Student-t Copula Research Articles

Multivariate Copula-Based Joint Probability Distribution of Water Supply and Demand in Irrigation District

Based on the data series of rainfall, reference crop evapotranspiration and irrigation water from 1970 to 2013 in the Luhun irrigation district of China, the multivariate joint probability of water supply and demand are constructed with student t-copula function. The results show that student t-copula function can indicate the associated dependence structure amongst these variables well, and the constructed multivariate copula-based joint probability distribution reveal the statistical characteristics and occurrence probability of different combinations of water supply and water demand. Moreover, the trivariate joint probability distribution is more reasonable than the bivariate distribution to reflect the water shortage risk, and these joint distribution values of different combinations of water supply and demand can provide the technological support for water shortage risk evaluation in the irrigation district.

Modeling spot price dependence in Australian electricity markets with applications to risk management

We examine the dependence structure of electricity spot prices across regional markets in Australia. One of the major objectives in establishing a national electricity market was to provide a nationally integrated and efficient electricity market, limiting market power of generators in the separate regional markets. Our analysis is based on a GARCH approach to model the marginal price series in the considered regions in combination with copulae to capture the dependence structure between the marginals. We apply different copula models including Archimedean, elliptical and copula mixture models. We find a positive dependence structure between the prices for all considered markets, while the strongest dependence is exhibited between markets that are connected via interconnector transmission lines. Regarding the nature of dependence, the Student-t copula provides a good fit to the data, while the overall best results are obtained using copula mixture models due to their ability to also capture asymmetric dependence in the tails of the distribution. Interestingly, our results also suggest that for the four major markets, NSW, QLD, SA and VIC, the degree of dependence has decreased starting from the year 2008 towards the end of the sample period in 2010. Examining the Value-at-Risk of stylized portfolios constructed from electricity spot contracts in different markets, we find that the Student-t and mixture copula models outperform the Gaussian copula in a backtesting study. Our results are important for risk management and hedging decisions of market participants, in particular for those operating in several regional markets simultaneously.

A MULTIVARIATE JUMP DIFFUSION PROCESS FOR COUNTERPARTY RISK IN CDS RATES

We consider counterparty risk in CDS rates. To do so, we use a multivariate jump diffusion process for obligors’ default intensity, where jumps (i.e. magnitude of contribution of primary events to default intensities) occur simultaneously and their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process. We apply copuladependent default intensities of multivariate Cox process to derive the joint Laplace transform that provides us with joint survival/default probability and other relevant joint probabilities. For that purpose, the piecewise deterministic Markov process (PDMP) theory developed in [7] and the martingale methodology in [6] are used. We compute survival/default probability using three copulas, which are Farlie-Gumbel-Morgenstern (FGM), Gaussian and Student-t copulas, with exponential marginal distributions. We then apply the results to calculate CDS rates assuming deterministic rate of interest and recovery rate. We also conduct sensitivity analysis for the CDS rates by changing the relevant parameters and provide their figures.

Estimating Risk of Natural Gas Portfolios by Using GARCH-EVT-Copula Model.

This paper concentrates on estimating the risk of Title Transfer Facility (TTF) Hub natural gas portfolios by using the GARCH-EVT-copula model. We first use the univariate ARMA-GARCH model to model each natural gas return series. Second, the extreme value distribution (EVT) is fitted to the tails of the residuals to model marginal residual distributions. Third, multivariate Gaussian copula and Student t-copula are employed to describe the natural gas portfolio risk dependence structure. Finally, we simulate N portfolios and estimate value at risk (VaR) and conditional value at risk (CVaR). Our empirical results show that, for an equally weighted portfolio of five natural gases, the VaR and CVaR values obtained from the Student t-copula are larger than those obtained from the Gaussian copula. Moreover, when minimizing the portfolio risk, the optimal natural gas portfolio weights are found to be similar across the multivariate Gaussian copula and Student t-copula and different confidence levels.

