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

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

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

A Semi-Analytical Parametric Model for Dependent Defaults

A semi-analytical parametric approach to modeling default dependency is presented. It is a multi-factor model based on instantaneous default correlation that also takes into account higher order default correlations. It is capable of accommodating a term structure of default correlations and has a dynamic formulation in the form of a continuous time Markov chain. With two factors and a constant hazard rate, it provides perfect fits to four tranches of CDX.NA.IG and iTraxx Europe CDOs of 5, 7 and 10 year maturities. With time dependent hazard rates, it provides perfect fits to all the five tranches for all three maturities.Default Risk; Default Correlation; CDO; Markov Chain; Semi-analytical; Parametric

Levy Density Based Intensity Modeling of the Correlation Smile

The jump distribution for the default intensities in a reduced form framework is modeled and calibrated to provide reasonable fits to CDX.NA.IG and iTraxx Europe CDOs, to 5, 7 and 10 year maturities simultaneously. Calibration is carried out using an efficient Monte Carlo simulation algorithm suitable for both homogeneous and heterogeneous collections of credit names. The underlying jump process is found to relate closely to a maximally skewed stable Levy process with index of stability alpha ~ 1.5.Default Risk; Default Correlation; Default Intensity; Intensity Model; Levy Density; CDO; Monte Carlo

Synthesis of a control model for a liquid nitrogen cooled, closed circuit, cryogenic nitrogen wind tunnel and its validation

The details of the efforts to synthesize a control-compatible multivariable model of a liquid nitrogen cooled, gaseous nitrogen operated, closed circuit, cryogenic pressure tunnel are presented. The synthesized model was transformed into a real-time cryogenic tunnel simulator, and this model is validated by comparing the model responses to the actual tunnel responses of the 0.3 m transonic cryogenic tunnel, using the quasi-steady-state and the transient responses of the model and the tunnel. The global nature of the simple, explicit, lumped multivariable model of a closed circuit cryogenic tunnel is demonstrated

Minimum Energy Test Direction Design in the Control of Cryogenic Wind Tunnels

The advent of the cryogenic wind tunnel concept is attributable to the need for high Reynolds number flow in wind tunnels. The cryogenic wind tunnel concept consists of operating the test medium of a conventional tunnel at cryogenic temperatures down to 80 K. Nitrogen gas, cooled by injected liquid nitrogen, proves to be ideal for the cryogenic tunnel test medium because of its near perfect behavior in insentropic flow. Cryogenic operation of a wind tunnel results in reduced fan power consumption and no penalty in flow dynamic pressure. In a cryogenic tunnel, the flow parameters (Reynolds number, Mach number and flow dynamic pressure) can be independently controlled by separately controlling the tunnel flow variables: total temperature, test section mass flow, and the tunnel total pressure. The problem of closed-loop control of the tunnel total temperature, flow Mach number, and total pressure is addressed and reported

Development and validation of a hybrid-computer simulator for a transonic cryogenic wind tunnel

A study was undertaken to model the cryogenic wind tunnel process, to validate the model by the use of experimental data from the Langley 0.3 Meter Transonic Cryogenic Tunnel, and to construct an interactive simulator of the cryogenic tunnel using the validated model. Additionally, this model was used for designing closed loop feedback control laws for regulation of temperature and pressure in the 0.3 meter cryogenic tunnel. The global mathematical model of the cryogenic tunnel that were developed consists of coupled, nonlinear differential governing equations based on an energy state concept of the physical cryogenic phenomena. Process equations and comparisons between actual tunnel responses and computer simulation predictions were examined. Also included are the control laws and simulator responses obtained using the feedback schemes for closed loop control of temperature and pressure were also included