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.     

Bayesian corporate bond pricing and credit default swap premium models for deriving default probabilities and recovery rates

This paper develops a Bayesian method by jointly formulating a corporate bond (CB) pricing model and credit default swap (CDS) premium pricing models to estimate the term structure of default probabilities and the recovery rate. These parameters are formulated by incorporating firm characteristics such as industry, credit rating and Balance Sheet/Profit and Loss information. A cross-sectional model valuing all given CB prices and CDS premiums is considered. The quantities derived are regarded as what market participants infer in forming CB prices and CDS premiums. We also develop a statistical significance test procedure without any distributional assumptions for the specified model. An empirical analysis is conducted using Japanese CB and CDS market data.     

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 discrete adapted hierarchical basis solver for radial basis function interpolation

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness.     

Modelling dependence structure with Archimedean copulas and applications to the iTraxx CDS index

In this paper we model the dependence structure between credit default swap (CDS) and jump risk using Archimedean copulas. The paper models and estimates the different relationships that can exist in different ranges of behaviour. It studies the bivariate distributions of CDS index spreads and the kurtosis of equity return distribution. To take into account nonlinear relationships and different structures of dependency, we employ three Archimedean copula functions: Gumbel, Clayton, and Frank. We adopt nonparametric estimation of copula parameters and we find an extreme co-movement of CDS and stock market conditions. In addition, tail dependence indicates the extreme co-movements and the potential for a simultaneous large loss in stock markets and a significant default risk. Ignoring the tail dependence would lead to underestimation of the default risk premium.     

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 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 Credit Risk in the Jump Threshold Framework

ABSTRACT

The jump threshold framework for credit risk modelling developed by Garreau and Kercheval enjoys the advantages of both structural- and reduced-form models. In their article, the focus is on multidimensional default dependence, under the assumptions that stock prices follow an exponential Lévy process (i.i.d. log returns) and that interest rates and stock volatility are constant. Explicit formulas for default time distributions and basket credit default swap (CDS) prices are obtained when the default threshold is deterministic, but only in terms of expectations when the default threshold is stochastic. In this article, we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds and CDS premia. The main idea is to make use of the Duffie–Pan–Singleton method of evaluating expectations of exponential integrals of affine diffusions.     

随机挽回率下第n次违约互换的蒙特卡罗模拟定价

考虑到挽回率是违约互换定价的重要因素,同时获得准确的挽回率也是极其困难的,于是假设挽回率是随机的,并与对应资产违约时间呈Copula类相依结构.在该假设条件下提出一种对第n次违约互换定价的模拟算法.通过实证模拟发现,恒定挽回率,独立随机挽回率和Copula结构的挽回率对应的定价结果相差较大.     

基于分数维Vasicek利率模型的CDS定价

本文研究CDS的定价问题, 其中涉及到利率风险和传染风险. 文中用分数维Vasicek利率模型刻画利率风险, 对公司的违约强度进行建模, 给出了违约与利率相关时风险债券的价格, 并在此基础上得到CDS的价格.     

Credit default swaps and systemic risk

We present a network model for investigating the impact on systemic risk of central clearing of over the counter (OTC) credit default swaps (CDS). We model contingent cash flows resulting from CDS and other OTC derivatives by a multi-layered network with a core-periphery structure, which is flexible enough to reproduce the gross and net exposures as well as the heterogeneity of market shares of participating institutions. We analyze illiquidity cascades resulting from liquidity shocks and show that the contagion of illiquidity takes place along a sub-network constituted by links identified as ’critical receivables’. A key role is played by the long intermediation chains inherent to the structure of the OTC network, which may turn into chains of critical receivables. We calibrate our model to data representing net and gross OTC exposures of large dealer banks and use this model to investigate the impact of central clearing on network stability. We find that, when interest rate swaps are cleared, central clearing of credit default swaps through a well-capitalized CCP can reduce the probability and the magnitude of a systemic illiquidity spiral by reducing the length of the chains of critical receivables within the financial network. These benefits are reduced, however, if some large intermediaries are not included as clearing members.     

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.     

关于双曲衰减的违约相关模型及CDS定价

引进一个双曲类型的衰减函数来表示一方违约对另一方违约强度的影响．若交易双方为竞争对手(合作公司),当一方的违约时,另一方的违约强度将减小(增大)．随着时间的推移,这种影响将逐渐减小,直至为零．在这个模型下,通过测度变换,可以得到两公司违约时间的联合分布及各自的边际分布,从而可以对违约互换进行定价．     

Restructuring risk in credit default swaps: An empirical analysis

This paper estimates the price for restructuring risk in the US corporate bond market during 1999–2005. Comparing quotes from default swap (CDS) contracts with a restructuring event and without, we find that the average premium for restructuring risk represents 6%–8% of the swap rate without restructuring. We show that the restructuring premium depends on firm-specific balance-sheet and macroeconomic variables. And, when default swap rates without a restructuring event increase, the increase in restructuring premia is higher for low-credit-quality firms than for high-credit-quality firms. We propose a reduced-form arbitrage-free model for pricing default swaps that explicitly incorporates the distinction between restructuring and default events. A case study illustrating the model’s implementation is provided.     

