Levy subordinator model: A two parameter model of default dependency

The May 2005 crisis and the recent credit crisis have indicated to us that any realistic model of default dependency needs to account for at least two risk factors, firm-specific and catastrophic. Unfortunately, the popular Gaussian copula model has no identifiable support to either of these. In this article, a two parameter model of default dependency based on the Levy subordinator is presented accounting for these two risk factors. Subordinators are Levy processes with non-decreasing sample paths. They help ensure that the loss process is non-decreasing leading to a promising class of dynamic models. The simplest subordinator is the Levy subordinator, a maximally skewed stable process with index of stability 1/2. Interestingly, this simplest subordinator turns out to be the appropriate choice as the basic process in modeling default dependency. Its attractive feature is that it admits a closed form expression for its distribution function. This helps in automatic calibration to individual hazard rate curves and efficient pricing with Fast Fourier Transform techniques. It is structured similar to the one-factor Gaussian copula model and can easily be implemented within the framework of the existing infrastructure. As it turns out, the Gaussian copula model can itself be recast into this framework highlighting its limitations. The model can also be investigated numerically with a Monte Carlo simulation algorithm. It admits a tractable framework of random recovery. It is investigated numerically and the implied base correlations are presented over a wide range of its parameters. The investigation also demonstrates its ability to generate reasonable hedge ratios

Organic carbon transport and C/N ratio variations in a large tropical river: Godavari as a case study, India

This study gives an insight into the source of organic carbon and nitrogen in the Godavari river and its tributaries, the yield of organic carbon from the catchment, seasonal variability in their concentration and the ultimate flux of organic and inorganic carbon into the Bay of Bengal. Particulate organic carbon/particulate organic nitrogen (POC/PON or C/N) ratios revealed that the dominant source of organic matter in the high season is from the soil (C/N = 8–14), while in the rest of the seasons, the river-derived (in situ) phytoplankton is the major source (C/N = l–8). Amount of organic materials carried from the lower catchment and flood plains to the oceans during the high season are 3 to 91 times higher than in the moderate and low seasons. Large-scale erosion and deforestation in the catchment has led to higher net yield of organic carbon in the Godavari catchment when compared to other major world rivers. The total flux of POC, and dissolved inorganic carbon (DIC) from the Godavari river to the Bay of Bengal is estimated as 756 · 109 and 2520 · 109 g yr1, respectively. About 22% of POC is lost in the main channel because of oxidation of labile organic matter, entrapment of organic material behind dams/sedimentation along flood plains and river channel; the DIC fluxes as a function of alkalinity are conservative throughout the river channel. Finally, the C/N ratios (12) of the ultimate fluxes of particulate organic carbon suggest the dominance of refractory/ stable soil organic matter that could eventually get buried in the coastal sediments on a geological time scale

Delayed Default Dependency and Default Contagion

Delayed, hence non-simultaneous, dependent defaults are discussed in a reduced form model. The model is a generalization of a multi-factor model based on simultaneous defaults to incorporate delayed defaults. It provides a natural smoothening of discontinuities in the joint probability densities in models with simultaneous defaults. It is a dynamic model that exhibits default contagion in a multi-factor setting. It admits an efficient Monte Carlo simulation algorithm that can handle heterogeneous collections of credit names. It can be calibrated to provide exact fits to CDX.NA.IG and iTraxx Europe CDOs just as its version with simultaneous defaults.Default Risk; Default Correlation; Default Contagion; Delayed Default; CDO; Monte Carlo

Combining phase field crystal methods with a Cahn-Hilliard model for binary alloys

During phase transitions certain properties of a material change, such as composition field and lattice-symmetry distortions. These changes are typically coupled, and affect the microstructures that form in materials. Here, we propose a 2D theoretical framework that couples a Cahn-Hilliard (CH) model describing the composition field of a material system, with a phase field crystal (PFC) model describing its underlying microscopic configurations. We couple the two continuum models via coordinate transformation coefficients. We introduce the transformation coefficients in the PFC method, to describe affine lattice deformations. These transformation coefficients are modeled as functions of the composition field. Using this coupled approach, we explore the effects of coarse-grained lattice symmetry and distortions on a phase transition process. In this paper, we demonstrate the working of the CH-PFC model through three representative examples: First, we describe base cases with hexagonal and square lattice symmetries for two composition fields. Next, we illustrate how the CH-PFC method interpolates lattice symmetry across a diffuse composition phase boundary. Finally, we compute a Cahn-Hilliard type of diffusion and model the accompanying changes to lattice symmetry during a phase transition process.Comment: 9 pages, 5 figure

Heat Equation on the Cone and the Spectrum of the Spherical Laplacian

Spectrum of the Laplacian on spherical domains is analyzed from the point of view of the heat equation on the cone. The series solution to the heat equation on the cone is known to lead to a study of the Laplacian eigenvalue problem on domains on the sphere in higher dimensions. It is found that the solution leads naturally to a spectral function, a `generating function' for the eigenvalues and multiplicities of the Laplacian, expressible in closed form for certain domains on the sphere. Analytical properties of the spectral function suggest a simple scaling procedure for estimating the eigenvalues. Comparison of the first eigenvalue estimate with the available theoretical and numerical results for some specific domains shows remarkable agreement.Comment: 16 page