Pricing and Hedging Credit Derivatives with Copulas

In this paper, we apply a copula function pricing technique to the evaluation of credit derivatives, namely a vulnerable default put option and a credit switch. Also in this case, copulas enable one to separate the specification of marginal default probabilities from their dependence structure. Their use is based here on no–arbitrage arguments, which provide pricing bounds and easy–to–implement super–replication strategies.
At a second stage, we specify the copula function to be a mixture one. In this case, we obtain closed form prices and hedges, which we calibrate on real market data. For the sake of comparison, we add a Clayton calibration.
(J.E.L: G11, G12).     

Dependence structure between nominal and index-linked bond returns: a bivariate copula and DCC-GARCH approach

This article investigates the dependence structure related to four French nominal and index-linked bonds with various maturities and reference indices. To achieve this aim, we estimate various copulas to select the appropriate one for our data. We also compare results obtained using the copula method with multivariate dynamic conditional correlation GARCH (DCC-GARCH) modelling. The major issue in this study is that the best copulas used to model the dependence among bond returns are the Plackett and Student models. We also find a dynamic correlation between bond returns. In particular, the relationship between nominal and indexed bonds is characterized by an asymmetric dependence. Moreover, the results obtained by the copula approach are confirmed by those obtained by multivariate GARCH modelling. Our empirical study provides a useful method that may be employed by decision-makers to quantitatively introduce dependence and spillover effects in their bond issuance policy. For investors, we propose optimal investment combinations in bonds with respect to their investment horizons.     

Regional and copula estimation effects on EU and US energy equity portfolios

ABSTRACT

The aim of this paper is to investigate the regional interdependence structure of energy equities in the US and in the EU. Based on weekly stock prices of 28 big energy firms in the two regions from 2008 to 2019, we compare the efficiency of using bivariate or multivariate copulas to describe the dependence structure of energy equities. Furthermore, we investigate the impact of the choice between these two methods on the performance of energy equity portfolios. Our empirical results show that multivariate copulas, such as C-Vine, allow to better describe the dependence structure of energy equities. We also find that there is a stronger and more complex dependence structure among EU energy equities than among US energy equities. Our scenario analysis also shows that the dependence structure is stronger during the GFC while being weaker during the ESDC. More importantly, the correlation matrix obtained from the multivariate copula method allows to obtain optimal mean-CVaR portfolios with a higher performance than that from the bivariate copula method. More importantly, optimal portfolios constituted with multivariate copulas allow to reduce the portfolio’s sensitivity to oil prices.     

A new test procedure for the choice of dependence structure in risk measurement: application to the US and UK stock market indices

The choice of an appropriate dependence structure in modelling multivariate risks is an important issue because different tail structure embedded in copula leads to a different capital requirement for the institution. We present how to select a well-specified dependence structure to given application data. Using a simple simulation technique, we develop a statistical test to assess the adequacy of a specific dependence structure. We examine the sensitivity of risk estimates to the choice of copulas using the S&P 500 and FTSE 100 stock indices.     

Energy portfolio risk management using time-varying extreme value copula methods

This work is concerned with the statistical modeling of the dependence structure between three energy commodity markets (WTI crude oil, natural gas and heating oil) using the concept of copulas and proposes a method for estimating the Value at risk (VaR) of energy portfolio based on the combination of time series models with models of the extreme value theory before fitting a copula. Each return series is modeled by AR-(FI) GARCH univariate model. Then, we fit the GPD distribution to the tails of the residuals to model marginal residuals distributions. The extreme value copula to the iid residuals is fitted and we simulate from it to construct N portfolios and estimate VaR. As a first step, the method is applied to a two-dimensional energy portfolio. In second step, we extend method in trivariate context to measure VaR of three-dimensional energy portfolio. Dependences between residuals are modeled using a trivariate nested Gumbel copulas. Methods proposed are compared with various univariate and multivariate conventional VaR methods. The reported results demonstrate that GARCH-t, conditional EVT and FIGARCH extreme value copula methods produce acceptable estimates of risk both for standard and more extreme VaR quantiles. Generally, copula methods are less accurate compared with their predictive performances in the case of portfolio composed of exchange market indices.     

