Modelling default contagion using multivariate phase-type distributions

We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with ten obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered default times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase-type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.     

Utility valuation of multi-name credit derivatives and application to CDOs

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyse resulting yield spreads in multi-name credit derivatives, particularly CDOs. We study first the idealized problem with constant intensities where solutions are essentially explicit. We also give the large portfolio asymptotics for this problem. We then analyse the case where the firms have stochastic default intensities driven by a common factor, which can be viewed as another extreme from the independent case. This involves the numerical solution of a system of reaction-diffusion PDEs. We observe that the nonlinearity of the utility-indifference valuation mechanism enhances the effective correlation between the times of the credit events of the various firms leading to non-trivial senior tranche spreads, as often seen from market data.     

Discrete credit barrier models

The model introduced in this article is designed to provide a consistent representation for both the real-world and pricing measures for the credit process. We find that good agreement with historical and market data can be achieved across all credit ratings simultaneously. The model is characterized by an underlying stochastic process that takes on values on a discrete lattice and represents credit quality. Rating transitions are associated with barrier crossings and default events are associated with an absorbing state. The stochastic process has state-dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift is estimated to consistently match the average spread curves corresponding to all the various ratings.     

A bridge from ruin theory to credit risk

The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance 25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.      

Credit Derivatives in an Affine Framework

An efficient method for valuing credit derivatives based on three entities is developed in an affine framework. This includes interdependence of market and credit risk, joint credit migration and counterparty default risk of three firms. As an application we provide closed form expressions for the joint distribution of default times, default correlations, and default swap spreads in the presence of counterparty default risk. Vienna Institute of Finance is funded by WWTF (Vienna Science and Technology Fund).     

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Portfolio credit derivatives are contracts that are tied to an underlying portfolio of defaultable reference assets and have payoffs that depend on the default times of these assets. The hedging of credit derivatives involves the calculation of the sensitivity of the contract value with respect to changes in the credit spreads of the underlying assets, or, more generally, with respect to parameters of the default-time distributions. We derive and analyze Monte Carlo estimators of these sensitivities. The payoff of a credit derivative is often discontinuous in the underlying default times, and this complicates the accurate estimation of sensitivities. Discontinuities introduced by changes in one default time can be smoothed by taking conditional expectations given all other default times. We use this to derive estimators and to give conditions under which they are unbiased. We also give conditions under which an alternative likelihood ratio method estimator is unbiased. We illustrate the application and verification of these conditions and estimators in the particular case of the multifactor Gaussian copula model, but the methods are more generally applicable.      

基于会计信息的KMV模型实证研究

本文借助KMV模型框架,运用统计方法对大量的公司财务报表数据进行模型参数估计,计算得到了非上市公司的违约距离和经验EDF函数,实现了违约概率的模型估计。实证表明,我国公司在违约距离或违约数量上的真实概率分布均呈现显著的T分布和肥尾特性;违约距离具有较高的风险区分能力;由会计信息进行参数估计的模型导出的EDF具有较高的风险标识精度;进而表明基于会计报表数据的违约风险模型和基于资本市场数据的模型在实证上的有效性非常近似。     

An Empirical Investigation of CDS Spreads Using a Regime-Switching Default Risk Model

Default risk in equity returns can be measured by structural models of default. In this article we propose a credit warning signal (CWS) based on the Merton Default Risk (MDR) model and a Regime-Switching Default Risk (RSDR) model. The RSDR model is a generalization of the MDR model, comprises regime-switching asset distribution dynamics, and thus produces more realistic default probability estimates in cases of deteriorating credit quality. Alternatively, it reduces to the MDR model. Using a dataset of U.S. credit default swap (CDS) contracts around the 2007-8 crisis we construct rating-based indices to investigate the MDR and RSDR implied probabilities of default in relation to the market-observed CDS spreads. The proposed CWS measure indicates an increase in implied default probabilities several months ahead of notable increases in CDS spreads.     

Valuation of collateralized debt obligations with hierarchical Archimedean copulae

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on hierarchical Archimedean copulae (HAC) with up to three parameters, with default intensities calibrated to market data and with random loss given defaults that are correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution. We also obtain a flat correlation smile across tranches thereby solving the implied correlation puzzle.     

Unilateral Counterparty Risk Valuation for CDS Under a Regime Switching Interacting Intensities Model

We consider the credit valuation adjustment (CVA) of credit default swap under an interacting intensities model. The default intensities of the protection seller and the reference entity are both influenced by an external shock event. The arrival of the shock event is a regime switching Poisson process, which is a special case of Cox processes. We give the explicit formula for the CVA of the credit and examine the regime switching effect on the premium and the CVA.     

Sovereign credit spreads

The paper develops a structural credit risk model to study sovereign credit risk and the dynamics of sovereign credit spreads. The model features endogenous default and recovery rates that both depend on the interaction between domestic output fluctuations and global macroeconomic conditions. We show that sovereigns choose to default at higher levels of economic output once global macroeconomic conditions are bad. This yields to default rates and credit spreads that are substantially higher compared to normal times. We derive closed-form expressions for sovereign debt values and default times and focus on the dynamics of sovereign credit spreads. As opposed to standard theories of sovereign debt, this paper’s structural model generates much richer default patterns and non-linearities through regime-shifts in the global macroeconomic environment. Moreover, changes in the global environment reveal the interconnectedness of the financial system.     

Valuation of default-sensitive claims under imperfect information

We propose a valuation method for financial assets subject to default risk, where investors cannot observe the state variable triggering the default but observe a correlated price process. The model is sufficiently general to encompass a large class of structural models and can be seen as a generalization of the model of Duffie and Lando (Econometrica 69:633–664, [2001]). In this setting we prove that the default time is totally inaccessible in the market’s filtration and derive the conditional default probabilities and the intensity process. Finally, we provide pricing formulas for default-sensitive claims and illustrate in particular examples the shapes of the credit spreads.      

