A reduced-form model with default intensities containing contagion and regime-switching Vasicek processes

The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps (CDSs). Numerical results are presented to show the impacts of the model parameters on the fair spreads.     

利用传染模型对含有对手风险的信用证券定价(英文)

本文利用传染模型研究了可违约债券和含有对手风险的信用违约互换的定价。我们在约化模型中引入具有违约相关性的传染模型,该模型假设违约过程的强度依赖于由随机微分方程驱动的随机利率过程和交易对手的违约过程.本文模型可视为Jarrow和Yu(2001)及Hao和Ye(2011)中模型的推广.进一步地,我们利用随机指数的性质导出了可违约债券和含有对手风险的信用违约互换的定价公式并进行了数值分析.     

Bilateral Counterparty Risk Valuation on a CDS with a Common Shock Model

In this paper, we study the counterparty risk on a CDS in a common shock model. We introduce the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk. Especially, we consider the pricing problem of credit default swap with counterparty risk under a common shock model with regime switching. The arrivals of the shock events are modeled by conditionally independent Cox processes whose stochastic intensities depend on the state of the economy described by a Markov chain. We give the explicit formula for the credit valuation adjustment (CVA) and examine the impact of the change of economic state on the CVA.     

农村信用社信贷违约识别模型及其应用

构建农村信用社信用风险模型对完善农村金融风险管理体系、提高农村信用社经营管理意义重大.基于还款意愿和还款能力两方面,系统分析了影响农信社贷款债务人违约率的主要因素,在此基础上应用logistic方法建立农信社债务人违约率预测模型,并通过Gini系数对模型区分能力和识别能力进行验证评估.实证结果表明,模型中债务人年龄、所在地区、贷款额所占家庭收入比例、与信用社信贷关系密切程度以及户口状况等因素都表现显著;违约率预测模型在样本内和样本外均有较好的违约识别能力,从而可为农信社放贷前的债务人信用评估、贷款发放和风险管理提供有力参考.     

Counterparty risk valuation on credit-linked notes under a Markov Chain framework

A credit-linked note(CLN) is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity. In this paper, we study the counterparty risk on CLNs under a Markov chain framework, and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer. Assuming that the respective default intensities are directly and inversely proportional to the interest rate, which follows a CIR process, we obtain the explicit formulae for CLN values through a PDE approach.Finally, credit valuation adjustment(CVA) formula is derived to price counterparty credit risk.     

Unilateral counterparty risk valuation of CDS using a regime-switching intensity model

We consider the unilateral credit valuation adjustment (CVA) of a credit default swap (CDS) under a contagion model with regime-switching interacting intensities. The model assumes that the interest rate, the recovery, and the default intensities of the protection seller and the reference entity are all influenced by macro-economy described by a homogeneous Markov chain. By using the idea of “change of measure” and some formulas for the Laplace transforms of the integrated intensity processes, we derive the semi-analytical formulas for the joint distribution of the default times and the unilateral CVA of a CDS.     

A Reduced-Form Model for Correlated Defaults with Regime-Switching Shot Noise Intensities

In this paper, we consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes that the intensities of the default times are driven by macro-economy described by a homogenous Markov chain and that the default of one firm may trigger a positive jump, associated with the state of Markov chain, in the default intensity of the other firm. The intensities before the default of the other firm are modeled by a two-dimensional regime-switching shot noise process with common shocks. By using the idea of “change of measure” and some closed-form formulas for the joint conditional Laplace transforms of the regime-switching shot noise processes and the integrated regime-switching shot noise processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we can express the single-name credit default swap (CDS) spread, the first and second-to-default CDS spreads on two underlyings in terms of fundamental matrix solutions of linear, matrix-valued, ordinary differential equations.     

A contagion model with Markov regime-switching intensities

We consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes the intensities of the default times are driven by macro-economy described by a homogeneous Markov chain as well as the other default. By using the idea of ＇change of measure＇ and some closed-form formulas for the Laplace transforms of the integrated intensity processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we give the explicit formulas for the fair spreads of the first-to-default and second-to-default credit default swaps （CDSs） on two underlyings.     

