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.     

Extending the intensity model with joint defaults to incorporate the lasting effects from common credit events

This paper studies how the lasting effects of common credit events influence default probability distribution and the prices of multiname credit derivatives. Based on a joint defaults model where common credit events are used to generate simultaneous defaults, we extend the model to allow for their impacts to last for a longer while. The default intensity of each entity is heightened significantly while the impact still has an influence, until some time later when this effect fades away. Incorporating these lasting effects helps to generate higher default correlation, which is more consistent with today's highly correlated financial markets. The proposed model can be either formulated as a Markov chain or implemented by Monte Carlo simulation in order to calculate the default probability distributions and multiname derivatives prices. Our numerical results demonstrate the strong influences from the lasting effects and provide a justification of their incorporation.     

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.     

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.     

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.     

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.     

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.     

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 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.     

Intensity models and transition probabilities for credit card loan delinquencies

We estimate the probability of delinquency and default for a sample of credit card loans using intensity models, via semi-parametric multiplicative hazard models with time-varying covariates. It is the first time these models, previously applied for the estimation of rating transitions, are used on retail loans. Four states are defined in this non-homogenous Markov chain: up-to-date, one month in arrears, two months in arrears, and default; where transitions between states are affected by individual characteristics of the debtor at application and their repayment behaviour since. These intensity estimations allow for insights into the factors that affect movements towards (and recovery from) delinquency, and into default (or not). Results indicate that different types of debtors behave differently while in different states. The probabilities estimated for each type of transition are then used to make out-of-sample predictions over a specified period of time.     

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.     

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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.     

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.     

Modelling the profitability of credit cards by Markov decision processes

This paper derives a Markov decision process model for the profitability of credit cards, which allows lenders to find an optimal dynamic credit limit policy. The states of the system are based on the borrower’s behavioural score and the decisions are what credit limit to give the borrower each period. In determining which Markov chain best describes the borrower’s performance, second order as well as first order Markov chains are considered and estimation procedures developed that deal with the low default levels that may exist in the data. A case study is given in which the optimal credit limit is derived and the results compared with the actual outcomes.     

Joint survival probability via truncated invariant copula

Given an intensity-based credit risk model, this paper studies dependence structure between default intensities. To model this structure, we use a multivariate shot noise intensity process, where jumps occur simultaneously and their sizes are correlated. Through very lengthy algebra, we obtain explicitly the joint survival probability of the integrated intensities by using the truncated invariant Farlie–Gumbel–Morgenstern copula with exponential marginal distributions. We also apply our theoretical result to pricing basket default swap spreads. This result can provide a useful guide for credit risk management.     

Sensitivity analysis for marked Hawkes processes: application to CLO pricing

This paper deals with a model for pricing Collateralized Loan Obligations, where the underlying credit risk is driven by a marked Hawkes process, involving both clustering effects on defaults and random recovery rates. We provide a sensitivity analysis of the CLO price with respect to the parameters of the Hawkes process using a change of probability and a variational approach. We also provide a simplified version of the model where the intensity of the Hawkes process is taken as the instantaneous default rate. In this setting, we give a moment-based formula for the expected survival probability.     

Randomization in the first hitting time problem

In this paper, we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F(t), and a linear boundary, b(t)=μt, find a distribution of the initial state such that the distribution of the first hitting time is F(t). This problem has important applications in credit risk modeling where the process represents the so-called distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.     

Copula方法与相依违约研究

目前信用风险研究的重点已经从单笔债务的违约概率研究转移到多笔债务的相依违约(Dependent Defaults)研究。Copula方法是研究相依违约的重要方法。这种方法是最近几年才被应用到信用领域研究中的一种新方法。本结合代表性献对Copula方法在相依违约研究中的应用进行了探讨。探讨的内容包括Copula方法被应用于相依违约研究的原因、该方法对于相依违约建模理论的改进以及在实证应用中使用Copula方法应该注意的问题。     

Modelling the evolution of credit spreads using the Cox process within the HJM framework: A CDS option pricing model

In this paper a simulation approach for defaultable yield curves is developed within the Heath et al. (1992) framework. The default event is modelled using the Cox process where the stochastic intensity represents the credit spread. The forward credit spread volatility function is affected by the entire credit spread term structure. The paper provides the defaultable bond and credit default swap option price in a probability setting equipped with a subfiltration structure. The Euler–Maruyama stochastic integral approximation and the Monte Carlo method are applied to develop a numerical scheme for pricing. Finally, the antithetic variable technique is used to reduce the variance of credit default swap option prices.     

The pricing of credit risky securities under stochastic interest rate model with default correlation

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R.A. Jarrow and F.Yu (2001), S.Y.Leung and Y.K.Kwok (2005), A.Wang and Z.Ye (2011)). By using the method of change of measure and the technology (H. S.Park (2008), R.Hao and Z.Ye (2011)) of dealing with jump diffusion processes, we obtain the analytic pricing formulas of defaultable zero-coupon bonds. Moreover, by the “total hazard construction”, we give the analytic pricing formulas of credit default swap (CDS).