Operational Risk Modelling in Insurance and Banking

The author of the presented paper is trying to develop and implement the model that can mimic the state of the art models of operational risk in insurance. It implements generalized Pareto distribution and Monte Carlo simulation and tries to mimic and construct operational risk models in insurance. At the same time, it compares lognormal, Weibull and loglogistic distribution and their application in insurance industry. It is known that operational risk models in insurance are characterized by extreme tails, therefore the following analysis should be conducted: the body of distribution should be analyzed separately from the tail of the distribution. Afterwards the convolution method can be used to put together the annual loss distribution by combining the body and tail of the distribution. Monte Carlo method of convolution is utilized. Loss frequency in operational risk in insurance and overall loss distribution based on copula function, in that manner using student-t copula and Monte Carlo method are analysed. The aforementioned approach represents another aspect of observing operational risk models in insurance. This paper introduces: 1) Tools needed for operational risk models; 2) Application of R code in operational risk modeling;3) Distributions used in operational risk models, specializing in insurance; 4) Construction of operational risk models.

Localization and Tracking of Objects Using Cross-Correlation of Shadow Fading Noise

Multipath and shadow fading are the primary cause for positioning errors in a Received Signal Strength Indicator (RSSI) based localization scheme. While fading, in general, is detrimental to localization accuracy, cross-correlation and divergence properties of shadow fading residuals may be utilized to improve localization and tracking accuracy of mobile IEEE 802.15.4 transmitters. Therefore, this paper begins by presenting a stochastic filter that models the fast changing multipath fading as a mean reverting Ornstein-Uhlenbeck (OU) process followed by a Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) filtering to isolate the slow changing shadow fading residuals from measured RSSI values. Subsequently, a novel wireless transmitter localization scheme that combines the measured cross-correlation in shadow fading residuals between adjacent receivers using a Student-t Copula likelihood function is proposed. However, the long convergence time for this highly non-convex copula function might render our method unsuitable for tracking applications. Therefore, we present a faster tracking method where the velocity and heading of a mobile transmitter are estimated from α-Divergence between shadow fading signals and an onboard gyroscope respectively. To bind the localization error in this tracking method, the transmitter location estimates are smoothed by a Bayesian particle filter. The performance of our proposed localization and tracking method is validated over simulations and hardware experiments.

Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River

As an alternative to the commonly used univariate flood frequency analysis, copula frequency analysis can be used. In this study, 58 flood events at the Litija gauging station on the Sava River in Slovenia were analysed, selected based on annual maximum discharge values. Corresponding hydrograph volumes and durations were considered. Different bivariate copulas from three families were applied and compared using different statistical, graphical and upper tail dependence tests. The parameters of the copulas were estimated using the method of moments with the inversion of Kendall's tau. The Gumbel–Hougaard copula was selected as the most appropriate for the pair of peak discharge and hydrograph volume (Q-V). The same copula was also selected for the pair hydrograph volume and duration (V-D), and the Student-t copula was selected for the pair of peak discharge and hydrograph duration (Q-D). The differences among most of the applied copulas were not significant. Different primary, secondary and conditional return periods were calculated and compared, and some relationships among them were obtained. Copyright © 2014 John Wiley & Sons, Ltd.

Simplified pair copula constructions—Limitations and extensions

So-called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption, that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on, is popular.We show that the only Archimedean copulas in dimension d≥3 which are of the simplified type are those based on the Gamma Laplace transform or its extension, while the Student-t copulas are the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on, and demonstrate a technique to assess the distance of a multivariate distribution from a nearby distribution that satisfies the simplifying assumption.

Hedging structured credit products during the credit crisis: A horse race of 10 models

Pricing and hedging structured credit products poses major challenges to financial institutions. This paper puts several valuation approaches through a crucial test: How did these models perform in one of the worst periods of economic history, September 2008, when Lehman Brothers went under? Did they produce reasonable hedging strategies? We study several bottom-up and top-down credit portfolio models and compute the resulting delta hedging strategies using either index contracts or a portfolio of single-name CDS contracts as hedging instruments. We compute the profit-and-loss profiles and assess the performances of these hedging strategies. Among all 10 pricing models that we consider the Student-t copula model performs best. The dynamical generalized-Poisson loss model is the best top-down model, but this model class has in general problems to hedge equity tranches. Our major finding is however that single-name and index CDS contracts are not appropriate instruments to hedge CDO tranches.

Forecasting Modeling of Heavy Tail Typed Demand using Student's t-Copula Fitting in Supply Chain Management

As the demand-oriented management has been getting important in Supply Chain Management (SCM), various forecasting methods have been suggested including regression analyses. However, dependency structures among variables have been captured by a correlation coefficient, only. It results in inaccurate demand predictions. This paper suggests a new and effective forecasting modeling framework using student's t-copula function. In order to show overall modeling procedures framework, heavy tail typed numerical data and its copula estimations are provided. The suggested methodology can contribute to decrease the bullwhip effect and to stabilize volatile environment in a supply chain network.

