A multinomial tree model for pricing credit default swap options

Among the traded credit derivatives, the market interest in credit default swap options (CDSwaptions) is enormous. We propose a multinomial tree model to price Bermudan CDSwaptions. Our basic rationale is that we distribute the occurring probability for each node in a branch proportional to the probability density function of the assumed (normal) distribution. Through this approach, without the need of solving a large number of equations simultaneously, only the first four moments are required to build an arbitrarily large N-branches tree. We also demonstrate the detailed model implementation procedure including the valuation and the estimation of critical prices through an empirical example in Tucker and Wei (J Fixed Income 15(1):88–95, 2005). Numerical results show that, in the valuation, the proposed multinomial tree model is accurate and can significantly save pricing time under the same degree of accuracy as the binomial tree model. In the estimation of critical prices, the results are less accurate than those in the valuation, but the relative errors are acceptable.     

Estimation of rating classes and default probabilities in credit risk models with dependencies

Let Y = m(X) + ε be a regression model with a dichotomous output Y and a one‐step regression function m . In the literature, estimators for the three parameters of m , that is, the breakpoint θ and the levels a and b , are proposed for independent and identically distributed (i.i.d.) observations. We show that these standard estimators also work in a non‐i.i.d. framework, that is, that they are strongly consistent under mild conditions. For that purpose, we use a linear one‐factor model for the input X and a Bernoulli mixture model for the output Y . The estimators for the split point and the risk levels are applied to a problem arising in credit rating systems. In particular, we divide the range of individuals' creditworthiness into two groups. The first group has a higher probability of default and the second group has a lower one. We also stress connections between the standard estimator for the cutoff θ and concepts prevalent in credit risk modeling, for example, receiver operating characteristic. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas

We introduce a new importance sampling method for pricing basket default swaps employing exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structures than existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of the proposed algorithms, we present numerical examples and compare them with the crude Monte Carlo simulation, and finally show that our proposed estimators produce considerably smaller variances.     

Using kalman filter and finite difference techniques in default free bond pricing models

A model to price default free bonds, similar to ones developed by Cox, Ingersoll and Ross, Langetieg, and Richard, is empirically examined. Calculation of model prices involves three disjoint tasks: (1) estimation of the values of the real interest rate and the inflation rate (which we will refer to as state variables or sources of uncertainty) as well as the parameters of the state stochastic differential equations, (2) estimation of the market prices of risk associated with the two state variables, and (3) the solution of the valuation partial differential equation. Task 1 is accomplished by using a Kalman Filter algorithm, task 2 uses a Fama/MacBeth approach, and task 3 utilizes an Alternating Direction Implicit finite difference technique. Model prices are compared to actual prices. The model performs better during a period of relatively stable economic conditions compared to a period associated with more volatile conditions. Pricing errors are smaller at short maturities, and increase as time to maturity increases.     

Modelling credit default swap spreads by means of normal mixtures and copulas

This paper develops a multivariate statistical model for the analysis of credit default swap spreads. Given the large excess kurtosis of the univariate marginal distributions, it is proposed to model them by means of a mixture of distributions. However, the multivariate extension of this methodology is numerically difficult, so that copulas are used to capture the structure of dependence of the data. It is shown how to estimate the parameters of the marginal distributions via the EM algorithm; then the parameters of the copula are estimated and standard errors computed through the nonparametric bootstrap. An application to credit default swap spreads of some European reference entities and extensive simulation results confirm the effectiveness of the method.     

Forecasting probabilities of default and loss rates given default in the presence of selection

This paper offers a joint estimation approach for forecasting probabilities of default and loss rates given default in the presence of selection. The approach accommodates fixed and random risk factors. An empirical analysis identifies bond ratings, borrower characteristics and macroeconomic information as important risk factors. A portfolio-level analysis finds evidence that common risk measurement approaches may underestimate bank capital by up to 17% relative to the presented model.     

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.     

Bayesian confidence intervals for probability of default and asset correlation of portfolio credit risk

We derive Bayesian confidence intervals for the probability of default (PD), asset correlation (Rho), and serial dependence (Theta) for low default portfolios (LDPs). The goal is to reduce the probability of underestimating credit risk in LDPs. We adopt a generalized method of moments with continuous updating to estimate prior distributions for PD and Rho from historical default data. The method is based on a Bayesian approach without expert opinions. A Markov chain Monte Carlo technique, namely, the Gibbs sampler, is also applied. The performance of the estimation results for LDPs validated by Monte Carlo simulations. Empirical studies on Standard & Poor’s historical default data are also conducted.     

Mixture cure models in credit scoring: If and when borrowers default

Mixture cure models were originally proposed in medical statistics to model long-term survival of cancer patients in terms of two distinct subpopulations - those that are cured of the event of interest and will never relapse, along with those that are uncured and are susceptible to the event. In the present paper, we introduce mixture cure models to the area of credit scoring, where, similarly to the medical setting, a large proportion of the dataset may not experience the event of interest during the loan term, i.e. default. We estimate a mixture cure model predicting (time to) default on a UK personal loan portfolio, and compare its performance to the Cox proportional hazards method and standard logistic regression. Results for credit scoring at an account level and prediction of the number of defaults at a portfolio level are presented; model performance is evaluated through cross validation on discrimination and calibration measures. Discrimination performance for all three approaches was found to be high and competitive. Calibration performance for the survival approaches was found to be superior to logistic regression for intermediate time intervals and useful for fixed 12 month time horizon estimates, reinforcing the flexibility of survival analysis as both a risk ranking tool and for providing robust estimates of probability of default over time. Furthermore, the mixture cure model’s ability to distinguish between two subpopulations can offer additional insights by estimating the parameters that determine susceptibility to default in addition to parameters that influence time to default of a borrower.     

