Default Boundary Research Articles

The Impact of Government Interventions on CDS and Equity Markets

We question the impact of government guarantees on the pricing of default risk in credit and stock markets and, using a Merton-type credit model, provide evidence of a structural break in the valuation of U.S. bank debt in the course of the 2007-2009 financial crisis, manifesting in a lowered default boundary, or, under the pre-crisis regime, in higher stock-implied credit spreads. A possible explanation is the asymmetric treatment of debt and equity in rescue measures, which tend to favor creditors. The discrepancies are driven by several factors including firm size, default correlation, and high ratings, thus corroborating our too-big-to-fail hypothesis.

Asymptotic Approximations to CEV and SABR Models

The problem of pricing, hedging and calibrating equity derivatives in a fast and consistent fashion is considered when the underlying asset does not follow the standard Black-Scholes model but instead the CEV or SABR models. The underlying process in the CEV model has volatility as a deterministic function of the asset price while in the SABR model the volatility as a stochastic function of the asset price. In such situations, trading desks often resort to numerical methods to solve the pricing and hedging problem. This can be problematic for complex models if real-time valuations, hedging and calibration are required. A more efficient and practical alternative is to use a formula even if it is only an approximation. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations yielding simple formulas for the pricing problem. For these models, default may be possible and the original ray approximation is not valid near the default boundary so a modifiedd asymptotic approximation or boundary layer correction is derived. New results are also derived for the standard CEV model and the SABR results. The applicability of the results is illustrated by deriving new analytical approximations for vanilla options based on the CEV and SABR models. The accuracy of the results is demonstrated numerically.

A numerical study of summer ozone concentration over the Kanto area of Japan using the MM5/CMAQ model

We assessed the ability of the MM5/CMAQ model to predict ozone (O 3) air quality over the Kanto area and to investigate the factors that affect simulation of O 3. We find that the coupled MM5/CMAQ model is a useful tool for the analysis of urban environmental problems. The simulation results were compared with observational data and were found to accurately replicate most of the important observed characteristics. The initial and boundary conditions were found to have a significant effect on simulated O 3 concentrations. The results show that on hot and dry days with high O 3 concentration, the CMAQ model provides a poor simulation of O 3 maxima when using initial and boundary conditions derived from the CMAQ default data. The simulation of peak O 3 concentrations is improved with the JCAP initial and boundary conditions. On mild days, the default CMAQ initial and boundary conditions provide a more realistic simulation. Meteorological conditions also have a strong impact on the simulated distribution and accumulation of O 3 concentrations in this area. Low O 3 concentrations are simulated during mild weather conditions, and high concentrations are predicted during hot and dry weather. By investigating the effects of different meteorological conditions on each model process, we find that advection and diffusion differ the most between the two meteorological regimes. Thus, differences in the winds that govern the transport of O 3 and its precursors are likely the most important meteorological drivers of ozone concentration over the central Kanto area.

Stochastic Evolution Equations in Portfolio Credit Modelling

We consider a structural credit model for a large portfolio of credit risky assets where the correlation is due to a market factor. By considering the large portfolio limit of this system we show the existence of a density process for the asset values. This density evolves according to a stochastic partial differential equation, and we establish existence and uniqueness for the solution taking values in a suitable function space. The loss function of the portfolio is then a function of the evolution of this density at the default boundary. We develop numerical methods for pricing and calibration of the model to credit indices and consider its performance before and after the credit crunch.

Asymptotic Approximations to Deterministic and Stochastic Volatility Models

The problem of pricing, hedging, and calibrating equity derivatives in a fast and consistent fashion is considered when the underlying asset does not follow the standard Black–Scholes model but instead the stochastic CEV (constant elasticity of variance) or SABR (stochastic alpha beta rho) model. The underlying process in the SABR model has the volatility as a stochastic function of the asset price. In such situations, trading desks often resort to numerical methods to solve the pricing and hedging problem. This can be problematic for complex models if real-time valuations, hedging, and calibration are required. A more efficient and practical alternative is to use a formula even if it is only an approximation. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations yielding simple formulas for the pricing problem. For the SABR model, default may be possible, and the original ray approximation is not valid near the default boundary, so a modified asymptotic approximation or boundary layer correction is derived. New results are also derived for the standard CEV model, which has deterministic volatility, as a special case of the SABR results. The applicability of the results is illustrated by deriving new analytical approximations for vanilla options based on the CEV and SABR models. The accuracy of the results is demonstrated numerically.

Comment on “Simulation of Surface Ozone Pollution in the Central Gulf Coast Region Using WRF/Chem Model: Sensitivity to PBL and Land Surface Physics”

A recently published meteorology and air quality modeling study has several serious deficiencies deserving comment. The study uses the weather research and forecasting/chemistry (WRF/Chem) model to compare and evaluate boundary layer and land surface modeling options. The most serious of the study's deficiencies is reporting WRF/Chem results for both meteorological and chemical quantities using the asymmetric convective model version 2 (ACM2). While the ACM2 is a valid model option for WRF, it has not yet been implemented for the chemical portion of the WRF/Chem model. Hence, the reported air quality modeling results using ACM2 are invalid. Furthermore, publication of these results gives the erroneous impression that the ACM2 model is not well suited for air quality applications when, in fact, it is the default boundary layer model in the community multiscale air quality (CMAQ) model.