An economic evaluation of stock–bond return comovements with copula-based GARCH models

Owing to their importance in asset allocation strategies, the comovements between the stock and bond markets have become an increasingly popular issue in financial economics. Moreover, the copula theory can be utilized to construct a flexible joint distribution that allows for skewness in the distribution of asset returns as well as asymmetry in the dependence structure between asset returns. Therefore, this paper proposes three classes of copula-based GARCH models to describe the time-varying dependence structure of stock–bond returns, and then examines the economic value of copula-based GARCH models in the asset allocation strategy. We compare their out-of-sample performance with other models, including the passive, the constant conditional correlation (CCC) GARCH and the dynamic conditional correlation (DCC) GARCH models. From the empirical results, we find that a dynamic strategy based on the GJR-GARCH model with Student-t copula yields larger economic gains than passive and other dynamic strategies. Moreover, a less risk-averse investor will pay higher performance fees to switch from a passive strategy to a dynamic strategy based on copula-based GARCH models.

Greek Crisis, Stock Market Volatility and Exchange Rates in the European Monetary Union: A VAR-GARCH-Copula Model

The main objectives of this study are twofold. The first objective is to examine the volatility spillover between seventeen European stock market returns and exchange rate, over the period 2007-2011, in a multivariate setting, using the VAR (1)-GARCH (1,1) model which allows for transmission in returns and volatility. The second is to investigate the dependence structure and to test the degree of the dependence between financial returns using two measures of dependence: correlations and copula functions. Five candidates, the Gaussian, the Student’s t, the Frank, the Clayton and the Gumbel copulas, are compared. Our empirical results for the first objective suggest that past own volatilities matter more than past shocks (news) and there are moderate cross market volatility transmission and shocks between the markets. Moreover, the result on the second objective implies that, considering all the financial returns together, the Student-t copula seems the best fitting model, followed by the Normal copula, both for the two sub-period. The dependence structure is symmetric and has non-zero tail dependence. However, if we examine the relationship between each pair of stock-FX returns, both of the degree of the dependence and the dependence structure vary when the financial Greek crisis occurs. Our findings have important implications for global investment risk management by taking into account joint tail risk.

Conditional dependence structure between oil prices and exchange rates: A copula-GARCH approach

We study the conditional dependence structure between crude oil prices and U.S. dollar exchange rates using a copula-GARCH approach. Various copula functions of the elliptical, Archimedean and quadratic families are used to model the underlying dependence structure in both bearish and bullish market phases. Over the 2000–2011 period, we find evidence of significant and symmetric dependence for almost all the oil-exchange rate pairs considered. The rise in the price of oil is found to be associated with the depreciation of the dollar. Moreover, we show that Student-t copulas best capture the extreme dependence, and that taking the extreme comovement into account leads to improve the accuracy of VaR forecasts. Our main results remain unchanged when considering alternative GARCH-type specifications and the crisis period, but are sensitive to the use of raw returns.

The Greek Financial Crisis, Extreme Co-Movements and Contagion Effects in the Emu: A Copula Approach

This paper examines the extent of the current financial Greek crisis and the contagion effects it concludes toward the euro zone by conducting an empirical investigation of the dependence structure between seventeen European stock markets during the period 2007-2011. In particular, several copula functions are used to model the degree of cross-market linkages. The model is implemented with a GARCH model for the marginal distributions and the student-t copula for the joint distribution which allows capturing nonlinear relationships and offers significant advantages over econometric techniques in analyzing the co-movement of financial time-series. Our empirical results show that there is strong evidence of market dependence in the euro area. However, the dependence remains significant but weaker, for the major of stock markets, after the occurrence of the Greek crisis.

General Multivariate Dependence Using Associated Copulas

This paper studies the general multivariate dependence of a random vector using associated copulas. We extend definitions and results of positive dependence to the general dependence case. This includes associated tail dependence functions and associated tail de- pendence coefficients. We derive the relationships among associated copulas and study the associated copulas of the perfect dependence cases and elliptically contoured distributions. We present the expression for the associated tail dependence function of the multivariate Student-t copula, which accounts for all types of tail dependence.

