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Some dynamical contagion models for default risk have been proposed in the literature, where a system (composed of individual debtors) evolves as a Markov process conditionally on the observation of its stochastic environment, with interacting intensities.The Markovian assumption necessitates that the environment evolves autonomously and is not influenced by the transitions of the system. We extend this classical literature and allow a default system to have a contagious impact on its environment. With a certain probability, the transition of a debtor to the default state has an impact on the system's environment. This in turn affects the transition intensities of the other debtors inside the system.Therefore, in our framework, contagion can either be contained within the default system (i.e., direct contagion from a counterparty to another) or spill from the default system over its environment (indirect contagion). This type of model is of interest whenever one wants to capture within a model possible impacts of the defaults of a class of debtors on the more global economy and vice versa. [ABSTRACT FROM AUTHOR]

In this paper we introduce a sublinear conditional operator with respect to a family of possibly nondominated probability measures in presence of multiple ordered default times. In this way we generalize the results of Biagini and Zhang (2019), where a reduced-form framework under model uncertainty for a single default time is developed. Moreover, we use this operator for the valuation of credit portfolio derivatives under model uncertainty. [ABSTRACT FROM AUTHOR]

We define a disastrous default as the default of a systemic entity. Such an event is expected to have a negative effect on the economy and to be contagious. Bringing macroeconomic structure to a no-arbitrage asset-pricing framework, we exploit prices of disaster-exposed assets (credit and equity derivatives) to extract information on (i) the expected influence of a disastrous default on consumption and (ii) the probability of a financial meltdown. Using European data, we find that the returns of disaster-exposed assets are consistent with a systemic default being followed by a 2% decrease in consumption. The recessionary influence of disastrous defaults implies that financial instruments whose payoffs are exposed to such credit events carry substantial risk premiums. We also produce systemic risk indicators based on the probability of observing a certain number of systemic defaults or a sharp drop of consumption. [ABSTRACT FROM AUTHOR]

A critical aspect in the valuation and risk management of multi-name credit derivatives is the modeling of the dependence among sources of credit risk. The dependence modeling poses difficulties in the pricing of a multi-name credit derivatives, in the estimation of the value-at-risk of a portfolio, or in the pricing of some other basket credit derivative as the description not only on the default arrival in an individual reference entity but on the default dependence among entities in the portfolio should be considered. Although the elliptical models have been widely used due to their mathematical tractability, the dependence modeling using the multi-dimensional Lévy process has shown growing interest among researchers despite its complexity. In this paper, we introduce one factor copula model for portfolio credit risk based on Normal Tempered Stable (NTS) distribution and calibrate the model through 5-year synthetic Collateralized Debt Obligation (CDO) tranche spreads under a large homogeneous portfolio approximation. The calibration results show that the one factor copula model based on NTS distribution is more flexible and provides a dependence structure fitting market CDO tranche spreads. As one of the major applications of the dependence modeling in credit risk, this model shares the advantage of the Gaussian one factor model, and all extensions and implementation methods used for it can be utilized. [ABSTRACT FROM AUTHOR]

For fixed income investment, the preponderant risk is the clustering of defaults in the portfolio. Accurate prediction of such clustering depends on the knowledge of default correlation. We develop models with exogenous debt and endogenous debt to predict default correlations from equity correlations based on a self-consistent structural framework. We also examine how taxes affect the prediction of default correlations based on the two models. The empirical analysis shows that the corporate taxes tend to decrease default correlations, while personal taxes could increase or decrease default correlations. Our default correlation model with exogenous debt does a better job of predicting default correlations for high quality bonds, while the one with endogenous debt predicts more accurately for lower rated bonds. Our studies not only theoretically improve the modeling of default correlation in the structural setting but also shed new light on various aspects of default correlations and thereby help financial practitioners price credit derivatives more accurately and formulate more effective strategies to manage default risk of credit portfolios. [ABSTRACT FROM AUTHOR]

We establish Markovian models in the Heath, Jarrow, and Morton (1992) paradigm that permit an exponential affine representation of riskless and risky bond prices while offering significant flexibility in the choice of volatility structures. Estimating models in our family is typically no more difficult than in the workhorse affine family. Besides diffusive and jump-induced default correlations, defaults can impact the credit spreads of surviving firms, allowing for a greater clustering of defaults. Numerical implementations highlight the importance of incorporating interest rate–credit spread correlations, credit spread impact factors, and the full credit spread curve when building a unified framework for pricing credit derivatives. [ABSTRACT FROM PUBLISHER]

