Valuing risky debt: A new model combining structural information with the reduced-form approach

A new model of credit risk is proposed in which the intensity of default is described by an additional stochastic differential equation coupled with the process of the obligor’s asset value. Such an approach allows us to incorporate structural information as well as to capture the effect of external factors (e.g. macroeconomic factors) in a both parsimonious and economically consistent way. From the practical standpoint, the proposed model offers great flexibility and allows us to obtain credit spread curves of many different shapes, including double humped term structures. Furthermore, an approximate closed-form solution is derived, which is accurate, easy to implement, and allows for an efficient calibration to realized credit spreads. Numerical experiments are presented showing that the novel approach provides a very satisfactory fitting to market data and outperforms the model developed by Madan and Unal (2000).     

一个多因子信用违约互换定价模型

考虑一个违约互换的多因子的简化模型,通过测度的转化并求解一个多变量Riccati常微分方程而得出了这个模型的解析解.此模型是Duffie-Pan-Singleton(2000)模型的特例,但是这个模型存在解析解,而Duffie-Pan-Singleton模型并不存在解析解.模型的解析解对获得直觉的判断和进行模型的计算都非常有帮助,而且模型的解析解对实证检验也是有帮助的.     

An incentive-compatible solution for trade credit term incorporating default risk

This paper studies the optimal trade credit term decision in an extended economic ordering quantity (EOQ) framework that incorporates a default risk component. A principal-agent bilevel programming model with costs minimization objectives is set up to derive the incentive-compatible credit term. The supplier determines the credit term as the leader in the first level programming, by balancing her/his financing capacity with the retailer’s default risk, order behavior and cost shifting. At the second level, the retailer makes decisions on ordering and payment time by reacting on the term offered by the supplier. A first order condition solution procedure is derived for the bilevel programming when credit term is confined within the practically feasible interval. Two key results are obtained – the condition to derive incentive-compatible credit term, and an equation system to derive threshold default risk criterion filtering retailers suitable for credit granting. Numerical experiments show that the capital cost of the supplier is the most important factor determining the credit term. Default risk acts like a filtering criterion for selecting retailers suitable for credit granting. Empirical evidence supporting our theoretical considerations is obtained by estimating three panel econometric models, using a dataset from China’s listed companies.     

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.     

Comparison results for exchangeable credit risk portfolios

This paper is dedicated to risk analysis of credit portfolios. Assuming that default indicators form an exchangeable sequence of Bernoulli random variables and as a consequence of de Finetti’s theorem, default indicators are Binomial mixtures. We can characterize the supermodular order between two exchangeable Bernoulli random vectors in terms of the convex ordering of their corresponding mixture distributions. Thus we can proceed to some comparisons between stop-loss premiums, CDO tranche premiums and convex risk measures on aggregate losses. This methodology provides a unified analysis of dependence for a number of CDO pricing models based on factor copulas, multivariate Poisson and structural approaches.     

Evaluating corporate bonds with complicated liability structures and bond provisions

This paper presents a general and numerically accurate lattice methodology to price risky corporate bonds. It can handle complex default boundaries, discrete payments, various asset sales assumptions, and early redemption provisions for which closed-form solutions are unavailable. Furthermore, it can price a portfolio of bonds that accounts for their complex interaction, whereas traditional approaches can only price each bond individually or a small portfolio of highly simplistic bonds. Because of the generality and accuracy of our method, it is used to investigate how credit spreads are influenced by the bond provisions and the change in a firm’s liability structure due to bond repayments.     

No arbitrage in a simple credit risk model

In a simple credit risk model we find an equivalent condition to the no-simple-arbitrage principle and show that it is insufficient for an equivalent martingale measure to exist.     

Valuing Credit Default Swap under a double exponential jump di?usion model简

This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.     

Pricing catastrophe options with counterparty credit risk in a reduced form model

In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.     

违约风险的交换期权的定价模型与解法

含交易对手违约风险的交换期权采用混合模型定价,借助公司价值模型中的补偿率,同时采用以强度为基础的违约函数来确定违约的发生.假定违约强度遵从均值回复的重随机Poisson过程：且违约强度过程与标的资产,企业价值都相关.利用等价鞅测度变换方法导出含有违约风险的交换期权的价格闭解.     

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.     

Random effects logistic regression model for default prediction of technology credit guarantee fund

Korean government has been funding the small and medium enterprises (SME) with superior technology based on scorecard. However high default rate of funded SMEs has been reported. In order to effectively manage such governmental fund, it is important to develop accurate scoring model for SMEs. In this paper, we provide a random effects logistic regression model to predict the default of funded SMEs based on both financial and non-financial factors. Advantage of such a random effects model lies in the ability of accommodating not only the individual characteristics of each SME but also the uncertainty that cannot be explained by such individual factors. It is expected that our study can contribute to effective management of government funds by proposing the prediction models for defaults of funded SMEs.     

