Confidence sets and confidence bands for a beta distribution with applications to credit risk management

Incorporating statistical multiple comparisons techniques with credit risk measurement, a new methodology is proposed to construct exact confidence sets and exact confidence bands for a beta distribution. This involves simultaneous inference on the two parameters of the beta distribution, based upon the inversion of Kolmogorov tests. Some monotonicity properties of the distribution function of the beta distribution are established which enable the derivation of an efficient algorithm for the implementation of the procedure. The methodology has important applications to financial risk management. Specifically, the analysis of loss given default (LGD) data are often modeled with a beta distribution. This new approach properly addresses model risk caused by inadequate sample sizes of LGD data, and can be used in conjunction with the standard recommendations provided by regulators to provide enhanced and more informative analyses.     

The effect of the subprime crisis on the credit risk in global scale

Credit default swap (CDS) has become one of the most actively traded credit derivatives, and its importance in finance markets has increased after the subprime crisis. In this study, we analyzed the correlation structure of credit risks embedded in CDS and the influence of the subprime crisis on this topological space. We found that the correlation was stronger in the cluster constructed according to the location of the CDS reference companies than in the one constructed according to their industries. The correlation both within a given cluster and between different clusters became significantly stronger after the subprime crisis. The causality test shows that the lead lag effect between the portfolios (into which reference companies are grouped by the continent where each of them is located) is reversed in direction because the portion of non-investable and investable reference companies in each portfolio has changed since then. The effect of a single impulse has increased and the response time relaxation has become prolonged after the crisis as well.     

基于贝叶斯方法的低违约组合违约率估计

现代信用风险建模的核心是估计违约率,违约率估计是否准确将直接影响信用风险建模的质量。在估计违约率的众多文献中,频率法或logistic回归等统计方法的运用非常广泛,此类统计模型的基础是大样本,它客观上需要最低数量或最优数量的违约数据,而低违约组合(LDP)是指只有很少违约数据甚至没有违约数据的组合,如何估计LDP的违约率、反映违约率的非预期波动是一个值得关注的现实问题。本文针对银行贷款LDP缺乏足够历史违约数据的情况,采用贝叶斯方法估计LDP的违约率,并进一步探讨了根据专家判断或者根据同类银行LDP违约数量的历史数据来确定先验分布的方法。在贝叶斯估计中,通过先验分布的设定,不仅可以实现违约率估计的科学性和合理性,而且可以反映违约的非预期波动,有助于银行实施谨慎稳健的风险管理。     

Pricing Options on Defaultable Stocks*

? ?. This work was inspired by the SAMSI workshops on Financial Mathematics, Statistics and Econometrics (Fall 2005, Spring 2006 North Carolina). The author wishes to thank the organizers for the travel grant to participate in this stimulating event. I also would like to thank Bo Yang for his research assistance and the two anonymous referees and an anonymous associate editor for their valuable suggestions. Stock option price approximations are developed for a model which takes both the risk of default and the stochastic volatility into account. The intensity of defaults is assumed to be influenced by the volatility. It is shown that it might be possible to infer the risk neutral default intensity from the stock option prices. The proposed option price approximation has a rich implied volatility surface structure and fits the data implied volatility well. A calibration exercise shows that an effective hazard rate from bonds issued by a company can be used to explain the impliedvolatility skew of the option prices issued by the same company. It is also observed that the implied yield spread obtained from calibrating all the model parameters to the option prices matches the observed yield spread.     

Sato Processes in Default Modelling

Abstract

In reduced form default models, the instantaneous default intensity is the classical modelling object. Survival probabilities are then given by the Laplace transform of the cumulative hazard defined as the integrated intensity process. Instead, recent literature tends to specify the cumulative hazard process directly. Within this framework we present a new model class where cumulative hazards are described by self-similar additive processes, also known as Sato processes. Furthermore, we analyse specifications obtained via a simple deterministic time change of a homogeneous Lévy process. While the processes in these two classes share the same average behaviour over time, the associated intensities exhibit very different properties. Concrete specifications are calibrated to data on all the single names included in the iTraxx Europe index. The performances are compared with those of the classical Cox–Ingersoll–Ross intensity and a recently proposed class of intensity models based on Ornstein–Uhlenbeck-type processes. It is shown that the time-inhomogeneous Lévy models achieve comparable calibration errors with fewer parameters and with more stable parameter estimates over time. However, the calibration performance of the Sato processes and the time-change specifications are practically indistinguishable.     

基于最优负债系数的上市银行违约概率测算模型与实证

当上市银行的长期负债系数γ的取值不同时,应用KMV模型测算出的银行违约概率大相径庭。根据债券的实际信用利差可以推算出上市银行的违约概率PDi,CS,根据长期负债系数γ可以运用KMV模型确定上市银行的理论违约概率PDi,KMV。本文通过理论违约率与实际违约率的总体差异∑ni=1|PDi,KMV-PDi,cs|最小的思路建立规划模型,确定了KMV模型的最优长期负债γ系数;通过最优长期负债系数γ建立了未发债上市银行的违约率测算模型、并实证测算了我国14家全部上市银行的违约概率。本文的创新与特色一是采用KMV模型计算的银行违约概率PDi,KMV与实际信用利差确定的银行违约概率PDi,CS总体差异∑ni=1|PDi,KMV-PDi,cs|最小的思路建立规划模型,确定了KMV模型中的最优长期负债γ系数;使γ系数的确定符合资本市场利差的实际状况,解决了现有研究中在0和1之间当采用不同的长期负债系数γ、其违约概率的计算结果截然不同的问题。二是实证研究表明,当长期负债系数γ=0.7654时,应用KMV模型测算出的我国上市银行违约概率与我国债券市场所接受的上市银行违约概率最为接近。三是实证研究表明国有上市银行违约概率最低,区域性的上市银行违约概率较高,其他上市银行的违约概率居中。     