Standard credit risk models cannot explain the observed clustering of default, sometimes described as “credit contagion.” This paper provides the first empirical analysis of credit contagion via direct counterparty effects. We find that bankruptcy announcements cause negative abnormal equity returns and increases in CDS spreads for creditors. In addition, creditors with large exposures are more likely to suffer from financial distress later. This suggests that counterparty risk is a potential additional channel of credit contagion. Indeed, the fear of counterparty defaults among financial institutions explains the sudden worsening of the credit crisis after the Lehman bankruptcy in September 2008. [ABSTRACT FROM AUTHOR]

The probability of extreme default losses on portfolios of U.S. corporate debt is much greater than would be estimated under the standard assumption that default correlation arises only from exposure to observable risk factors. At the high confidence levels at which bank loan portfolio and collateralized debt obligation (CDO) default losses are typically measured for economic capital and rating purposes, conventionally based loss estimates are downward biased by a full order of magnitude on test portfolios. Our estimates are based on U.S. public nonfinancial firms between 1979 and 2004. We find strong evidence for the presence of common latent factors, even when controlling for observable factors that provide the most accurate available model of firm-by-firm default probabilities. [ABSTRACT FROM AUTHOR]

Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economy-wide impact, this paper generalizes existing reduced-form models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firm-specific risks that are termed "counterparty risks." Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps. [ABSTRACT FROM AUTHOR]

Recent empirical evidences indicate that default rates are influenced not only by the observable or latent risk factors, but also depend on the history of past defaults. Motivated by this empirical finding, we consider in this paper a reduced-form, intensity-based credit risk model, which allows for both frailty and default contagion, using a so-called 'self-exciting' intensity, in the sense that the default intensity varies not only with the risk factors, but also depends on the previous default history of all the firms. With 'self-exciting' default intensity, we are able to obtain closed-form expressions for the pricing of credit derivative securities in our model. The estimation of parameters using the EM algorithm is considered as well. [ABSTRACT FROM AUTHOR]

We propose a factor contagion model with the Marshall–Olkin copula for correlated default times and develop an analytic approach for finding the th default time distribution based on our model. We combine a factor copula model with a contagion model under the assumption that the individual default intensities follow contagion processes, and that the default times have a dependence structure with the Marshall–Olkin copula. Then, we derive an analytic formula for the th default time distribution and apply it to compute the price of portfolio credit derivatives, such as th-to-default swaps and single-tranche CDOs. To test efficiency and accuracy of our formula, we compare the theoretical prediction with existing methods. [ABSTRACT FROM AUTHOR]

The global financial crisis has shown that many financial institutions dealing with credit derivatives were exposed to severe unexpected losses. This indicates that systematic influences are decisively underestimated particularly with regard to structured products like securitized tranches of collateralized debt obligations. Our analytical study addresses these systematic effects: We provide a simple model which allows a closed-form comparison of both bonds and tranches with respect to their systematic risk. We demonstrate that the exposure to systematic risk of tranches may be many times higher than the exposure of bonds, even if both products share the same rating grade, e.g., an 'AAA' rating, measured by either default probability or expected loss. Particularly in economic downturns, default rates of tranches may be multiples of those of bonds. Our results help understand high default rates of tranches during the financial crisis and show that classical ratings are insufficient metrics for measuring risks of structured products. [ABSTRACT FROM AUTHOR]

The article focuses on the estimation of the probability of default (PD) as an important parameter in risk management, request for loans, rating estimation, pricing of credit derivatives and other key financial fields. It estimates the PD of U.S. banks using statistical models. Linear discriminant analysis and regression models are the two main categories of credit scoring models. About 300 U.S. commercial banks which were separated into non-default and default are examined.