Verification and validation of numerical solutions of two-dimensional reactive flow in rocket engine nozzles

Two-dimensional mathematical models for gaseous H₂/O₂ reactive flows are solved for two geometries: a conical and a parabolic one. Five different physical models are studied: two one-species and three multi-species models (frozen, equilibrium and non-equilibrium flows). In the mathematical model, temperature is used as unknown in the energy equation and velocity is obtained for all speed flows. For all analyses, a non-orthogonal finite volume code was implemented, taking into account first (UDS) and second (CDS) order interpolation schemes and co-located grid arrangement. Model predictions of the pressure distribution and Mach number in the nozzle with a conical geometry, calculated using a CDS scheme, were found to agree well with experimental results. For both geometries, numerical results for apparent orders of convergence agreed well with the asymptotic (expected) ones for one-species flows. Some other analyses were provided for mixture of gases flows; in this case, for frozen flow, the apparent order values tend to the asymptotic ones in all cases; for local equilibrium flow, the use of CDS degenerated the apparent order to unity; this fact can be associated to the use of UDS interpolation scheme in the source term of the energy equation. Numerical solutions, including their error estimates, are provided for UDS and CDS schemes. Their analysis shows that global variables of interest (such as thrust and specific impulse) are less affected by the chosen physical model than are local variables of interest (such as the temperature at the symmetry line).     

Semi-analytical Formula for Pricing Bilateral Counterparty Risk of CDS with Correlated Credit Risks

Based on the framework of [7], we discuss pricing bilateral counterparty risk of CDS, where each individual default intensity is modeled by a shifted CIR process with jump (JCIR++), and the correlation between the default times is modeled by a copula function. We present a semi-analytical formula for pricing bilateral counterparty risk of CDS, which is more convenient to compute through calculating multiple numerical integration or using Monte-Carlo simulation without simulating default times. Moreover, we obtain simpler formulae under FGM copulas, Bernstein copulas and C^A,B copulas, which can be applied for speeding up the computation and reducing the pricing error. Numerical results under FGM copulas and C^A,B copulas show that our method performs better both in computation speed and accuracy.     

Domain decomposition for radial basis meshless methods

Both overlapping and nonoverlapping domain decomposition methods (DDM) on matching and nonmatching grid have been developed to couple with the meshless radial basis function (RBF) method. Example shows that overlapping DDM with RBF can achieve much better accuracy with less nodal points compared to FDM and FEM. Numerical results also show that nonmatching grid DDM can achieve almost the same accuracy within almost the same iteration steps as the matching grid case; hence our method is very attractive, because it is much easier to generate nonmatching grid just by putting blocks of grids together (for both overlapping and nonoverlapping), where each block grid can be generated independently. Also our methods are showed to be able to handle discontinuous coefficient. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 450–462, 2004.     

Adaptive radial basis function interpolation using an error indicator

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.     

一般均衡下基于产品市场和资本市场的单名CDS定价

次贷危机呼吁新的信用衍生品定价模型, 因此为存在产品市场和资本市场的经济结构建立一般均衡的单名CDS定价模型, 使用最优化求解一般均衡下的商品价格和CDS价格. 可以发现一般均衡的CDS定价具有资本市场和产品市场的因素, 这表示CDS的价格不再是由单纯的资本市场因素决定的, 而是由无风险利率、资本产出弹性、违约率、回收率同时决定的. 通过数量约束用模拟的方式研究多个均衡的动态变化, 发现违约风险的增加使得价格剧烈波动且市场交易萎缩. 在为以中国工商银行为参考资产的CDS定价过程中, 发现各种因素在不同的时期都可能成为定价的主要影响因素. 可以发现, 次贷危机的定价体系存在着信用调整问题和定价与实体经济脱节的问题. 可以认为, 一般均衡下基于产品市场和资本市场的单名CDS定价可以囊括多个市场的交叉影响, 为衍生品定价提供一个新的方向.     

一篮子信用违约互换定价的偏微分方程方法

通过对一篮子信用违约互换的结构性分析,在约化法框架下,用PDE方法提出一个新的计算具有违约相关性的多个公司联合生存概率的方法,在此基础上得到信用互换到期之前一篮子中违约数量的概率分布.应用这个概率分布,在条件独立的假定下,先后建立了首次违约、二次违约的信用违约互换定价模型,并用PDE方法给出了定价的显性表达式,并进一步扩展到解决m次违约的信用违约互换的定价问题.