Hazard Rate Process Research Articles

Cointegration analysis of hazard rates and CDSs: Applications to pairs trading strategy

This study examines the cointegration relationship between multiple credit default swap (CDS) spreads by constructing the cointegrated hazard rate model, which assumes the structure of the vector error correction model (VECM) in the drift term of hazard rate processes. We merge the cointegration nature into the framework of arbitrage-free pricing and thereby derive the theoretical spread formula of multiple cointegrated CDSs. For the estimation of hazard rate dynamics, we develop a Bayesian statistical inference method combined with the numerical ordinary differential equation solver because the theoretical CDS spread cannot be expressed in closed form. In the empirical study of Japanese corporate CDSs, we find that the overall term structures of cointegrated CDSs can be explained by a simple two-dimensional VECM of cointegrated hazard rates. Furthermore, we study the pairs trading strategy of CDSs.

On a hazard (failure) rate process with delays after shocks

Distinct from conventional shock models when a failure of a system or accumulation of the corresponding damage occurs immediately after a shock, we consider a setting when these consequences appear with a random delay. In our model, a shock acts directly upon the failure rate of a system. This, after a random time, can result either in a failure or in the increase in the failure rate by a random amount. We derive expressions for survival probability and the failure rate for a system. Asymptotic behavior of the failure rate is also studied in detail.

LOCAL RISK MINIMIZATION OF CONTINGENT CLAIMS SIMULTANEOUSLY EXPOSED TO ENDOGENOUS AND EXOGENOUS DEFAULT TIMES

We study the local risk minimization approach for contingent claims that might be simultaneously prone to both endogenous (or structural) and exogenous (or reduced form) default events. The exogenous default time is defined through a hazard rate process that can depend on both the underlying risky asset values and its running infimum process. On the other hand, the endogenous default time could be modeled by a first-passage-time approach. In particular, our framework provides a unification of structural and reduced form credit risk modeling. In our work, the evolution of the underlying risky asset values is modeled by an exponential Lévy process, for example exponential jump-diffusion models. Our aim is to determine locally risk minimizing hedging strategies of the contingent claims that are affected by both structural and reduced form default events, through solutions of either partial differential equations or partial-integro differential equations. Finally, we show that these solutions are numerically implementable, and we provide some numerical examples.

Modeling stochastic mortality for joint lives through subordinators

There is a burgeoning literature on mortality models for joint lives. In this paper, we propose a new model in which we use time-changed Brownian motion with dependent subordinators to describe the mortality of joint lives. We then employ this model to estimate the mortality rate of joint lives in a well-known Canadian insurance data set. Specifically, we first depict an individual’s death time as the stopping time when the value of the hazard rate process first reaches or exceeds an exponential random variable, and then introduce the dependence through dependent subordinators. Compared with existing mortality models, this model better interprets the correlation of death between joint lives, and allows more flexibility in the evolution of the hazard rate process. Empirical results show that this model yields highly accurate estimations of mortality compared to the baseline non-parametric (Dabrowska) estimation.

DURATION MODELS FOR CREDIT RATING MIGRATION: EVIDENCE FROM THE FINANCIAL CRISIS

We introduce a specific duration model to analyze the prediction of the credit rating migration. We consider hazard rate processes based on multi‐state autoregressive conditional duration models. To take account of the economic context, we model the conditional mean of the duration between two ratings by means of a latent process. To this purpose, a dynamic‐ordered probit model is developed to describe the directions taken by the ratings in the presence of multiple states. As an illustration, we study the migration of credit rating during periods before and after the financial crisis. (JEL C14, C41, G24)

Does Modeling Framework Matter? A Comparative Study of Structural and Reduced-Form Models

