Normal Copula Model Research Articles

A Time-Varying Multivariate Noncentral Contaminated Normal Copula Model and Its Application to the Visualized Dependence Analysis of Hong Kong Stock Markets

Financial data usually have the features of complexity and interdependence structure, such as asymmetric, tail, and time-varying dependence. This study constructs a new multivariate skewed fat-tailed copula, namely, noncentral contaminated normal (NCCN) copula, to analyze the dependent structure of financial market data. The dynamic conditional correlation (DCC) model is also incorporated into constructing the time-varying NCCN copula model. This study comprehensively examines the effects of the DCC-NCCN copula and related models on fitting dependence structures of Hong Kong stock markets. The results show that the DCC-NCCN copula model can better depict the dependence structures of returns. Considering the flexibility and complexity, the DCC-NCCN copula model is a relatively ideal, time-varying, multivariate skewed fat-tailed copula model.

Weighted scores estimating equations and CL1 information criteria for longitudinal ordinal response

Available extensions of generalized estimating equations for longitudinal ordinal response require a conversion of the ordinal response to a vector of binary category indicators. That leads to a rather complicated working correlation structure and to large matrices when the number of categories and dimension of the clusters are large. Weighted scores estimating equations are constructed to overcome the aforementioned problems. Similar to generalized estimating equations which construct unbiased equations weighting the residuals, the weighted scores weight the univariate score functions. To specify the weight matrices, the weighted scores estimating equations use a working dependence model, namely the multivariate normal (MVN) copula model with univariate ordinal probit or logit regressions as the marginals. There is no need to convert the ordinal response to binary indicators, thus the weight matrices have smaller dimensions. Composite likelihood information criteria are further proposed as an intermediate step for selecting both the covariates in the mean function modelling and the structure of the latent correlation matrix induced by the MVN latent variables. The weighted scores estimating equations and composite likelihood information criteria are illustrated by analysing a rheumatoid arthritis clinical trial. Our modelling framework is implemented in the package weightedScores within the open source statistical environment R.

Computation of credit portfolio loss distribution by a cross entropy method

Quantification and management of credit risk is always crucial for the financial industry. Computing credit risk is generally a challenging task while correlated defaults exist. Traditional approaches such as exponential twisting are model specific and often involve difficult analysis, therefore computational methods are sought to estimate the credit risk when analysis is unavailable. The accurate measurement of credit risk is often a rare-event simulation problem, i.e., calculating probabilities (which are usually small) of extreme losses. It is well-known that the Monte Carlo (MC) method may become slow and expensive for such problems. Importance sampling (IS), a variance-reduction technique, can then be utilized for rare-event simulation for credit risk management. In this work, we propose the implementation of a special IS procedure, the cross-entropy (CE) method, to simulate credit risk models. More specifically, we obtain iteratively biasing probability density functions (PDF’s) for credit portfolio losses by the CE method, and then combine the results from each stage by the technique of multiple importance sampling to obtain a complete PDF. The main advantage of this method is that it can avoid the nontrivial analysis required by a general IS method, and therefore simplifies the estimation of loss distributions. Moreover, this approach is generic and can be applied to a wide variety of models with little modifications. In particular, we apply this approach to a normal copula model and a t-copula model to estimate the probabilities of extreme portfolio losses under the models. Numerical examples are provided to demonstrate the performance of our method.

Conditionalization of Copula-Based Models

Copula models are becoming increasingly popular in engineering and financial applications. They provide a flexible way of constructing joint distributions with arbitrary one-dimensional margins and a wide variety of dependence structures. This paper studies different types of conditionalization of copula-based models. We conditionalize these models in the usual way on point values but also propose a different type of conditionalization on a new margin. This new type of conditionalization is motivated by experience with a model built for application in modeling risk in civil aviation. Changing one margin in a copula model is very easy; however, it is not equivalent to conditionalizing on a new margin. For this purpose, the technique presented by Holland and Wang [Holland PW, Wang YJ (1987) Dependence function for continuous bivariate densities. Commun. Statist. Theory Methods 16(3):867–876], called marginal replacement, is used. The main result of this paper allows a simplified way of conditionalizing on more than one univariate margin in a normal copula model.