Numerous incidents in the financial world have exposed the need for the design and analysis of models for correlated default timings. Some models have been studied in this regard which can capture the feedback in case of a major credit event. We extend the research in the same direction by proposing a new family of models having the feedback phenomena and capturing the effects of regime switching economy on the market. The regime switching economy is modeled by a continuous time Markov chain. The Markov chain may also be interpreted to represent the credit rating of the firm whose bond we seek to price. We model the default intensity in a pool of firms using the Markov chain and a risk factor process. We price some single-name and multi-name credit derivatives in terms of certain transforms of the default and loss processes. These transforms can be calculated explicitly in case the default intensity is modeled as a linear function of a conditionally affine jump diffusion process. In such a case, under suitable technical conditions, the price of credit derivatives are obtained as solutions to a system of ODEs with weak coupling, subject to appropriate terminal conditions. Solving the system of ODEs numerically, we analyze the credit derivative spreads and compare their behavior with the nonswitching counterparts. We show that our model can easily incorporate the effects of business cycle. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low floating interest rate, high default intensity rate, and high volatility. We also model the effects of firm restructuring on the credit spread, in case of a default. [ABSTRACT FROM AUTHOR]

This paper proposes a model for pricing credit derivatives in a defaultable HJM framework. The model features hump-shaped, level dependent, and unspanned stochastic volatility, and accommodates a correlation structure between the stochastic volatility, the default-free interest rates, and the credit spreads. The model is finite-dimensional, and leads (a) to exponentially affine default-free and defaultable bond prices, and (b) to an approximation for pricing credit default swaps and swaptions in terms of defaultable bond prices with varying maturities. A numerical study demonstrates that the model captures stylized various features of credit default swaps and swaptions. [ABSTRACT FROM AUTHOR]

The financial crisis set off by the default of Lehman Brothers in 2008 leading to disastrous consequences for the global economy has focused attention on regulation and pricing issues related to credit derivatives. Credit risk refers to the potential losses that can arise due to the changes in the credit quality of financial instruments. These changes could be due to changes in the ratings, market price (spread) or default on contractual obligations. Credit derivatives are financial instruments designed to mitigate the adverse impact that may arise due to credit risks. However, they also allow the investors to take up purely speculative positions. In this article we provide a succinct introduction to the notions of credit risk, the credit derivatives market and describe some of the important credit derivative products. There are two approaches to pricing credit derivatives, namely the structural and the reduced form or intensity-based models. A crucial aspect of the modelling that we touch upon briefly in this article is the problem of calibration of these models. We hope to convey through this article the challenges that are inherent in credit risk modelling, the elegant mathematics and concepts that underlie some of the models and the importance of understanding the limitations of the models. [ABSTRACT FROM AUTHOR]

Up to the 2007 crisis, research within bottom-up CDO models mainly concentrated on the dependence between defaults. Since then, due to substantial increases in market prices of systemic credit risk protection, more attention has been paid to recovery rate assumptions. In this paper, we use stochastic orders theory to assess the impact of recovery on CDOs and show that, in a factor copula framework, a decrease of recovery rates leads to an increase of the expected loss on senior tranches, even though the expected loss on the portfolio is kept fixed. This result applies to a wide range of latent factor models and is not specific to the Gaussian copula model. We then suggest introducing stochastic recovery rates in such a way that the conditional on the factor expected loss (or, equivalently, the large portfolio approximation) is the same as in the recovery markdown case. However, granular portfolios behave differently. We show that a markdown is associated with riskier portfolios than when using the stochastic recovery rate framework. As a consequence, the expected loss on a senior tranche is larger in the former case, whatever the attachment point. We also deal with implementation and numerical issues related to the pricing of CDOs within the stochastic recovery rate framework. Due to differences across names regarding the conditional (on the factor) losses given default, the standard recursion approach becomes problematic. We suggest approximating the conditional on the factor loss distributions, through expansions around some base distribution. Finally, we show that the independence and comonotonic cases provide some easy to compute bounds on expected losses of senior or equity tranches. [ABSTRACT FROM PUBLISHER]

Systemic default risk--that is, the risk of the simultaneous default of multiple institutions--has caused great concern in the recent past. The aim of this article is to estimate the joint probability of default for multiple financial institutions. Both bond and credit derivative markets convey information on the default process: The former provides information on the marginal, the latter, on the joint default probabilities. We consider the corporate bond and the credit default swap (CDS) markets. The over-the-counter nature of the CDS market implies the presence of counterparty risk, or the risk that the protection seller will fail to fulfill its obligations. The counterparty risk is reflected in the CDS price through the joint default probability of the reference entity and the protection seller. Applying a no-arbitrage argument, we extract from market data forward-looking joint default probabilities of financial institutions operating in the CDS market from January 3, 2005-Mar 15, 2010. [ABSTRACT FROM AUTHOR]