Convergence rates of trinomial tree methods for option pricing under regime-switching models

Recently trinomial tree methods have been developed to option pricing under regime-switching models. Although these novel trinomial tree methods are shown to be accurate via numerical examples, it needs to give a rigorous proof of the accuracy which can theoretically guarantee the reliability of the computations. The aim of this paper is to prove the convergence rates (measure of the accuracy) of the trinomial tree methods for the option pricing under regime-switching models.     

Upper bounds for ruin probabilities in two dependent risk models under rates of interest

In this article, we consider two discrete‐time risk models, in which dependent structures of the payments and the interest force are considered. Two autoregressive moving‐average (ARMA) models are introduced to model the premiums and rates of interest, and the claims are assumed to be independent. Generalized Lundberg inequalities for the ruin probabilities are derived by using renewal recursive technique, which extend some known results. Copyright © 2009 John Wiley & Sons, Ltd.     

Closed-form solutions for pricing credit-risky bonds and bond options

This paper proposes closed-form solutions for pricing credit-risky discount bonds and their European call and put options in the intensity-based reduced-form framework, assuming the stochastic dynamics of both the risk-free interest rate and the credit-spread are driven by two correlated Ho-Lee models [T.S.Y. Ho, S.B. Lee, Term structure movements and pricing interest rates contingent claims, Journal of Finance 41 (5) (1986) 1011-1029]. The results are easily to implement, and require very few parameters which are directly implied from market data.     

Bound and collapse Bayesian reject inference for credit scoring

Reject inference is a method for inferring how a rejected credit applicant would have behaved had credit been granted. Credit-quality data on rejected applicants are usually missing not at random (MNAR). In order to infer credit-quality data MNAR, we propose a flexible method to generate the probability of missingness within a model-based bound and collapse Bayesian technique. We tested the method's performance relative to traditional reject-inference methods using real data. Results show that our method improves the classification power of credit scoring models under MNAR conditions.     

Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit

This study models a finite horizon inventory problem for deteriorating and fashion goods under trade credit and partial backlogging conditions. Demand may vary with price or time. The supplier can extend credit to the retailer. As a result, the retailer does not have to pay for goods immediately upon acquiring them, and can instead earn interest on the retail price of the goods between the time they are sold and the end of the credit period. The proposed model considers two-phase pricing and inventory decisions. In other words, it determines both the optimal prices and the lengths of the in-stock and stock-out period. This paper is the first to consider different price decisions for in-stock and stock-out periods under trade credit. We develop an algorithm to determine the optimal pricing and replenishment strategy while still maximizing the total profit. Further, this study shows that the proposed two-phase pricing strategy is superior to a one-phase pricing strategy in terms of profit maximization. Computational analysis illustrates the solution procedures and the impacts of the related parameters on decisions and profits. The results of this study can serve as references for business managers or administrators.     

Bayesian estimation and inference for log-ACD models

This paper adapts Bayesian Markov chain Monte Carlo methods for application to some auto-regressive conditional duration models. Subsequently, the properties of these estimators are examined and assessed across a range of possible conditional error distributions and dynamic specifications, including under error mis-specification. A novel model error distribution, employing a truncated skewed Student-t distribution is proposed and the Bayesian estimator assessed for it. The results of an extensive simulation study reveal that favourable estimation properties are achieved under a range of possible error distributions, but that the generalised gamma distribution assumption is most robust and best preserves these properties, including when it is incorrectly specified. The results indicate that the powerful numerical methods underlying the Bayesian estimator allow more efficiency than the (quasi-) maximum likelihood estimator for the cases considered.     

Importance sampling for integrated market and credit portfolio models

A sophisticated approach for computing the total economic capital needed for various stochastically dependent risk types is the bottom-up approach. In this approach, usually, market and credit risks of financial instruments are modeled simultaneously. As integrating market risk factors into standard credit portfolio models increases the computational burden of calculating risk measures, it is analyzed to which extent importance sampling techniques previously developed either for pure market portfolio models or for pure credit portfolio models can be successfully applied to integrated market and credit portfolio models. Specific problems which arise in this context are discussed. The effectiveness of these techniques is tested by numerical experiments for linear and non-linear portfolios.     

Homogeneous groups and multiscale intensity models for multiname credit derivatives

We discuss a computationally tractable approach to the valuation of multiname credit derivatives employing name grouping for dimension reduction, and singular perturbation approximations for model robustness. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cost-based pricing model with value-added tax and corporate income tax for a supply chain network

This article aims to propose the short-term cost-based pricing method of supply chain network with the consideration of value-added tax (VAT) and corporate income tax. First, the average cost function of each business unit in supply chain network is given, and the average cost function is taken as the monotone mapping in n-dimensional space. According to Kantorovich theorem, the existence and uniqueness of equilibrium point where the cost equals the income is discussed. When the demand function satisfies certain conditions, there generally exist many equilibrium points for cost-based pricing. Moreover, the iteration method for finding one of the equilibrium solutions is given. Then, tax burden of producers and consumers is described and illustrated with an example.     

Dependence properties and bounds for ruin probabilities in multivariate compound risk models

In risk management, ignoring the dependence among various types of claims often results in over-estimating or under-estimating the ruin probabilities of a portfolio. This paper focuses on three commonly used ruin probabilities in multivariate compound risk models, and using the comparison methods shows how some ruin probabilities increase, whereas the others decrease, as the claim dependence grows. The paper also presents some computable bounds for these ruin probabilities, which can be calculated explicitly for multivariate phase-type distributed claims, and illustrates the performance of these bounds for the multivariate compound Poisson risk models with slightly or highly dependent Marshall-Olkin exponential claim sizes.