Credit risk model with contagious default dependencies affected by macro-economic condition

We consider a credit risk model with two industrial sectors, where defaults of corporations would be influenced by two factors. The first factor represents the macro economic condition which would affect the default intensities of the two industrial sectors differently. The second factor reflects the influences of the past defaults of corporations against other active corporations, where such influences would affect the two industrial sectors differently. A two-layer Markov chain model is developed, where the macro economic condition is described as a birth-death process, while another Markov chain represents the stochastic characteristics of defaults with default intensities dependent on the state of the birth-death process and the number of defaults in two sectors. Although the state space of the two-layer Markov chain is huge, the fundamental absorbing process with a reasonable state space size could capture the first passage time structure of the two-layer Markov chain, thereby enabling one to evaluate the joint probability of the number of defaults in two sectors via the uniformization procedure of Keilson. This in turn enables one to value a variety of derivatives defined on the underlying credit portfolios. In this paper, we focus on a financial product called CDO, and a related option.     

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Credit valuation adjustment is the price adjustment of financial contract considering possible default of counterparty and it is an important way to measure counterparty risk. It is the key to establish a reasonable default dependence structure model. We introduce an economic state variable and shot noise processes in a Markov copula model and establish a regime switching Markov copula model with shot noise, where we can not only describe the impact of common economic conditions characteristics but also describe the credit name's characteristic. In this proposed model, we study martingale property of the model and the collateralized CVA of credit default swaps, and furthermore, we perfer some numerical calculations on the collateralized CVA and examine the impact of some model parameters on the CVA.     

马氏调制强度的传染模型下CDS的CVA计算

本文考虑了具有马氏调制强度的传染模型下,信用违约互换(CDS)的双边信用估值调整(CVA).在我们考虑的模型中,利率、回收率以及CDS的买方、卖方和参照实体三方的违约强度均受宏观经济环境的影响,该经济状况由一连续时间状态的齐次马氏链所刻画.利用测度变换和累积强度的Laplace变换,我们给出了CDS合同的双边CVA的表达公式,该公式可以表示为线性常微分方程组的基本解的形式.利用所得到的公式,我们数值分析了马氏调制和违约相关性对双边CVA的影响.     

A Reduced Model with Thinning-Dependence Structure

The class of reduced form models is a very important class of credit risk models, and the modelling of the default dependence structure is essential in the reduced form models. This paper models dependent defaults under a thinning-dependent structure in the reduced form framework. In our tractable model, the joint survival probability for correlated defaults can be derived, and hence the CDS premium rates (with or without counterparty risk) are given in closed form. The numerical result shows that the thinning-dependent structure is effective to model the default dependence.     

A limit distribution of credit portfolio losses with low default probabilities

This paper employs a multivariate extreme value theory (EVT) approach to study the limit distribution of the loss of a general credit portfolio with low default probabilities. A latent variable model is employed to quantify the credit portfolio loss, where both heavy tails and tail dependence of the latent variables are realized via a multivariate regular variation (MRV) structure. An approximation formula to implement our main result numerically is obtained. Intensive simulation experiments are conducted, showing that this approximation formula is accurate for relatively small default probabilities, and that our approach is superior to a copula-based approach in reducing model risk.     

Modelling credit risk of portfolio of consumer loans

One of the issues that the Basel Accord highlighted was that, though techniques for estimating the probability of default and hence the credit risk of loans to individual consumers are well established, there were no models for the credit risk of portfolios of such loans. Motivated by the reduced form models for credit risk in corporate lending, we seek to exploit the obvious parallels between behavioural scores and the ratings ascribed to corporate bonds to build consumer-lending equivalents. We incorporate both consumer-specific ratings and macroeconomic factors in the framework of Cox Proportional Hazard models. Our results show that default intensities of consumers are significantly influenced by macro factors. Such models then can be used as the basis for simulation approaches to estimate the credit risk of portfolios of consumer loans.     

A Multivariate Regime-Switching Mean Reverting Process and Its Application to the Valuation of Credit Risk

In this article, we study the counterparty risk on a credit default swap (CDS) and the valuation of a first-to-default basket swap on three underlyings under a common shock model with regime-switching intensities. We assume that the defaults of all the names are driven by some shock events, whose arrivals are governed by a multivariate regime-switching shot noise process. Based on some expressions for the joint Laplace transform of the regime-switching shot noise processes, we give explicit formulas for the spread of the CDS contract with and without counterparty risk and the spread of the first-to-default basket swap on the three underlyings.     