The increasing popularity of credit default swaps (CDSs) necessitates understanding their various features. In this study, we analyse the capability of CDSs in predicting CDS prices of other companies in the same risk class or CDS prices of further time horizons. In doing so, we employ the basic forms of structural (the Merton model) and reduced-form (constant intensity) models in a cross-sectional and a time series setup. By utilizing a credit default swap dataset exclusively for estimation and out-of-sample prediction, our study also serves as a comparison between the basic forms of credit risk models. Finally, it contrasts the results with the performance of a new supervised learning forecasting technique, the Support Vector Machines Regression. We show that although the Merton and the constant intensity models handle default timing and interest rates differently, the prediction performance in cross-sectional and time series analyses is, on average, similar. In one-, five-, and 10-step-ahead predictions of time series, the machine learning algorithm significantly outperforms financial models. [ABSTRACT FROM AUTHOR]

In this study, we consider short interest rate models of the Vascicek type with stochastic volatility and obtain formulas for default risk in interest rate derivatives. Corrections from a fast mean-reverting stochastic volatility are computed to show how they can affect the term structure of the interest rate derivatives. Our results for the defaultable bonds as well as the corresponding bond options are obtained as an extension of the non-defaultable case studied by Cotton et al. [Math. Finance, 2004, 14(2), 173–200]. [ABSTRACT FROM AUTHOR]

We propose an alternative method for finding the kth default time distribution in a homogeneous portfolio with dependency. Analysing order statistics of default times with a one-factor Gaussian copula model, we explicitly derive the probability distribution. Moreover, we compute the prices of basket default swaps such as the kth to default swaps and m out of n default swaps within our framework. To test the efficiency and accuracy of our method we compare the theoretical prediction with existing methods. [ABSTRACT FROM AUTHOR]

This paper presents a modified version of Leland and Toft's [J. Finance, 1996, 51, 987–1019] structural credit risk model, together with a novel calibration methodology based on stock and CDS data: the firm asset value and volatility are consistently derived from equity prices; the default barrier is calibrated from CDS premia. It empirically shows that as long as the appropriate default barrier is selected, the model generates time series of stock market implied credit spreads that fit the times series of CDS spreads. Moreover, CDS implied default barriers prove to be consistent with stockholders’ rationality, with predictions made by structural models with endogenous default, and with historical recovery rates. [ABSTRACT FROM AUTHOR]

For the valuation of a CDO (collateralized debt obligation), a standard approach in practice is to employ the Gaussian copula model of (Li, 7) [J. Fixed Income, 2000, 9, 43–54]. However, this model is limited in that its framework is completely static, failing to capture the dynamic evolution of the CDO. In general, portfolio credit derivatives are subject to two kinds of risk, a default event risk, when any underlying firm involved in the CDO fails to fulfill its obligations, and credit spread risk, due to the change of the default intensity over time. In dealing with either type of risk, it is absolutely necessary to develop a dynamic model incorporating the stochastic behavior of the macro-economic conditions and their influence on the default intensity. In this paper, a dynamic stochastic model is developed where the macro-economic conditions are assumed to follow a birth–death process, which would affect loss distributions characterized by a Markov-modulated Poisson process (MMPP). By exploiting the stochastic structure of the MMPP, efficient computational procedures are established for evaluating time-dependent loss distributions and prices of the CDO. Numerical results are presented, demonstrating the potential usefulness of the model by estimating the underlying parameters based on real market data. [ABSTRACT FROM AUTHOR]

The need for an accurate representation of tail risk has become increasingly acute in the wake of the credit crisis. We introduce a hyper-cuboid normal mixture copula that permits the representation of complex tail-dependence structures in a multi-dimensional setting. We outline an efficient pattern-recognition calibration methodology that can identify tail dependencies independent of the number of risk factors considered. This model is used to develop a new framework for pricing credit derivative instruments, and we derive semi-analytical and analytical pricing formulae for a first-to-default swap and illustrate with an example valuation. Model assumptions are validated against iTraxx Series 5 equity data over an 8-year period. Identification and representation of tail dependencies is crucial to further the study of contagion dynamics, and our model provides a basis for future research in this area. [ABSTRACT FROM AUTHOR]