FPGA acceleration of CDO pricing based on correlation expansions

Because of the significant growth in the financial market, faster and accurate pricing of widespread instruments is becoming more important. In this paper, we describe an FPGA implementation of an analytical method for collateralized debt obligation (CDO) pricing in the multifactor Normal Copula model. Our experiments show that the FPGA system is about 40 times faster than corresponding software on a single core 3 GHz Intel Core2 processor.

A normal copula model for the arrival process in a call center

AbstractWe propose and examine a probabilistic model for the multivariate distribution of the number of calls in each period of the day (e.g., 15 or 30 min) in a call center, where the marginal distribution of the number of calls in any given period is arbitrary, and the dependence between the periods is modeled via a normal copula. Conditional on the number of calls in a period, their arrival times are independent and uniformly distributed over the period. This type of model has the advantage of being simple and reasonably flexible, and can match the correlations between the arrival counts in different periods much better than previously proposed models. For the situation where the number of periods is large, so the number of correlations to estimate can be excessive, we propose simple parametric forms for the correlations, defined as functions of the time lag between the periods. We test our proposed models on three data sets taken from real‐life call centers and compare their goodness of fit to the best previously proposed methods that we know. In the three cases, the new models provide a much better match of the correlations and coefficients of variation of the arrival counts in individual periods.

GRAPHICAL MODELS FOR CORRELATED DEFAULTS

A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for default dependence: it represents any given marginal distribution for single firms and pairwise correlation matrix. These techniques also provide a calibration algorithm based on maximum likelihood estimation. Finally, the model is compared with standard normal copula model in terms of tails of the loss distribution and implied correlation smile.

Use of observed scaled daily storm profiles in a copula based rainfall disaggregation model

It has been demonstrated in this paper that observed scaled daily storm profiles (OSDSPs) can be used at a different climatic site to disaggregate daily rainfall data into fine timescales by following some selection rules. For a given daily rainfall depth, the normal copula model predicts the total daily wet periods’ duration, capturing the seasonal variation of the copula and marginal lognormal model parameters by a first harmonic Fourier series. An OSDSP type, defined according to whether the first and/or last periods of the day are wet, is selected based on whether the day is an isolated wet day or in a cluster of wet days. For two consecutive wet Julian days, a transition probability is used to ascertain whether rainfall will cross midnight and spread to the next Julian day. A pool of the selected OSDSP type having the same, or close, numerical value of the copula predicted total daily wet periods’ duration is created. From the pool is selected one OSDSP that has the closest value of lag-1 autocorrelation of the natural logarithm of the wet period depths to a simulated value conditioned on the total daily wet periods’ duration. Multiplication of the OSDSP’s period depths by the daily rainfall depth gives the complete daily storm profile. The applicability of the OSDSP based disaggregation model is demonstrated by using two case study sites and OSDSPs from Australian State capital cities situated in different climates. Using the capital cities’ OSDSPs independently, and also together, predicted quite well the monthly gross rainfall statistics down to 6min.

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

Reliable risk measurement is a key problem for financial institutions and regulatory authorities. The current industry standard Value-at-Risk has several deficiencies. Improved risk measures have been suggested and analyzed in the recent literature, but their computational implementation has largely been neglected so far. We propose and investigate stochastic approximation algorithms for the convex risk measure Utility-Based Shortfall Risk. Our approach combines stochastic root-finding schemes with importance sampling. We prove that the resulting Shortfall Risk estimators are consistent and asymptotically normal, and provide formulas for confidence intervals. The performance of the proposed algorithms is tested numerically. We finally apply our techniques to the Normal Copula Model, which is also known as the industry model CreditMetrics. This provides guidance for future implementations in practice.