Modeling credit portfolio derivatives,including both a default and a prepayment feature

Apart from heteronomy exit events such as, for example credit default or death, several financial agreements allow policy holders to voluntarily terminate the contract. Examples include callable mortgages or life insurance contracts. For the contractual counterpart, the result is a cash‐flow uncertainty called prepayment risk. Despite the high relevance of this implicit option, only few portfolio models consider both a default and a cancellability feature. On a portfolio level, this is especially critical because empirical observations of the mortgage market suggest that prepayment risk is an important determinant for the pricing of mortgage‐backed securities. Furthermore, defaults and prepayments tend to occur in clusters, and there is evidence for a negative association between the two risks. This paper presents a realistic and tractable portfolio model that takes into account these observations. Technically, we rely on an Archimedean dependence structure. A suitable parameterization allows to fit the likelihood of default and prepayment clusters separately and accounts for the postulated negative interdependence. Moreover, this structure turns out to be tractable enough for real‐time evaluation of portfolio derivatives. As an application, the pricing of loan credit default swaps, an example of a portfolio derivative that includes a cancellability feature, is discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

基于KMV模型的我国中小上市公司信用风险研究

经过提高股权价值波动率精度的KMV模型对我国中小上市公司有很强的识别信用风险状况的能力.中小上市公司违约的可能性大于我国大型企业,信用状况不容乐观,整体信用状况在近3年间表现波澜不惊,到2006年违约风险有增大趋势.通过设定两条信用预警线,来监控中小上市公司的信用危机.资产规模对信用风险有显著影响,2004年之后资产规模与违约风险显著负相关,总资产小于3亿元的小公司抗风险能力最差.股份分置改革引起了中小上市公司信用风险短时间的波动,是2006年中小上市公司违约风险变大的重要原因。     

Kolmogorov–Smirnov‐type testing for the partial homogeneity of Markov processes—with application to credit risk

In banking, the default behaviour of the counterpart is not only of interest for the pricing of transactions under credit risk but also for the assessment of a portfolio credit risk. We develop a test against the hypothesis that default intensities are chronologically constant within a group of similar counterparts, e.g. a rating class. The Kolmogorov–Smirnov‐type test builds up on the asymptotic normality of counting processes in event history analysis. The right censoring accommodates for Markov processes with more than one no‐absorbing state. A simulation study and two examples of rating systems demonstrate that partial homogeneity can be assumed, however occasionally, certain migrations must be modelled and estimated inhomogeneously. Copyright © 2007 John Wiley & Sons, Ltd.     

A classification problem of credit risk rating investigated and solved by optimisation of the ROC curve

Estimation of probability of default has considerable importance in risk management applications where default risk is referred to as credit risk. Basel II (Committee on Banking Supervision) proposes a revision to the international capital accord that implies a more prominent role for internal credit risk assessments based on the determination of default probability of borrowers. In our study, we classify borrower firms into rating classes with respect to their default probability. The classification of firms into rating classes necessitates the finding of threshold values separating the rating classes. We aim at solving two problems: to distinguish the defaults from non-defaults, and to put the firms in an order based on their credit quality and classify them into sub-rating classes. For using a model to obtain the probability of default of each firm, Receiver Operating Characteristics (ROC) analysis is employed to assess the distinction power of our model. In our new functional approach, we optimise the area under the ROC curve for a balanced choice of the thresholds; and we include accuracy of the solution into the program. Thus, a constrained optimisation problem on the area under the curve (or its complement) is carefully modelled, discretised and turned into a penalized sum-of-squares problem of nonlinear regression; we apply the Levenberg–Marquardt algorithm. We present numerical evaluations and their interpretations based on real-world data from firms in the Turkish manufacturing sector. We conclude with a discussion of structural frontiers, parametrical and computational features, and an invitation to future work.     

On the simulation of portfolios of interest rate and credit risk sensitive securities

We discuss extensions of reduced-form and structural models for pricing credit risky securities to portfolio simulation and valuation. Stochasticity in interest rates and credit spreads is captured via reduced-form models and is incorporated with a default and migration model based on the structural credit risk modelling approach. Calculated prices are consistent with observed prices and the term structure of default-free and defaultable interest rates. Three applications are discussed: (i) study of the inter-temporal price sensitivity of credit bonds and the sensitivity of future portfolio valuation with respect to changes in interest rates, default probabilities, recovery rates and rating migration, (ii) study of the structure of credit risk by investigating the impact of disparate risk factors on portfolio risk, and (iii) tracking of corporate bond indices via simulation and optimisation models. In particular, we study the effect of uncertainty in credit spreads and interest rates on the overall risk of a credit portfolio, a topic that has been recently discussed by Kiesel et al. [The structure of credit risk: spread volatility and ratings transitions. Technical report, Bank of England, ISSN 1268-5562, 2001], but has been otherwise mostly neglected. We find that spread risk and interest rate risk are important factors that do not diversify away in a large portfolio context, especially when high-quality instruments are considered.