This study provides a rigorous empirical comparison of structural and reduced-form credit risk frameworks. The literature differentiates between structural models that are based on modeling of the evolution of the balance sheet of the issuer, and reduced-form models that specify credit risk exogenously by a hazard rate process. Until now, there has been no common agreement in academia and practice on which model framework better captures credit risk. As major difference we focus on the discriminative modeling of the default time. In contrast to the previous literature, we calibrate both approaches to the same data set, apply comparable estimation techniques, and assess the out-of-sample prediction quality on the same time series of CDS prices. As our empirical implementations of both approaches rely on the same market information we are able to judge whether empirically the model structure itself makes an important difference. Interestingly, our study shows that the models' prediction power are quite close on average indicating that for pricing purposes the modeling type does not greatly matter compared to the input data used. Still, the reduced-form approach outperforms the structural for investment-grade names and longer maturities. In contrast the structural approach performs better for shorter maturities and sub-investment grade names.

On some conditional characteristics of hazard rate processes induced by external shocks

Stochastic failure models for systems under randomly variable environment (dynamic environment) are often described using hazard rate process. In this paper, we consider hazard rate processes induced by external shocks affecting a system that follow the nonhomogeneous Poisson process. The sample paths of these processes monotonically increase. However, the failure rate of a system can have completely different shapes and follow, e.g., the upside-down bathtub pattern. We describe and study various ‘conditional properties’ of the models that help to analyze and interpret the shape of the failure rate and other relevant characteristics.

Evaluating Double-Default Risk with a Regime Switching Hazard Rate Process and a Chaining Copula Model

Since the financial crises in 2008, the default risk of counter-parties has drawn increased attention. One way of mitigating this default risk is the so-called credit support annex (CSA). The CSA denies rules under which collateral is received -- respectively posted -- in order to protect against losses in case of the default of the counter-party. However, for credit derivatives there is still a residual risk, when the derivative's reference entity and the counter-party default at the same time. As will be outlined in due course, standard techniques known from default correlation derivatives are inappropriate to evaluate that gap risk. Hence, we will present new techniques, namely a copula model for simultaneous defaults and a regime-switching hazard rate model to cope with that evaluation problem.

Characterization of extendible distributions with exponential minima via processes that are infinitely divisible with respect to time

We present a stochastic representation for multivariate extendible distributions with exponential minima (exEM), whose components are conditionally iid in the sense of de Finetti’s theorem. It is shown that the “exponential minima property” is in one-to-one correspondence with the conditional cumulative hazard rate process being infinitely divisible with respect to time (IDT). The Laplace exponents of non-decreasing IDT processes are given in terms of a Bernstein function applied to the state space variable and are linear in time. Examples for IDT processes comprise killed Levy subordinators, monomials whose slope is randomized by a stable random variable, and several combinations thereof. As a byproduct of our results, we provide an alternative proof (and a mild generalization) of the important conclusion in Genest and Rivest (Stat. Probab. Lett. 8:207211, 1989), stating that the only copula which is both Archimedean and of extreme-value kind is the Gumbel copula. Finally, we show that when the subfamily of strong IDT processes is used in the construction leading to exEM, the result is the proper subclass of extendible min-stable multivariate exponential (exMSMVE) distributions.

Does modeling framework matter? A comparative study of structural and reduced-form models

This study provides a rigorous empirical comparison of structural and reduced-form credit risk frameworks. The literature differentiates between structural models that are based on modeling of the evolution of the balance sheet of the issuer, and reduced-form models that specify credit risk exogenously by a hazard rate process. Until now, there has been no common agreement in academia and practice on which model framework better captures credit risk. As major difference we focus on the discriminative modeling of the default time. In contrast to the previous literature, we calibrate both approaches to the same data set, apply comparable estimation techniques, and assess the out-of-sample prediction quality on the same time series of credit default swap prices. As our empirical implementations of both approaches rely on the same market information we are able to judge whether empirically the model structure itself makes an important difference. Interestingly, our study shows that the models’ prediction power are quite close on average indicating that for pricing purposes the modeling type does not greatly matter compared to the input data used. Still, the reduced-form approach outperforms the structural for investment-grade names and longer maturities. In contrast the structural approach performs better for shorter maturities and sub-investment grade names.