Modeling CDS spreads: A comparison of some hybrid approaches

According to the credit risk model proposed by Cathcart and El-Jahel (2006), default can occur either expectedly, when a certain signaling variable breaches a lower barrier, or unexpectedly, as the first jump of a Poisson process, whose intensity depends on the signaling variable itself and on the interest rate. In the present paper we test the performances of such a model and of other three models generalized by it in fitting the term structure of credit default swap (CDS) spreads. In order to do so, we derive a semi-analytical formula for pricing CDSs and we use it to fit the observed term structures of 65 different CDSs. The analysis reveals that all the model parameters yield a relevant contribution to credit spreads. Moreover, if the dependence of the default intensity on both the signaling variable and the interest rate is removed, the pricing of CDSs becomes very simple, from both the analytical and the computational standpoint, while the goodness-of-fit is reduced by only a few percentage points. Therefore, when using the credit risk model proposed by Cathcart and El-Jahel (2006), assuming a constant default intensity provides an interesting and efficient compromise between parsimony and goodness-of-fit. Furthermore, by fitting the term structure of CDS spreads on a period of about twelve years, we find that the parameters of the model with constant default are rather stable over time, and the goodness-of-fit is maintained high.     

A first-passage-time model under regime-switching market environment

In this paper, we suggest a first-passage-time model which can explain default probability and default correlation dynamics under stochastic market environment. We add a Markov regime-switching market condition to the first-passage-time model of Zhou [Zhou, C., 2001. An analysis of default correlations and multiple defaults. Review of Financial Studies 14, 555–576]. Using this model, we try to explain various relationship between default probability, default correlation, and market condition. We also suggest a valuation method for credit default swap (CDS) with (or without) counterparty default risk (CDR) and basket default swap under this model.Our numerical results provide us with several meaningful implications. First, default swap spread is higher in economic recession than in economic expansion across default swap maturity. Second, as the difference of asset return volatility between under bear market and under bull market increases, CDS spread increases regardless of maturity. Third, the bigger the intensity shifting from bull market to bear market, the higher the spread for both CDS without CDR and basket default swap.     

Valuation of a Credit Swap of the Basket Type

This article provides a simple model to value a credit swap ofthe basket type. Unlike the previous literature, we considerthe joint survival probability of occurrence times of creditevents in terms of stochastic intensity processes under the assumptionof conditional independence. Based on the joint survival probability,such a credit swap can be valued under the risk-neutral valuationframework. Assuming that the default intensity processes followthe extended Vasicek model with a correlation structure, an analyticexpression of the valuation formula is derived. Some numericalexample is given to demonstrate the usefulness of our model.     

Modeling the evolution of implied CDO correlations

CDO tranche spreads (and prices of related portfolio-credit derivatives) depend on the market’s perception of the future loss distribution of the underlying credit portfolio. Applying Sklar’s seminal decomposition to the distribution of the vector of default times, the portfolio-loss distribution derived thereof is specified through individual default probabilities and the dependence among obligors’ default times. Moreover, the loss severity, specified via obligors’ recovery rates, is an additional determinant. Several (specifically univariate) credit derivatives are primarily driven by individual default probabilities, allowing investments in (or hedging against) default risk. However, there is no derivative that allows separately trading (or hedging) default correlations; all products exposed to correlation risk are contemporaneously also exposed to default risk. Moreover, the abstract notion of dependence among the names in a credit portfolio is not directly observable from traded assets. Inverting the classical Vasicek/Gauss copula model for the correlation parameter allows constructing time series of implied (compound and base) correlations. Based on such time series, it is possible to identify observable variables that describe implied correlations in terms of a regression model. This provides an economic model of the time evolution of the market’s view of the dependence structure. Different regression models are developed and investigated for the European CDO market. Applications and extensions to other markets are discussed.     

Contagion models a la carte: which one to choose?

We propose a copula contagion mixture model for correlated default times. The model includes the well-known factor, copula, and contagion models as its special cases. The key advantage of such a model is that we can study the interaction of different models and their pricing impact. Specifically, we model the default times of the underlying names in a reference portfolio to follow contagion intensity processes with exponential decay coupled with a copula dependence structure. We also model the default time of the counterparty and its dependence structure with the reference portfolio. Numerical tests show that correlation and contagion have an enormous joint impact on the rates of CDO tranches and the corresponding credit value adjustments are extremely high to compensate for the wrong-way risk.     

Implied migration rates from credit barrier models

The risk-neutral credit migration process captures quantitative information which is relevant to the pricing theory and risk management of credit derivatives. In this article, we derive implied migration rates by means of a recently introduced credit barrier model which is calibrated on the basis of aggregate information such as credit migration rates and credit spread curves. The model is characterized by an underlying stochastic process that represents credit quality, and default events are associated to barrier crossings. The stochastic process has state dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift and forward liquidity spreads are estimated to consistently match the average spread curves corresponding to all the various ratings. The implied migration rates obtained with our credit barrier model are then compared with those obtained via the Kijima–Komoribayashi model.     

Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.     

Default risk premium and asset prices

We estimate a standard structural model of credit risk to draw insights about the premium demanded by investors for bearing default risk, using data on credit default swaps and market capitalization. We pin down the daily market value of assets for a set of non-financial firms and uncover cross-sectional heterogeneity in terms of the magnitude and time variation of the premium. By exploring the link between asset and default risk premia, we show that this heterogeneity closely depends on the relationship between the firm-specific market value of the assets and the business cycle.