Sovereign bond market dependencies and crisis transmission around the eurozone debt crisis: a dynamic copula approach

We examine the dependency between the European government bond markets around the recent sovereign debt crisis. A dynamic copula approach is used to model the time-varying dependence structure of those government bond markets, evaluate the nature and strength of their dependencies over time, and gauge the transmission of the crisis shocks. Our results can be summarized as follows: i) the eurozone sovereign bond markets under consideration have a significant and positive dependence with the Greek and the EMU benchmark sovereign bond markets; ii) the dynamic-BB7 copula function best describes the dependence structure between these sovereign bond markets and provides evidence of asymmetric tail dependence; iii) the conditional probability of crisis transmission from Greece to other eurozone countries is higher than the other way around; and iv) Greece is the most vulnerable country when the eurozone entered into the sovereign debt crisis.     

Analyzing the dependence structure of various sectors in the Brazilian market: A Pair Copula Construction approach

In this paper we estimate the dependence structure between economic sectors in the Brazilian financial market through Pair Copula Construction. We use daily data from indices which represent telecommunications, energy, industrials, consumer, financial, basic materials and real estate sectors in BM&F/Bovespa. Results indicate predominance of student's t copula in structure. BB1, BB7, BB8, Frank and Joe copulas also fit into some relationships. Regarding dependence, tail measures obtain relevant values in most relationships. Lower tail dependence exceeds absolute, measured by Kendall's Tau, and upper tail in many cases, reflecting the asymmetry in some relationships. Further, in order to give robustness to these results, we forecast daily Value at Risk, considering distinct significance levels, of a portfolio composed of studied sectors through the estimated structure. Results allow one to conclude that VaR predictions are correct. These results permit business industry participants to construct portfolios with assets of these sectors under a proper diversification structure. Moreover, from an international point of view, investors who are interested in diversification could perform more sophisticated strategies in this country rather than simply trading the index.     

A copula–GARCH model for macro asset allocation of a portfolio with commodities

Many authors have suggested that the mean-variance criterion, conceived by Markowitz (The Journal of Finance 7(1):77–91, 1952), is not optimal for asset allocation, because the investor expected utility function is better proxied by a function that uses higher moments and because returns are distributed in a non-Normal way, being asymmetric and/or leptokurtic, so the mean-variance criterion cannot correctly proxy the expected utility with non-Normal returns. In Riccetti (The use of copulas in asset allocation: when and how a copula model can be useful? LAP Lambert, Saarbrücken 2010), a copula–GARCH model is applied and it is found that copulas are not useful for choosing among stock indices, but can be useful in a macro asset allocation model, that is, for choosing stock and bond composition of portfolios. In this paper I apply that copula–GARCH model for the macro asset allocation of portfolios containing a commodity component. I find that the copula model appears to be useful and better than the mean-variance one for the macro asset allocation also in presence of a commodity index, even if it is not better than GARCH models on independent univariate series, probably because of the low correlation of the commodity index returns to the stock, the bond and the exchange rate returns.     

How to maximize golden opinions

In this article, we aim to model the level and structure of the dependence between the world's leading stock markets and those of the emerging market groups?–?Europe, Latin America and Far East. To this end we use a mixture model of Gaussian, Gumbel and Gumbel survival copulas. Our results indicate that none of the pairs of stock markets exhibit a right-tail dependence structure. All valid models exhibit a mixture of Gaussian and left-tail dependence structure. Our findings imply that Gaussian dependence structure is dominant in most of the models. The emerging equity markets in the European region exhibit the most significant dependence structure with the world leaders. Furthermore, most of the emerging equity markets have a significant dependence structure with the US stock market. We further compare our findings with the results of the conventional correlation coefficients and conclude the importance of using copula models in analysing the portfolio diversification opportunities. Our findings overall indicate two important remarks: First, the copula models reveal better indicators for global investors to establish a diversified portfolio; Second, international equity markets exhibit significant dependence, which leaves a smaller opportunity to benefit from international portfolio diversification.     