Basket CDS pricing with interacting intensities

We propose a factor contagion model for correlated defaults. The model covers the heterogeneous conditionally independent portfolio and the infectious default portfolio as special cases. The model assumes that the hazard rate processes are driven by external common factors as well as defaults of other names in the portfolio. The total hazard construction method is used to derive the joint distribution of default times. The basket CDS rates can be computed analytically for homogeneous contagion portfolios and recursively for general factor contagion portfolios. We extend the results to include the interacting counterparty risk and the stochastic intensity process.

A Credit Risk Model with Dynamic Frailties for Default Intensity Estimation

In the spirit of reduced-form models, default is treated as an unpredictable jump with an exogenously specific hazard rate process. Most researchers using reduced-form models assume doubly stochastic properties. In practice, this assumption is annulled in the presence of frailty factors, which are unobservable explanatory variables correlated across firms. However, few if any methods are used to estimate a credit risk model with frailty factors. This paper proposes a maximization likelihood estimation method using a proportional hazard model and an algorithm for implementation. The proportional hazard model is a popular method for analyzing the effect of covariates on the hazard rate. An application to real data is then presented, with the results showing that the model with frailty is more accurate than the model without it.

From Default Correlation to Implied Hazard Rate Densities

In this paper I show that the standard deviation of a CDO implied hazard rate density increases with increased correlation between firms' default intensity processes. This is done by generating CDO tranches spreads from a model where firms' defaults are modelled by jump-diffusion hazard rate processes that are correlated through systematic jumps, correlation in the processes' random shocks, or both. The increase in the standard deviation of a CDO implied hazard rate density holds irrespective of how the jump-diffusion hazard rate processes' are correlated. In addition, model simulations indicate that systematic jumps that are rare and with large mean size are required to yield senior CDO tranche model spreads that are consistent with market quotes. This observations supports similar conclusions reported in the existing literature. Further to this, I propose a model where the default of one firm triggers the default of other firms. When the number of firms hit by this default is approximately 1/3 of a CDO's reference portfolio, I am able to produce model spreads that are within range of the extreme spreads observed during the 2007-2008 credit crisis. In addition, using a vector autoregression on daily observations, I provided evidence that lags of the standard deviation of the implied hazard rate density and the spread of the 3-month Eurodollar and 3-month T-Bill rates (TED spread) are factors for one-period ahead prediction of each other.

Pricing Credit Default Swaps when Interest Rate Process and Hazard Rate Process are Stochastic

This paper models the dynamics of the building blocks for a plain vanilla credit default swap. We use a normal process for the interest rate (Vasicek model) to model the price of zero-coupon bonds (or the discount factor) and a log-normal process for the hazard rate to model the survival probabilities in a java applet. Model-generated data are used to calculate a fair market price for such a contract once survival probabilities and values of the discount function have been estimated using stochastic models. The article concludes by presenting numerical results from the models. Combining a constant interest rate and a stochastic hazard rate process, this one-factor model shows that the level of hazard rate rather than the volatility affecting the change in CDS spreads the most. With a constant survival probability and a stochastic interest rate, a lower CDS spread is associated with a higher spot rate due to the protection buyer's higher opportunity cost on the premium payments. Results also show that a higher survival probability of the reference entity indicates a lower CDS spread as expected. A two-factor model gives a much higher CDS price than the one-factor model especially when the drift parameters are higher in the credit events process.

Pricing contingent claims with credit risk: Asymptotic expansion approach

The pricing problems of credit derivatives have received much attention in the last decade. An important unresolved problem, however, is the pricing of credit derivatives under the general environment in which the interest rate process and the hazard rate process are stochastic. This article addresses the pricing problems of credit derivatives by the asymptotic expansion approach. This approach has only recently been introduced to mathematical finance, and it enables us to evaluate credit derivatives under a widely adapted class of models. We also present a numerical study.