Equity default swaps (EDS) are hybrid credit-equity products that provide a bridge from credit default swaps (CDS) to equity derivatives with barriers. This paper develops an analytical solution to the EDS pricing problem under the jump-to-default extended constant elasticity of variance model (JDCEV) of Carr and Linetsky. Mathematically, we obtain an analytical solution to the first passage time problem for the JDCEV diffusion process with killing. In particular, we obtain analytical results for the present values of the protection payoff at the triggering event, periodic premium payments up to the triggering event, and the interest accrued from the previous periodic premium payment up to the triggering event, and we determine arbitrage-free equity default swap rates and compare them with CDS rates. Generally, the EDS rate is strictly greater than the corresponding CDS rate. However, when the triggering barrier is set to be a low percentage of the initial stock price and the volatility of the underlying firm's stock price is moderate, the EDS and CDS rates are quite close. Given the current movement to list CDS contracts on organized derivatives exchanges to alleviate the problems with the counterparty risk and the opacity of over-the-counter CDS trading, we argue that EDS contracts with low triggering barriers may prove to be an interesting alternative to CDS contracts, offering some advantages due to the unambiguity, and transparency of the triggering event based on the observable stock price. [ABSTRACT FROM AUTHOR]

We study the pricing of credit derivatives with asymmetric information. The managers have complete information on the value process of the firm and on the default threshold, while the investors on the market have only partial observations, especially about the default threshold. Different information structures are distinguished using the framework of enlargement of filtrations. We specify risk neutral probabilities and we evaluate default sensitive contingent claims in these cases. [ABSTRACT FROM AUTHOR]

We present a methodology for valuing portfolio credit derivatives under a reduced form model for which the default intensity processes of risk assets follow the one-factor Vasicek model. A closed-form solution of joint survival time distribution is obtained. The solution is applied to value credit derivatives of a credit default swap index and collateralized debt obligation. The limitation of methods using the Vasicek model is discussed. We propose that the method is valid and efficient for a portfolio with small-scale correlated risk assets, for which the acceptable size is much greater than for the traditional method. Numerical examples and parameter analysis are also presented. [ABSTRACT FROM AUTHOR]

Under Basel II framework, credit risk assessment is of high significance in the light of correlation risk. Correlation risk is often envisioned along with business conditions and financial market’s impact. We employ copula methodology to identify the dependence structures that may exist between market risk fundamentals and credit risk fundamentals. Considering credit derivative spreads as credit risk fundamentals and market data as market risk determinants, we describe and quantify the asymmetric link prevailing between credit risk and market risk. Credit risk is negatively linked with market price risk whereas it becomes positively linked with market volatility risk. Such patterns give rise to interesting asymmetric dependence structures between both risk sources. We are then able to balance reliably market price risk with market volatility feedback, the market trend supporting a common correlation between securities. In the light of the previous trade-off, we propose also a simple credit risk management rule. [ABSTRACT FROM AUTHOR]

Contrary to the traditional pools of financial instruments such as bonds and loans, the pools of credit derivatives that back synthetic CDOs include credit derivatives, credit default swaps, future contracts and options. This paper extends Gaussian factor copula modelling by using a stochastic hazard rate. The process of capturing instantaneous credit spread dynamics can then be investigated by a stochastic intensity model. CDO-Squared (CDO²) denotes CDOs backed primarily by securities issued by other CDO vehicles. This paper also uses a probability bucketing technique for evaluating CDO². A practical illustration is provided to price fair credit spreads for various tranches. The findings indicate that the level of risk for portfolio management are improved by a greater credit mean-reverting speed, which will further enhance the ability of the internal control by obligors. [ABSTRACT FROM AUTHOR]

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Credit derivatives have been popular instruments for hedging of credit risks by banks and financial institutions. The notional value outstanding of credit default swap contracts, a type of credit derivative most in use, increased from US$6.4 trillion in December 2004 to US$57.89 trillion in December, 2007. However, this instrument, which was once 'apple of the eye' of market players, lost its sheen in the wake of the sub-prime crisis when it was perceived to have played a major role in igniting the crisis and spreading it across the global financial system. This article presents how this came about and the after thought of the regulators of developed countries in regulating these instruments. It then looks at what lessons India can draw from the experience of the Western nations before considering introduction of credit derivatives in the Indian markets. [ABSTRACT FROM AUTHOR]