Rare-event probability estimation with conditional Monte Carlo

Estimation of rare-event probabilities in high-dimensional settings via importance sampling is a difficult problem due to the degeneracy of the likelihood ratio. In fact, it is generally recommended that Monte Carlo estimators involving likelihood ratios should not be used in such settings. In view of this, we develop efficient algorithms based on conditional Monte Carlo to estimate rare-event probabilities in situations where the degeneracy problem is expected to be severe. By utilizing an asymptotic description of how the rare event occurs, we derive algorithms that involve generating random variables only from the nominal distributions, thus avoiding any likelihood ratio. We consider two settings that occur frequently in applied probability: systems involving bottleneck elements and models involving heavy-tailed random variables. We first consider the problem of estimating ℙ(X 1+⋅⋅⋅+X n >γ), where X 1,…,X n are independent but not identically distributed (ind) heavy-tailed random variables. Guided by insights obtained from this model, we then study a variety of more general settings. Specifically, we consider a complex bridge network and a generalization of the widely popular normal copula model used in managing portfolio credit risk, both of which involve hundreds of random variables. We show that the same conditioning idea, guided by an asymptotic description of the way in which the rare event happens, can be used to derive estimators that outperform existing ones.

Portfolio Credit Risk with Extremal Dependence: Asymptotic Analysis and Efficient Simulation

We consider the risk of a portfolio comprising loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in performance measures such as the probability that the portfolio incurs large losses over a fixed time horizon, and the expected excess loss given that large losses are incurred during this horizon. Contrary to the normal copula that is commonly used in practice (e.g., in the CreditMetrics system), we assume a portfolio dependence structure that is semiparametric, does not hinge solely on correlation, and supports extremal dependence among obligors. A particular instance within the proposed class of models is the so-called t-copula model that is derived from the multivariate Student t distribution and hence generalizes the normal copula model. The size of the portfolio, the heterogeneous mix of obligors, and the fact that default events are rare and mutually dependent make it quite complicated to calculate portfolio credit risk either by means of exact analysis or naïve Monte Carlo simulation. The main contributions of this paper are twofold. We first derive sharp asymptotics for portfolio credit risk that illustrate the implications of extremal dependence among obligors. Using this as a stepping stone, we develop importance-sampling algorithms that are shown to be asymptotically optimal and can be used to efficiently compute portfolio credit risk via Monte Carlo simulation.

A Normal Copula Model for the Economic Risk Analysis of Correlated Failures in Communications Networks

The reliability of a communications network is often evaluated without taking into account the economic consequence of failures. Here a new approach is proposed to assess the economic consequences of failures as a figure of merit of reliable networks. For this purpose a partition of the network operator’s market into service basins is proposed, which includes the presence of correlation between the subsystems needed to serve different service basins as well as within the same service basin. A simulation algorithm, based on the Cross-Entropy method, is fully described to evaluate the probability that the economic loss exceeds a given threshold. An application of the method to a simple scenario is finally reported.

Correlation expansions for CDO pricing

This paper develops numerical approximations for pricing collateralized debt obligations (CDOs) and other portfolio credit derivatives in the multifactor Normal Copula model. A key aspect of pricing portfolio credit derivatives is capturing dependence between the defaults of the elements of the portfolio. But, compared with an independent-obligor model, pricing in a model with correlated defaults is more challenging. Our approach strikes a balance by reducing the problem of pricing in a model with correlated defaults to calculations involving only independent defaults. We develop approximations based on power series expansions in a parameter that scales the underlying correlations. These expansions express a CDO tranche price in a multifactor model as a series of prices in independent-obligor models, which are easy to compute. The approach builds on a classical approximation for multivariate Gaussian probabilities; we introduce an alternative representation that greatly reduces the number of terms required to evaluate the coefficients in the expansion. We also apply this method to the underlying problem of computing joint probabilities of multivariate normal random variables for which the correlation matrix has a factor structure.

