level is still relatively low for EMs. We propose new measures of dynamic diversification benefits that take into account higher order moments and nonlinear dependence. The benefits from international diversification have reduced over time, drastically so for DMs. EMs still offer significant diversification benefits, especially during large market downturns.

Small sample properties of copula-GARCH modelling: a Monte Carlo study

Copula-GARCH models have been recently proposed in the financial literature as a statistical tool to deal with flexible multivariate distributions. Our extensive simulation studies investigate the small sample properties of these models and examine how misspecification in the marginals may affect the estimation of the dependence function represented by the copula. We show that the use of Normal marginals when the true Data Generating Process (DGP) is leptokurtic or asymmetric, produces negatively biased estimates of the Normal copula correlations. A striking result is that these biases reach their highest value when correlations are strongly negative, and viceversa. This result remains unchanged with both positively skewed and negatively skewed data, while no biases are found if the variables are uncorrelated. Besides, the effect of marginals asymmetry on correlations is smaller than that of leptokurtosis. We finally analyse the performance of these models in terms of numerical convergence and positive de...

Double T Copula Pricing of Structured Credit Products: Practical Aspects of a Trustworthy Implementation

In spite of its simplicity, the popular One Factor Gaussian Copula model remains the market standard for the valuation of CDO tranches and $n$-th to default. It suffers however from well-known weaknesses, mainly due to the tail behavior of the Normal distribution (namely: the tails are too light, and there is no tail dependence, whatever is the copula correlation). Alternative models have been proposed, among those is the double t copula, which does not share the Gaussian copula drawbacks while not being much more complex. In spite of its nice features, this framework suffers from some technical problems related to its implementation. Without a lot of care, this technique could easily lead to inconsistent results. In our opinion, these difficulties have prevent practitioners to really turn that theoretically sounding model into a workable pricing tool. This paper aims at filling this gap by giving routes toward a reliable implementation of the double t copula framework, throwing away the drawbacks of this framework compared to the Gaussian one. The first purpose of this letter is to show that the implementation issues related to the double t model actually reduce to the estimation of integrals with respect to some student-t distributions. The second part of this note presents an efficient numerical method to perform this tedious task.

Double-t copula pricing of structured credit products: practical aspects of a trustworthy implementation

In spite of its simplicity, the popular One Factor Gaussian Copula model remains the market standard for the valuation of CDO tranches and n-th to default. It suffers however from well-know weaknesses, mainly due to the tail behavior of the Normal distribution (namely: the tails are too light, and there is no tail dependence, whatever is the copula correlation). Alternative models have been proposed, among those is the double t copula, which does not share the Gaussian copula drawbacks while not being much more complex. In spite of its nice features, this framework suffers from some technical problems related to its implementation. Without a lot of care, this technique could easily lead to inconsistent results. In our opinion, these difficulties have prevent practitioners to really turn that theoretically sounding model into a workable pricing tool. This paper aims at filling this gap by giving routes toward a reliable implementation of the double t copula framework, throwing away the drawbacks of this framework compared to the Gaussian one. The first purpose of this letter is to show that the implementation issues related to the double t model actually reduce to the estimation of integrals with respect to some student-t distributions. The second part of this note presents an efficient numerical method to perform this tedious task.

Modeling Dependence Between Loss Triangles

A critical problem in property and casualty insurance is forecasting incurred but as yet unpaid losses. Forecasts and risk margins are often based on individual loss triangles with each triangle corresponding to a different line of business. However lines of business are often related and an overall risk margins must reflect dependence between triangles. This article develops, implements and applies a model for loss triangle dependence. The model relates payments in different triangles in the same calendar year. Dependence is modeled with a Gaussian copula correlation matrix. Correlations are moderated by quantities called communalities which measure the relative impact of cross dependence in each triangle. Correlations can be structured in terms of factor models. Methods reduce to relatively simple calculations in the case of marginal normal distribution. Procedures are applied to US loss triangle data and the impact of loss triangle dependence on risk margins is considered.

Three-stage semi-parametric estimation of [formula omitted]-copulas: Asymptotics, finite-sample properties and computational aspects

A two-stage semi-parametric estimation procedure for a broad class of copulas satisfying minimal regularity conditions has been recently proposed. In addition, a three-stage semi-parametric estimation method based on Kendall’s tau in order to estimate the Student’s t copula has also been designed. Its major advantage is to allow for greater computational tractability when dealing with high dimensional issues, where two-stage procedures are no more a viable choice. The asymptotic properties of this methodology are developed and its finite-sample behavior are examined via simulations. The advantages and disadvantages of this methodology are analyzed in terms of numerical convergence and positive definiteness of the estimated T -copula correlation matrix.

One Factor Models for the ABS Correlation Market: Pricing TABX Tranches

In this paper we look at one factor models for TABX, the tranches of ABX.HE. Both the Gaussian copula and Levy base correlation method are applied to price the tranches. We describe adaptations made to the standard recursive approach for pricing TABX. next we compare the gaussian copula formulation with the Levy base correlation method. We show that ABX.HE and TABX reveal important information when compared to the traditional subordination levels used by the rating agencies before the credit crunch. Finally we present some results obtained with our pricing methodology and we finish by showing the necessity for market participants to be more transparent on the prepayment assumptions.