Discriminatory power and predictions of defaults of structural credit risk models

This paper studies the discriminatory power and calibration quality of the structural credit risk models under the "exogenous default boundary" approach including those proposed by Longstaff and Schwartz (1995) and Collin-Dufresne and Goldstein (2001), and the "endogenous default boundary" approach in Leland and Toft (1996) based on 2,050 non-financial companies in 46 economies during the period 1998-2005. Their discriminatory power in terms of differentiating defaulting and non-defaulting companies is adequate and the differences among them are not material. In addition, the calibration quality of the three models is similar, although limited evidence is found that the Longstaff and Schwartz model marginally outperforms the others in some subsamples. Overall, no significant difference in the capability of measuring credit risk between the "exogenous default boundary" and "endogenous default boundary" approaches is found.

Prediction of contractor default probability using structural models of credit risk: an empirical investigation

Structural models of credit risk can apply for quantitatively predicting contractor defaults and pricing performance guarantees. However, the application involves crucial empirical issues. Some of the empirical issues are investigated using market and accounting data of public construction firms in Taiwan. Statistical analyses are conducted using the Wilcoxon rank sum test, Shumway's discrete‐time hazard model, and the receiver operating characteristic curve. Structural models are viable, and market value tends to dominate other measures of economic or financial distress in terms of prediction accuracy. However, when calibrated to minimize Type I and Type II errors, the default boundary of market value produces substantial residual errors. In addition, the calibrated boundary is at 151% of face debt, much higher than those suggested by previous empirical studies. This seems to reflect the idiosyncratic short‐term debt structures of Taiwanese construction firms. Leland and Toft's model is recommended for further investigations, because their theory explains the higher than expected calibrated boundary.

Intended boundaries detection in topic change tracking for text segmentation

This paper presents a topical text segmentation method based on intended boundaries detection and compares it to a well known default boundaries detection method, c99. We compared the two methods by running them on two different corpora of French texts and results are evaluated by two different methods: one using a modified classic measure, the FScore, the other based on a manual evaluation one the Internet. Our results showed that algorithms that are close when automatically evaluated can be quite far when manually evaluated.

A Structural Model with Unobserved Default Boundary

A firm‐value model similar to the one proposed by Black and Cox (1976) is considered. Instead of assuming a constant and known default boundary, the default boundary is an unobserved stochastic process. This process has a Brownian component, reflecting the influence of uncertain effects on the precise timing of the default, and a jump component, which relates to abrupt changes in the policy of the company, exogenous events or changes in the debt structure. Interestingly, this setup admits a default intensity, so the reduced form methodology can be applied. Part of this work was done while the first author stayed at the Isaac Newton Institute in Cambridge. Financial support from Isaac Newton Institute and Deutsche Forschungsgemeinschaft is gratefully acknowledged. The authors would like to thank M. Davis, L.C.G. Rogers, K. Giesecke and T. Bjrk for their inspiring comments and Ling Xu for her excellent help with the simulations. Moreover, the authors are grateful to N. Kordzakhia and two anonymous referees for their suggestions which helped to improve the paper considerably.

When Do Firms Default? A Study of the Default Boundary

In structural models of risky debt default is triggered when the market value of the firm's assets falls below a certain solvency boundary. Based on market values of defaulting firms, I estimate the default boundary to be 66% of the face value of debt, and find support for models in which the default timing is chosen endogenously to maximize the value of equity. But although default predictions based on the value of assets can match observed average default frequencies, they misclassify a substantial number of firms in cross-section, affecting the accuracy of boundary-based models. I show that using empirical specifications for the boundary can noticeably improve the models' ability to explain observed credit spreads.

On the Probabilities of Correlated Defaults: a First Passage Time Approach

This article investigates the joint probability of correlated defaults in the first passage time approach of credit risk subject to condition that the underlying firms’ assets values and the default boundaries follow geometric Brownian motion processes. The exact analytical expression of joint probability of two correlated defaults in the case of stochastic default boundaries is presented. Also, some properties of this solution are provided.

Present value of firm asset in case of correlated defaults: a generalized structural approach of credit risk

This article investigatesthe present value of a firm’s asset in the case of n \geq 2 correlateddefaults. The structural approach of credit risk is developed in the case when default boundaries follow geometric Brownian motions. Correlated defaults are defined by the implied correlation of Brownian motions. The operational risk and the risk of financial market changes are allowed in this model. Also, the impact of implied correlation to the present value of firm’s asset is shown numerically.

Can tax convexity be ignored in corporate financing decisions?