On the hazard rate process for imperfectly monitored multi-unit systems

The aim of this paper is to present a stochastic model to characterize the failure distribution of multi-unit systems when the current units state is imperfectly monitored. The definition of the hazard rate process existing with perfect monitoring is extended to the realistic case where the units failure time are not always detected (non-detection events). The so defined observed hazard rate process gives a better representation of the system behavior than the classical failure rate calculated without any information on the units state and than the hazard rate process based on perfect monitoring information. The quality of this representation is, however, conditioned by the monotony property of the process. This problem is mainly discussed and illustrated on a practical example (two parallel units). The results obtained motivate the use of the observed hazard rate process to characterize the stochastic behavior of the multi-unit systems and to optimize for example preventive maintenance policies.

Correlated Default Processes: A Criterion-Based Copula Approach

Modeling correlated default risk is a new phenomenon currently sweeping through the credit markets. In this paper, we develop a methodology to model, simulate and assess the joint default process of hundreds of issuers. Our study is based on a data set of default probabilities supplied by Moody's Risk Management Services. We undertake an empirical examination of the joint stochastic process of default risk over the period of 1987-2000 using copula functions. To determine the appropriate choice of the joint default process, we propose a new metric. This metric accounts for different aspects of default correlation, namely (i) level, (ii) asymmetry and (iii) tail-dependence and extreme behavior. Our model, based on estimating a joint system of over 600 issuers, is designed to replicate the empirical joint distribution of defaults. A comparison of a jump model and a regime-switching model shows that the latter provides a better representation of the properties of correlated default. We also find that the skewed double-exponential distribution is the best choice for the marginal distribution of each issuer's hazard rate process, and combines well with the normal, Gumbel, Clayton and student's t copulas in the joint dependence relationship amongst issuers. As a complement to the methodological innovation, we show that (a) appropriate choices of marginal distributions and copulas are essential in modeling correlated default, (b) accounting for regimes is an important aspect of joint specifications of default risk, and (c) misspecification of credit portfolio risk can occur easily if joint distributions are inappropriately chosen. The empirical evidence suggests that improvements are indeed possible over the standard Gaussian copula used in practice.

Valuation of Defaultable Claims - A Survey

The literature on default-claim pricing falls into three categories. Building on the classical Merton model, the structural approach models the dynamics of the asset value and assumes that default is triggered when the equity value reaches an exogenous asset level. In a second class of structural models, the firm itself derives a default boundary endogenously. Finally, in the reduced-form approach default occurs according to an exogenous hazard rate process. In this paper I survey the default-claim literature. I provide a general valuation framework for default-claim pricing. I then give an example designed to clarify the main difference between the structural and the reduced-form approach. For each model category I show how current pricing models fit into my general framework, describe the applicable papers in some detail, and discuss related extensions.

Simple bounds for terminating Poisson and renewal shock processes

A system subject to a point process of shocks is considered. The shocks occur in accordance with a renewal process or a nonhomogeneous Poisson process. Each shock independently of the previous history leads to a system failure with probability θ and is survived with a complimentary probability θ ̄ . A number of problems in reliability and safety analysis can be interpreted by means of this model. The exact solution for the probability of survival W ̄ (t,θ) can be obtained only in the form of infinite series (renewal process of shocks). Approximate solutions and new simple bounds for the probability of survival are obtained. The introduced method is based on the notion of a stochastic hazard rate process. A supplementary characteristic in this analysis is the mean of the hazard rate process. This method makes it possible to consider a generalization important in practical applications when the probability of a system failure under the effect of a current shock depends on the time since the previous one.

Valuation of Defaultable Claims — A Survey

The literature on default-claim pricing falls into three categories. Building on the classical Merton model, the structural approach models the dynamics of the asset value and assumes that default is triggered when the equity value reaches an exogenous asset level. In a second class of structural models, the firm itself derives a default boundary endogenously. Finally, in the reduced-form approach default occurs according to an exogenous hazard rate process. In this paper I survey the default-claim literature. I provide a general valuation framework for default-claim pricing. I then give an example designed to clarify the main difference between the structural and the reduced-form approach. For each model category I show how current pricing models fit into my general framework, describe he applicable papers in some detail, and discuss related extensions.