Calibration of the default probability model

In this paper, we study the calibration problem for the Merton–Vasicek default probability model [Robert Merton, On the pricing of corporate debt: the risk structure of interest rate, Journal of Finance 29 (1974) 449–470]. We derive conditions that guarantee existence and uniqueness of the solution. Using analytical properties of the model, we propose a fast calibration procedure for the conditional default probability model in the integrated market and credit risk framework. Our solution allows one to avoid numerical integration problems as well as problems related to the numerical solution of the nonlinear equations.     

A multicriteria approach for rating the credit risk of financial institutions

Within the new bank regulatory context, the assessment of the credit risk of financial institutions is an important issue for supervising authorities and investors. This study explores the possibility of a developing risk assessment model for financial institutions using a multicriteria classification method. The analysis is based on publicly available financial data for UK firms. The results indicate that the proposed multicriteria methodology provides promising results compared to well known statistical methods.     

Model combination for credit risk assessment: A stacked generalization approach

The development of credit risk assessment models is often considered within a classification context. Recent studies on the development of classification models have shown that a combination of methods often provides improved classification results compared to a single-method approach. Within this context, this study explores the combination of different classification methods in developing efficient models for credit risk assessment. A variety of methods are considered in the combination, including machine learning approaches and statistical techniques. The results illustrate that combined models can outperform individual models for credit risk analysis. The analysis also covers important issues such as the impact of using different parameters for the combined models, the effect of attribute selection, as well as the effects of combining strong or weak models.     

Random effects model for credit rating transitions

This paper proposes a random effects multinomial regression model to estimate transition probabilities of credit ratings. Unlike the previous studies on the rating transition, we applied a random effects model, which accommodates not only the environmental characteristics of the exposures of a rating but also the uncertainty not explained by such factors. The rating category specific factors such as retained earning and market equity are included in our proposed model. The random effects model provides less diagonally dominant matrix, where the transition probabilities are over-dispersed from the diagonal elements. Our study is expected to incorporate potential chances of rating transitions due to extra random variations.     

Constant barrier strategies in a two-state Markov-modulated dual risk model

In this paper,we consider the dividend problem in a two-state Markov-modulated dual risk model,in which the gain arrivals,gain sizes and expenses are influenced by a Markov process.A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived.In the case of exponential gain sizes,the equations are solved and the best barrier is obtained via numerical example.Finally,using numerical example,we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model.Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.     

On the parameterization of the CreditRisk model for estimating credit portfolio risk

The CreditRisk⁺ model is one of the industry standards for estimating the credit default risk for a portfolio of credit loans. The natural parameterization of this model requires the default probability to be apportioned using a number of (non-negative) factor loadings. However, in practice only default correlations are often available but not the factor loadings. In this paper we investigate how to deduce the factor loadings from a given set of default correlations. This is a novel approach and it requires the non-negative factorization of a positive semi-definite matrix which is by no means trivial. We also present a numerical optimization algorithm to achieve this.     

Optimization of nearest neighbor classifiers via metaheuristic algorithms for credit risk assessment

The classification problem consists of using some known objects, usually described by a large vector of features, to induce a model that classifies others into known classes. The present paper deals with the optimization of Nearest Neighbor Classifiers via Metaheuristic Algorithms. The Metaheuristic Algorithms used include tabu search, genetic algorithms and ant colony optimization. The performance of the proposed algorithms is tested using data from 1411 firms derived from the loan portfolio of a leading Greek Commercial Bank in order to classify the firms in different groups representing different levels of credit risk. Also, a comparison of the algorithm with other methods such as UTADIS, SVM, CART, and other classification methods is performed using these data.     

A latent process model for the pricing of corporate securities

We propose a structural model with a joint process of tangible assets (marker) and firm status for the pricing of corporate securities. The firm status is assumed to be latent or unobservable, and default occurs when the firm status process reaches a default threshold at the first time. The marker process is observable and assumed to be correlated with the latent firm status. The recovery upon default is a fraction of tangible assets at the time of default. Our model can evaluate both the corporate debt and equity to fit their market prices in a unified framework. When the two processes are perfectly correlated, our model is reduced to the seminal Black–Cox model. Numerical examples are given to support the usefulness of our model. A previous version of this paper was presented at the Tsukuba–Stanford workshop held at Stanford University on March 2006. The authors are grateful to participants of the workshop for helpful discussions.