Supplier default dependencies: Empirical evidence from the automotive industry

Common ways to mitigate the detrimental consequences of supplier bankruptcies are to install redundancy and to pursue a multiple-sourcing strategy. This is based on the assumption that the adverse event of one supplier going out of business is largely independent from the default of other suppliers. However, this implicit assumption does not hold in all cases. This study – based on empirical data from automotive suppliers – reveals that default dependencies among suppliers do often exist and can have significant consequences. We use copula functions, a method of representing joint distribution functions with particular marginals, to capture the default dependence between automotive suppliers and to simulate various default dependence scenarios. We also conduct a comparative static analysis illustrating the significant impact of default correlation in a supplier portfolio. Our findings should spur managers to analyze their supplier portfolios with respect to default dependencies, and to take this phenomenon into consideration when making sourcing decisions.     

基于信用利差与Logistic回归的公司违约概率测算模型与实证研究

用Logistic模型计算公司违约概率在实际应用中存在两个问题:一是在缺乏公司违约记录数据库或违约记录数据库不典型的情况下,无法应用该模型或模型计算结果不准确;二是现有Logistic违约概率模型忽视了不同行业财务指标分布特征的差异性,导致公司违约概率计算结果的准确性降低。针对问题一,本文通过公司债券信用利差计算市场隐含的公司违约概率,在Logistic变换的基础上进一步确定Logistic线性回归的参数,使得公司违约概率的计算结果符合债券市场的实际状况。针对问题二,通过不同行业关键财务指标的单因子方差分析,证实了行业间财务指标的分布特征具有显著性差异,通过拟合优度证实了区分行业建立Logistic违约概率模型可显著提高违约概率测算的准确性。本文Logistic违约概率模型的构建过程如下:通过初选财务指标的相关性分析,删除反映信息重复的财务指标;通过Logistic回归中财务指标系数的显著性检验,删除对违约概率解释能力弱的财务指标;以Logistic回归的拟合优度为标准,选取各样本行业Logistic违约概率模型的关键财务指标,建立了机械设备等5个样本行业的Logistic违约概率模型,为样本内行业公司违约概率的准确测算提供模型与方法。本文的创新与特色:一是在无套利条件下,通过公司债券信用利差计算市场隐含的公司违约概率,并对其进行Logistic变换,作为Logistic线性回归的被解释变量,解决了在缺乏公司违约记录数据情况下Logistic违约概率模型的参数估计问题;二是通过单因子方差分析方法,证实了行业间财务指标的分布特征具有显著性差异,说明应区分行业建立Logistic违约概率模型;三是通过财务指标间的相关分析删除反映信息重复的财务指标,通过财务指标系数的显著性检验删除对公司违约概率解释能力弱的财务指标,保证了Logistic违约概率模型中关键财务指标选取的合理性;四是实证研究结果表明,不同行业的Logistic违约概率模型的关键财务指标不同,同一财务指标的参数也存在显著差异。实证研究结果还表明,区分行业建立Logistic违约概率模型与不区分行业相比,前者可将拟合优度及调整后的拟合优度提高近1倍。本文研究结果对于提高公司违约概率测算的准确性具有重要参考意义,对于商业银行贷款定价、公司债券发行定价、银行信用风险管理具有重要参考意义。     

人脑默认模式网络的动力学行为

大脑具有自适应、自组织、多稳态等重要特征,是典型的复杂系统.人脑在静息态下的关键功能子网络--默认模式网络(DMN)的激活处于多状态间持续跳转的非平衡过程,揭示该过程背后的动力学机制具有重要的科学意义和临床应用前景.本文基于功能磁共振获得的血氧水平依赖(BOLD)信号,建立了DMN吸引子跳转非平衡过程的能量图景、吸引子非联通图、跳转关系网络等;以高级视觉皮层和听觉等皮层活动为例,通过对应激活DMN状态空间的分布,以及XGBoost、深度神经网络等算法验证了DMN状态变化与外部脑区状态的密切依赖关系;通过偏相关、收敛交叉映射等方法分析了DMN内各个脑区之间的相互作用.本文结果有助于理解静息态下大脑内在非平衡过程的动力学机制,以及从动力学的角度探索具有临床意义的脑功能障碍生物标志物.     

Pricing credit default swaps with a random recovery rate by a double inverse Fourier transform

We evaluate the par spread for a single-name credit default swap with a random recovery rate. It is carried out under the framework of a structural default model in which the asset-value process is of infinite activity but finite variation. The recovery rate is assumed to depend on the undershoot of the asset value below the default threshold when default occurs. The key part is to evaluate a generalized expected discounted penalty function, which is a special case of the so-called Gerber–Shiu function in actuarial ruin theory. We first obtain its double Laplace transform in time and in spatial variable, and then implement a numerical Fourier inversion integration. Numerical experiments show that our algorithm gives accurate results within reasonable time and different shapes of spread curve can be obtained.