农民工如何实现从土地保障向制度保障转变

目前,农村土地的保障功能正在不断弱化,但是中国农民工作为一个特殊的群体对农村土地仍然存在着很强的依赖性,这一矛盾的产生有其根本的制度性原因。以户籍制度为基础的一系列相关制度是阻止农民工融入城市的根本性障碍,要使农民工摆脱对农村土地的依赖,需要变革现有的制度环境,为他们提供制度上的保障。     

Directional time-varying partial correlation with the Gaussian copula–DCC–GARCH model

This article suggests a directional time-varying partial correlation based on the dynamic conditional correlation (DCC) method. A recent study proposed the copula DCC based on the vine structure. Due to the arbitrary variable selection, their method can produce unnecessary dependence in the multivariate structure, with extra economic and computational burdens. To overcome this limitation, we incorporate directional dependence by copula to track the causal relationship among multiple variables and then extend the copula bivariate DCC method to a directional time varying partial correlation in the multivariate structure. Our proposed method provides a reasonable and efficient conditional dependence structure, without the trial and error process. We offer an application of our method to the U.S. stock market as an illustrated example.     

Extremal Economic (Inter)Dependence Studies: A Case of the Eastern European Countries

This paper considers the application of copula models to study the shifts in extremal economic dependence of the Eastern European countries, i.e., Ukraine and its neighbouring countries, from 1969 to 2014. Extremal economic dependence is analysed in terms of poverty and affluence and with regard to growth rate. This paper contributes to the previous literature by applying the copula approaches to derive the measurements of the economic interdependence in terms of poverty and affluence. The received results depict the pattern of the (inter)dependence and its evolution across the analysed countries. Dependence on other countries in the extreme values can potentially be useful in adjustments of the economic policy of a country to minimize poverty and prevent high inequality.

     

Using Copulas to Model Switching Regimes with an Application to Child Labour*

The copula approach to econometric modelling involves the specification of the separate components of the joint distribution of the random variables of interest: models built for each margin are bound together using a copula function. In this paper, the copula approach is used to construct models for switching regimes. The construct is illustrated by fitting a wage earnings model for child workers in the early 1900s, with regimes governed according to literacy. The results improve on earlier modelling efforts by Poirier and Tobias (2003), finding that a child worker may on average expect a reduction in earnings from being literate.     

Modeling the dependence structure between default risk premium,equity return volatility and the jump risk: Evidence from a financial crisis

This paper investigates the dependence structure between default risk premium, equity return volatility and jump risk in the equity market before and during the subprime crisis. Using iTraxx CDS index spreads from Japanese and Australian markets, the paper models the different relationships that can exist in different ranges of behavior. We consider several Archimedean copula models with different tail dependence structures, namely, Gumbel, Clayton, Frank, AMH and Joe copulas. Although the dramatic change in the levels of the iTraxx CDS index, we find strong evidence that the dependence structure between CDS and stock market conditions is asymmetric and orienting toward the upper side. In addition, we find that the Japanese CDS market is more sensitive to the stock return volatility than the jump risk and the magnitude of this sensitivity is related to the market circumstances. However, Australian CDS market is more sensitive to the jump risk than stock return volatility before and during the financial crisis. This result has important implications for both global financial stability and default risk management. Specifically, the heterogeneity of markets, coupled with the diversity in the risk exposures cause the default risk premium and equity markets to exhibit different levels of sensitivity.     