The standard modeling practice in corporate finance has been to assume a linear tax schedule. This paper extends the structural contingent-claim model of corporate finance to incorporate a more realistic convex tax schedule. It is shown that tax convexity raises the optimal default boundary and thus increases the likelihood of default, and also reduces the optimal leverage ratio. While the former effect seems insignificant in general, the effect of tax convexity on the optimal leverage ratio can be quantitatively significant. We conclude that tax convexity should not be ignored in corporate financing decisions, and theoretical models should use a convex tax schedule instead of a linear one. Thus, the short answer to the question in the title is “No”.

Technical note: Overprediction of dose with default PRESTA‐I boundary crossing in DOSXYZnrc and BEAMnrc

Use of the PRESTA-I (current default) boundary crossing algorithm for charged particles in DOSXYZnrc and BEAMnrc is shown to result in dose overprediction by up to 2.5% in photon and electron beams (less in high-energy electron beams). The effect is most pronounced when charged particle equilibrium does not exist or when the phantom is not divided into uniform voxels throughout. As a result, the default boundary crossing algorithm in BEAMnrc will be changed to "exact" (single scattering).

: Merton (Bondholderss Optimal Strategic Behavior and Credit Spreads: Merton Model and Its Extension)

Korean Abstract: 본 논문은 기업의 청산비용이 존재하는 경우에 채권자의 최적의사결정이 기업의 신용위험에 미치는 영향에 대해 연구하였다. 기업의 파산이 기업 가치에 의해 외생적으로 주어지는 Merton 모형과는 달리 청산비용이 상당한 경우 기업 가치가 부도경계에 다다르더라도 청산을 연기하는 것이 채권자에게 최적의사결정임을 보여준다. 본 논문에서는 채권자의 전략적 행동에 따라 결정되는 부도프로세스로부터 위험채권의 가격을 계산하기 위해 미국형옵션 가격 계산과 유사한 방식의 유한차분법을 이용하였다. 부채비율과 변동성, 청산비용의 크기에 따라 신용스프레드가 어떻게 변화하는지 보여주고 있으며 기존의 모형과의 차이점에 대해 연구하였다. English Abstract: This paper investigates how the optimal decisions of bond holders affect the credit risk of firms when bankruptcy cost exists. Previous literature assumes that bankruptcy of a firm depends only on the firm’s value. We relax this assumption in our paper so that bondholders can delay the liquidation of the firm even though the firm value is lower than the exogenous default boundary. Our results show that delaying the liquidation can be the optimal behavior for bond holders when the bankruptcy cost is considerable. We implement the Crank-Nicolson finite difference method which is used in pricing American style options in order to value the risky bonds of which the default process is determined by the strategic behaviors of the bondholders. We investigate how credit spreads vary depending on the debt ratio, volatility and amount of the bankruptcy cost.

Application of a variable default boundary to the pricing of vulnerable European options: the use of a three dimensional binomial tree

This paper investigates the significance of using a variable default boundary when pricing European Black-Scholes options that are subjected to credit risks. We apply numerical method and combine Klein [1], and Johnson and Stulz [2] to link the payout ratio proportionally to the assets of the option writer. We also link the payout ratio to the value of the assets of the writer. Numerical examples compare our results with Klein [1], and Johnson and Stulz [2] based on alternative assumptions, and illustrate when the application of variable default boundary becomes important.

Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.

First Passage and Excursion Time Models for Valuing Defaultable Bonds

In this contribution, we study structural models of defaultable bond pricing in which default occurs at the first time a relevant process either reaches the default boundary or has spent continuously (or cumulatively) a fixed time period below that threshold. Unlike first-passage time approaches, excursion time models allow for a non-absorbing state of default. Both the first-passage time and the excursion time approaches can be generalized by defining the default time as the first instant at which the firm value process (or another signaling process) either remains a certain time below the default threshold or hits a lower barrier. This corresponds, for instance, to a situation in which a firm is allowed temporarily to be short of funds, but enters default immediately when the financial distress becomes severe. Moreover, we examine the effects of different default time specifications on bond prices and credit spreads.

Structural RFV: Fundamental Credit Risk Pricing and Hedging with Recovery of Face Value at Default

Receiving the same fractional recovery of par at default for bonds issued by a particular company regardless of maturity has been labelled in the academic literature as a Recovery of Face Value at Default (RFV). Such a recovery form results from language found in typical bond indentures and is supported by empirical evidence from defaulted bond values. We incorporate RFV into a exogenous boundary structural credit risk model and compare the pricing and hedging implications versus other recovery forms more often seen in such models. We find that the recovery form can significantly affect the pricing and sensitivities produced by these models. This has important implications for any practical use of these models as well as for empirical studies attempting to validate structural credit risk models. In general, we provide convincing evidence that choosing the recovery form is not a trivial assumption to make within this class of models. Some of our results complement those found in the literature which examines the endogeneity of the default boundary. We show that some features that may have been solely attributed to modelling the boundary as an optimal decision by the firm can obtain in an exogenous boundary framework which assumes an RFV recovery form. We extend our results to incorporate a multifactor default-free term structure model and examine the impact of the recovery form in estimating the cost of debt capital within a structural model framework.