Importance Sampling for Portfolio Credit Risk

Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and other instruments subject to possible default. The accurate measurement of credit risk is often a rare-event simulation problem because default probabilities are low for highly rated obligors and because risk management is particularly concerned with rare but significant losses resulting from a large number of defaults. This makes importance sampling (IS) potentially attractive. But the application of IS is complicated by the mechanisms used to model dependence between obligors, and capturing this dependence is essential to a portfolio view of credit risk. This paper provides an IS procedure for the widely used normal copula model of portfolio credit risk. The procedure has two parts: One applies IS conditional on a set of common factors affecting multiple obligors, the other applies IS to the factors themselves. The relative importance of the two parts of the procedure is determined by the strength of the dependence between obligors. We provide both theoretical and numerical support for the method.

Portfolio Credit Risk with Extremal Dependence

We consider the risk of a portfolio comprised of loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in the probability that the portfolio will incur large losses over a fixed time horizon. Contrary to the normal copula that is commonly used in practice (e.g., in the CreditMetrics system), we assume a portfolio dependence structure that supports {\it extremal dependence} among obligors and does not hinge solely on correlation. A particular instance within this model class is the so-called $t$-copula model that is derived from the multivariate Student $t$ distribution and hence, generalizes the normal copula model. The size of the portfolio, the heterogenous mix of obligors, and the fact that default events are rare and mutually dependent makes it quite complicated to calculate portfolio credit risk either by means of exact analysis or naive Monte Carlo simulation. The main contributions of this paper are twofold. We first derive sharp asymptotics for portfolio credit risk that illustrate the implications of extremal dependence among obligors. Using this as a stepping stone, we develop multi-stage importance sampling algorithms that are shown to be asymptotically optimal and can be used to efficiently compute portfolio credit risk via Monte Carlo simulation.

Modeling Default Dependence with Threshold Models

Default risk is one of the most important and fastest growing areas in derivatives. Single-name credit default swaps are now well established, and markets for contracts based on the default experience of a credit portfolio, like collateralized debt obligations (CDOs) and basket default swaps, are developing rapidly. Portfolio credit risk requires default correlation as a critical input. But defaults are rare, so default correlation must be estimated in another way, typically from correlations in stock returns. In a „structural” credit risk model, a credit event occurs when the net asset value of the firm falls below a critical threshold. Transitions among credit rating classes can be modeled with a set of thresholds corresponding to the different ratings. The widely used „normal copula” model treats the default probability for a single issuer, and default correlations among pairs of issuers, as arising from exposure to a set of common stochastic risk factors plus an idiosyncratic term. The resulting models are intuitive, but calibration to (implied) correlations embedded in market CDO prices can be difficult, especially as the number of factors increases. In this article, Overbeck and Schmidt offer an alternative structural approach that produces similar values to the normal copula model, with considerably less difficulty. Their trick is to alter the time scales of the Wiener processes driving the underlying asset values, which is equivalent to allowing time-varying volatilities, and permits a much easier calibration of the correlations.

Efficient Estimation in the Bivariate Normal Copula Model: Normal Margins Are Least Favourable

Consider semi-parametric bivariate copula models in which the family of copula functions is parametrized by a Euclidean parameter 0 of interest and in which the two unknown marginal distributions are the (infinite-dimensional) nuisance parameters. The efficient score for 6' can be characterized in terms of the solutions of two coupled Sturm-Liouville equations. Where the family of copula functions corresponds to the normal distributions with mean 0, variance 1 and correlation 0, the solution of these equations is given, and we thereby show that the normal scores rank correlation coefficient is asymptotically efficient. We also show that the bivariate normal model with equal variances constitutes the least favourable parametric submodel. Finally, we discuss the interpretation of 6' 1 in the normal copula model as the maximum (monotone) correlation coefficient.