Hedging Structured Credit Products During the Credit Crunch: A Horse Race of 10 Models

Pricing and hedging structured credit products poses major challenges to financial institutions. This has become very clear during the recent credit crisis. This paper puts several valuation approaches through a crucial test: How did these models perform in one of the worst periods of economic history, September 2008, when Lehman Brothers went down? Did they produce reasonable hedging strategies? We study several bottom-up and top-down credit portfolio models and compute the resulting delta hedging strategies using either index contracts or a portfolio of single-name CDS contracts as hedging instruments. We compute the profit-and-loss profiles and assess the performances of these hedging strategies. Among all 10 pricing models that we consider the Student-t copula model performs the best. The dynamical generalized-Poisson loss model is the best top-down model, but this model class has in general problems to hedge equity tranches. Our major finding is however that single-name and index CDS contracts are not appropriate instruments to hedge CDO tranches.

Using Dynamic Copulae for Modeling Dependency in Currency Denominations of a Diversified World Stock Index

Abstract The aim of this paper is to model the dependency among log-returns when security account prices are expressed in units of a well diversified world stock index. The dependency in log-returns of currency denominations of the index is modeled using time-varying copulae, aiming to identify the best fitting copula family. The Student-t copula turns generally out to be superior to e.g. the Gaussian copula, where the dependence structure relates to the multivariate normal distribution. It is shown that merely changing the distributional assumption for the log-returns of the marginals from normal to Student-t leads to a significantly better fit. The Student-5 copula with Student-t marginals is able to better capture dependent extreme values than the other models considered. Furthermore, the paper applies copulae to the estimation of the Value-at-Risk and the expected shortfall of a portfolio constructed of savings accounts of different currencies. The proposed copula-based approach allows to split market...

The economic value of co-movement between oil price and exchange rate using copula-based GARCH models

The US dollar is used as the primary currency of international crude oil trading; as such, the recent substantial depreciation in the US dollar has resulted in a corresponding increase in crude oil prices. In addition, oil price and exchange-rate returns have been shown to be skewed and leptokurtic, and to exhibit an asymmetric or tail dependence structure. Therefore, this study proposes dynamic copula-based GARCH models to explore the dependence structure between the oil price and the US dollar exchange rate. More importantly, an asset-allocation strategy is implemented to evaluate economic value and confirm the efficiency of the copula-based GARCH models. In terms of out-of-sample forecasting performance, a dynamic strategy based on the CGARCH model with the Student-t copula exhibits greater economic benefits than static and other dynamic strategies. In addition, the positive feedback trading activities are statistically significant within the oil market, but this information does not enhance the economic benefits from the perspective of an asset-allocation decision. Finally, a more risk-averse investor generates a higher fee for switching from a static strategy to a dynamic strategy based on copula-based GARCH models.

The Dependence Structure between Carbon Emission Allowances and Financial Markets - A Copula Analysis

This paper applies different copulas in order to investigate the complex dependence structure between EU emission allowance (EUA) futures returns and those of other commodities, equity and energy indices. The analysis yields important insights into the relationship between carbon, commodities and financial markets. First of all, we find a significant relationship between EUA returns and those of the other considered variables that is most appropriately modeled by a Gaussian and Student-t copula. These results contradict some earlier studies that report no statistically significant or even negative correlations between returns of emission allowances and other financial variables. Secondly, considering time-varying copulas shows that the estimated copula parameters are not constant over time. We find in particular that the dependence is stronger during the period of the financial crisis. In a Value-at-Risk (VaR) analysis, finally, we further illustrate the advantages of copula methods. In particular the Student-t copula provides an appropriate quantification of VaR at different confidence levels while other models fail to specify the risk correctly. This analysis shows that ignoring the actual nature of dependence might lead to an underestimation of the risk for portfolios combining EUAs with commodities or equity investments.

Tails of correlation mixtures of elliptical copulas

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is generally larger than in the limit, which can have serious consequences for estimation and evaluation of extreme risk. Finally, although correlation mixtures of Gaussian copulas inherit the property of asymptotic independence, at the same time they fall in the newly defined category of near asymptotic dependence. The consequences of these findings for modeling are assessed by means of a simulation study and a case study involving financial time series.