Econometrics and Risk Management: Volume 22

The main theme of this volume is credit risk and credit derivatives. Recent developments in financial markets show that appropriate modeling and quantification of credit risk is fundamental in the context of modern complex structured financial products. Moreover, there is a need for further developments in our understanding of this important area. In particular modeling defaults and their correlation has been a real challenge in recent years, and still is. This problem is even more relevant after the so-called subprime crisis that hit in the summer of 2007. This makes the volume very timely and hopefully useful for researchers in the area of credit risk and credit derivatives.

This article describes a new approach to compute values and sensitivities of synthetic collateralized debt obligation (CDO) tranches in the market-standard, single-factor, Gaussian copula model with base correlation. We introduce a novel decomposition of the conditional expected capped portfolio loss process into “intrinsic value” and “time value” components, derive a closed form solution for the intrinsic value, and describe a very efficient computational scheme for the time value, taking advantage of a curious time stability of this quantity.

This paper studies the pricing of collateralized debt obligation (CDO) using Monte Carlo and analytic methods. Both methods are developed within the framework of the reduced form model. One-factor Gaussian Copula is used for treating default correlations amongst the collateral portfolio. Based on the two methods, the portfolio loss, the expected loss in each CDO tranche, tranche spread, and the default delta sensitivity are analyzed with respect to different parameters such as maturity, default correlation, default intensity or hazard rate, and recovery rate. We provide a careful study of the effects of different parametric impact. Our results show that Monte Carlo method is slow and not robust in the calculation of default delta sensitivity. The analytic approach has comparative advantages for pricing CDO. We also employ empirical data to investigate the implied default correlation and base correlation of the CDO. The implication of extending the analytical approach to incorporating Levy processes is also discussed.

Portfolio credit derivatives, such as basket credit default swaps (basket CDS), require for their pricing an estimation of the dependence structure of defaults, which is known to exhibit tail dependence as reflected in observed default contagion. A popular model with this property is the (Student's) t-copula; unfortunately there is no fast method to calibrate the degree of freedom parameter.

In this paper, within the framework of Schönbucher's copula-based trigger-variable model for basket CDS pricing, we propose instead to calibrate the full multivariate t distribution. We describe a version of the expectation-maximization algorithm that provides very fast calibration speeds compared to the current copula-based alternatives.

The algorithm generalizes easily to the more flexible skewed t distributions. To our knowledge, we are the first to use the skewed t distribution in this context.

We have developed a new family of Archimedean copula processes for modeling the dynamic dependence between default times in a large portfolio of names and for pricing synthetic CDO tranches. After presenting a general procedure for constructing these processes, we focus on a specific one with lower tail dependence as in the Clayton copula. Using CDS data as on July 2005, we show that the base correlations given by this model at the standard detachment points are very similar to those quoted in the market for a maturity of 5 years.

Gaussian copula is by far the most popular copula used in the financial industry in default dependency modeling. However, it has a major drawback – it does not exhibit tail dependence, a very important property for copula. The essence of tail dependence is the interdependence when extreme events occur, say, defaults of corporate bonds. In this paper, we show that some tail dependence can be restored by introducing stochastic volatility on a Gaussian copula. Using perturbation methods we then derive an approximate copula – called perturbed Gaussian copula in this paper.

This chapter analyses the ability of some structural models to predict corporate bankruptcy. The study extends the existing empirical work on default risk in two ways. First, it estimates the expected default probabilities (EDPs) for a sample of bankrupt companies in the USA as a function of volatility, debt ratio, and other company variables. Second, it computes default correlations using a copula function and extracts common or latent factors that drive companies’ default correlations using a factor-analytical technique. Idiosyncratic risk is observed to change significantly prior to bankruptcy and its impact on EDPs is found to be more important than that of total volatility. Information-related tests corroborate the results of prediction-orientated tests reported by other studies in the literature; however, only a weak explanatory power is found in the widely used market-to-book assets and book-to-market equity ratio. The results indicate that common factors, which capture the overall state of the economy, explain default correlations quite well.

Survival (default) data are frequently encountered in financial (especially credit risk), medical, educational, and other fields, where the “default” can be interpreted as the failure to fulfill debt payments of a specific company or the death of a patient in a medical study or the inability to pass some educational tests.

This paper introduces the basic ideas of Cox's original proportional model for the hazard rates and extends the model within a general framework of statistical data mining procedures. By employing regularization, basis expansion, boosting, bagging, Markov chain Monte Carlo (MCMC) and many other tools, we effectively calibrate a large and flexible class of proportional hazard models.

The proposed methods have important applications in the setting of credit risk. For example, the model for the default correlation through regularization can be used to price credit basket products, and the frailty factor models can explain the contagion effects in the defaults of multiple firms in the credit market.

The credit migration process contains important information about the dynamics of a firm's credit quality, therefore, it has a significant impact on its relevant credit derivatives. We present a jump diffusion approach to model the credit rating transitions which leads to a partial integro-differential equation (PIDE) formulation, with defaults and rating changes characterized by barrier crossings. Efficient and reliable numerical solutions are developed for the variable coefficient equation that result in good agreement with historical and market data, across all credit ratings. A simple adjustment in the credit index drift converts the model to be used in the risk-neutral setting, which makes it a valuable tool in credit derivative pricing.

We analyze stochastic volatility effects in the context of the bond market. The short rate model is of Vasicek type and the focus of our analysis is the effect of multiple scale variations in the volatility of this model. Using a combined singular-regular perturbation approach we can identify a parsimonious representation of multiscale stochastic volatility effects. The results are illustrated with numerical simulations. We also present a framework for model calibration and look at the connection to defaultable bonds.

We propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. The dynamics of the default intensity are governed by the equation dλt=κ(ρ(Lt,t)−λt)dt+σλtdWt. The function ρ depends both on time t and accumulated loss Lt, providing sufficient freedom to calibrate the model to a generic distribution of loss. We develop a computationally efficient method for model calibration to the market of synthetic single tranche collateralized debt obligations (CDOs). The method is based on the Markovian projection technique which reduces the full model to a one-step Markov chain having the same marginal distributions of loss. We show that once the intensity function of the effective Markov chain consistent with the loss distribution implied by the tranches is found, the function ρ can be recovered with a very moderate computational effort. Because our model is Markovian and has low dimensionality, it offers a convenient framework for the pricing of dynamic credit instruments, such as options on indices and tranches, by backward induction. We calibrate the model to a set of recent market quotes on CDX index tranches and apply it to the pricing of tranche options.

We discuss the valuation of credit derivatives in extreme regimes such as when the time-to-maturity is short, or when payoff is contingent upon a large number of defaults, as with senior tranches of collateralized debt obligations. In these cases, risk aversion may play an important role, especially when there is little liquidity, and utility-indifference valuation may apply. Specifically, we analyze how short-term yield spreads from defaultable bonds in a structural model may be raised due to investor risk aversion.