Correlation Mapping Under Levy and Gaussian Base Correlation

In an earlier paper we treated the concept of Base Expected Loss (BEL) (both for the Gaussian Copula and Levy Base Correlation models) as an arbitrage free approach to interpolate the base correlation curves for pricing non-standard tranches of the standardized credit indices. In this paper we extend the approach further by using the technique to price CDO's of non-standardized portfolios (so called bespoke CDO's). That is the framework is developed in the context of correlation mapping. Additionally we compare the correlation mapping methodology developed here with alternative approaches largely used with practitioners.

The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study

The effect on the estimation of the Value at Risk when dealing with multivariate portfolios when there is a misspecification both in the marginals and in the copulas is investigated. It is first shown that, when there is skewness in the data and symmetric marginals are used, the estimated elliptical (normal or t) copula correlations are negatively biased, reaching values as high as 70% of the true values. Besides, the bias almost doubles if negative correlations are considered, compared to positive correlations. As for the t copula degrees of freedom parameter, the use of wrong marginals delivers large positive biases, instead. If the dependence structure is represented by a copula which is not elliptical, e.g. the Clayton copula, the effects of marginal misspecifications on the copula parameter estimation can be rather different, depending on the sign of marginal skewness. Extensive Monte Carlo studies then show that the misspecifications in the marginal volatility equation more than offset the biases in copula parameters when VaR forecasting is of concern, small samples are considered and the data are leptokurtic. The biases in the volatility parameters are much smaller, whereas those ones in the copula parameters remain almost unchanged or even increase when the sample dimension increases. In this case, copula misspecifications do play a role for VaR estimation. However, these effects depend heavily on the sign of the dependence.

Correlated Default Modeling with a Forest of Binomial Trees

This paper exploits the endogenous default function framework of Das and Sundaram (2007) to develop an approach for modeling correlated default on binomial trees usually used for pricing equity options. We show how joint default contracts may be valued on these trees. The model accommodates different correlation assumptions and practical implementation considerations. Credit portfolio characteristics are examined within the model and found to be consistent with stylized empirics. Risk premia for default are computable and shown to be relatively higher for poor quality firms. Equity volatility is shown to impact correlated credit loss distributions substantially. Two kinds of default dependence are explored, one coming from default intensity correlations, and the other from further conditional dependence in defaults after accounting for intensity correlations (residual copula correlation). Both are found to impact credit loss distributions, though the absence of either makes these distributions less sensitive to correlation assumptions; on balance intensity correlations are more critical.

Geospatial–temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America

A quantification of the spatio-temporal dependence among precipitation extremes is important for investigating the properties of intense storms as well as flood or flash-flood related hazards. Extreme value theory has been widely applied to the hydrologic sciences and hydraulic engineering. However, rigorous approaches to quantify dependence structures among extreme values in space and time have not been reported in the literature. Previous researchers have quantified the dependence among extreme values through the concept of (pairwise bivariate) tail dependence coefficients. For estimation of the tail dependence coefficients, we apply a recently developed method [Kuhn G. On dependence and extremes. PhD thesis (Advisor: C. Klüppelberg), Munich University of Technology, 2006] which utilized the multivariate tail dependence function of a subclass of elliptical copulas. This study extends the previous approach in the context of space and time by considering pairs of spatial grids in South America and quantifying the dependence among precipitation extremes based on the time series at each spatial grid. In addition, Kendall’s τ is used to estimate the pairwise copula correlation (for an elliptical copula) of precipitation between all grids in South America. The geospatial–temporal dependence measures are applied to precipitation observations from 1940 to 2005 as well as simulations from the Community Climate System Model version 3 (CCSM3) for 1940–2099. New insights are obtained regarding the spatio-temporal dependence structures for precipitation over South America both with regard to correlation as well as tail dependence.

Credit default swaps with counterparty risk: a calibrated Markov model

This article describes a continuous-time Markov approach to the riskneutral pricing of a credit default swap with counterparty risk. The key parameters in the approach are the transition rates, which naturally incorporate the ideas of contagion. Correlation (which is time-dependent) is a derived quantity, which results from contagion. An expansion in powers of a small parameter (a risk-neutral default probability) allows analytic formulae to be obtained for all relevant quantities. Thus, the problem of the calibration of the model to the market prices of the bonds of the reference entity and the counterparty, as well as to the credit default swap spreads of the reference entity (which are independent of the bond spreads in the case of counterparty risk), is solved. This is done with the help of analytic results for the spread of a credit default swap with counterparty risk, as well as for other derivative prices. A comparison with results produced by the market-standard Gaussian copula approach indicates that the market-standard approach could be significantly improved by allowing the copula correlation coefficient to be time-dependent.