Measuring and Optimizing Portfolio Credit Risk: A Copula‐based Approach*

In this work, we present a methodology for measuring and optimizing the credit risk of a loan portfolio taking into account the non‐normality of the credit loss distribution. In particular, we aim at modelling accurately joint default events for credit assets. In order to achieve this goal, we build the loss distribution of the loan portfolio by Monte Carlo simulation. The times until default of each obligor in portfolio are simulated following a copula‐based approach. In particular, we study four different types of dependence structure for the credit assets in portfolio: the Gaussian copula, the Student's t‐copula, the grouped t‐copula and the Clayton n‐copula (or Cook–Johnson copula). Our aim is to assess the impact of each type of copula on the value of different portfolio risk measures, such as expected loss, maximum loss, credit value at risk and expected shortfall. In addition, we want to verify whether and how the optimal portfolio composition may change utilizing various types of copula for describing the default dependence structure. In order to optimize portfolio credit risk, we minimize the conditional value at risk, a risk measure both relevant and tractable, by solving a simple linear programming problem subject to the traditional constraints of balance, portfolio expected return and trading. The outcomes, in terms of optimal portfolio compositions, obtained assuming different default dependence structures are compared with each other. The solution of the risk minimization problem may suggest us how to restructure the inefficient loan portfolios in order to obtain their best risk/return profile. In the absence of a developed secondary market for loans, we may follow the investment strategies indicated by the solution vector by utilizing credit default swaps.     

Mixed copula model with stochastic correlation for CDO pricing

This paper deals with the problem of pricing credit derivatives portfolio—CDO. The article assumes that the systematic factor and idiosyncratic factors subject to the fat-tailed mixed G-VG distribution instead of the traditional Gaussian distribution in the framework of factor model. Thus, the G-VG copula model is established. Stochastic correlation is also incorporated to account for the correlation skew problem. The semi-analytical expressions for conditional default probability, cumulative loss distribution function and expected tranche loss are explicitly derived in the G-VG copula models under large homogeneous portfolio approximation. Thus the CDO price can be determined. The numerical analysis is carried out and the properties of the new models with those of the traditional models are compared. Results show that new models not only provide a closer fit to the market quotes, but also bring more flexibility into the dependence structure.     

The dependence structure between the Canadian stock market and the USD/CAD exchange rate: a copula approach

Abstract This paper investigates the dependence structure between the real Canadian stock returns and the real USD/CAD exchange rate returns, using the Symmetrized Joe‐Clayton (SJC) copula function. We estimate the SJC copula with monthly data over the period 1995:1 to 2006:12. Our results show significant asymmetric static and dynamic tail dependence between the real stock returns and the real exchange rate returns, such that the two returns are more dependent in the left than in the right tail of their joint distribution. We explain this asymmetric dependence in terms of an asymmetric interest rate policy by Canadian monetary authorities in response to changes in the real exchange rate during sub‐periods of falling and rising commodity prices.     

Estimating Lower Tail Dependence Between Pairs of Poverty Dimensions in Europe

This article provides an innovative method for measuring the dependence between pairs of poverty dimensions using a semiparametric copula approach that permits us to account for the importance of extreme low values. The association between pairs of poverty dimensions at the lower tail is easily measured using the parameter estimates of the specified parametric copula, and no further calculations are needed. This approach is used to measure the bivariate lower tail dependence between the dimensions of the AROPE rate in Europe at two time points (2009 and 2018). The findings reveal a statistically significant increase in the lower tail dependence between 2009 and 2018 in several European countries.     

MODELLING ASYMMETRIC EXCHANGE RATE DEPENDENCE*

We test for asymmetry in a model of the dependence between the Deutsche mark and the yen, in the sense that a different degree of correlation is exhibited during joint appreciations against the U.S. dollar versus during joint depreciations. We consider an extension of the theory of copulas to allow for conditioning variables, and employ it to construct flexible models of the conditional dependence structure of these exchange rates. We find evidence that the mark–dollar and yen–dollar exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating.     

Value-at-risk Trade-off and Capital Allocation with Copulas

This paper uses copula functions to evaluate tail probabilities and market risk trade-offs at a given confidence level, dropping the joint normality assumption on returns. Copulas enable one to represent distribution functions separating the marginal distributions from the association structure. We present an application to two stock market indices: for each market we recover the marginal probability distribution. We then calibrate copula functions and recover the joint distribution. The estimated copulas directly give the joint probabilities of extreme losses. Their level curves measure the trade-off between losses over different desks. This trade-off can be exploited for capital allocation and is shown to depend on fat tails.
(J.E.L.: C14, G19, G29).